\section{Introduction}

This is a continuation of my previous work \cite{Ka16}. In the present paper, additional $K$-theoretic tools are developed in order to treat index theory for leaf-wise (i.e. orbital) and transverse elliptic operators on manifolds with a proper Lie group action. Under a group action we almost everywhere understand a smooth, proper, isometric Lie group action. The two index theories, orbital and transverse, are very much intertwined and interdependent, and will be treated together. The proofs of index theorems for elliptic, t-elliptic and leaf-wise elliptic operators (based on the $KK$-theoretic approach) are given in sections 9, 10, 11 of the present paper.

One of the new features of the present paper is a coarse approach to the pseudo-differential calculus. A significant part of index theory uses only the PDO calculus \emph{modulo compact operators}. The proposed coarse PDO calculus allows to treat those index theory problems which do not require very elaborate analytical tools. This is not a replacement of H\" ormander's PDO calculus, but it allows significant simplifications in many cases. 

The main index theory results for the transversally elliptic case were presented in sections 6, 7 and 8 of \cite{Ka16}. Unfortunately, there were errors in sections 7 and 8 of \cite{Ka16}. The corrections are given in the present paper. The main index theorem 8.18 of \cite{Ka16} is correct and is reproved in section 10 below.

The theory of leaf-wise operators (when the leaves are orbits of a Lie group action) is developed in the present paper from the basic definitions to the final index theorem. Although there are some common points in the group action case and the case of leaf-wise operators on general singular foliations (\cite{A-Sk II, A-Sk III}), methods available in the group action case give much more concrete results.  

\bigskip

\noindent{\bf\large Brief contents:}

\medskip

\noindent-- The basics of the coarse PDO calculus: sections 2--3.

\smallskip

\noindent-- {$C^*$}-algebras associated with a group action: section 4.

\smallskip

\noindent-- Leaf-wise symbols and the construction of leaf-wise operators: section 5.

\smallskip

\noindent-- Poincare duality and $K$-theory of symbol algebras: sections 6--7.

\smallskip

\noindent-- Elliptic symbols and index groups: section 8.

\smallskip

\noindent-- Index theorems for elliptic, t-elliptic and leaf-wise elliptic operators: sections 9, 10, 11.

\smallskip

\noindent-- Examples: t-elliptic operators on ${\bf R}^n$: section 12.

\smallskip

\noindent-- Comments on my paper \cite{Ka16}: Appendix.

\bigskip

\noindent{\bf\large Notational references:}

\medskip

We will use certain definitions from \cite{Ka88,Ka16}. Some (but not all) of these definitions will be repeated in the present paper. For  the reader's convenience, we give here the list of references. 

\medskip

\noindent-- $C_0(X)$-algebras and ${\cal R} KK(X;A,B)$ groups - \cite{Ka88}, sections 1, 2.

\smallskip

\noindent-- The algebra of $G$-invariant elements $B^G$ for a proper $G$-algebra $B$ - \cite{Ka88}, 3.2.

\smallskip

\noindent-- Clifford algebras of vector bundles, especially $Cl_\tau(X)$ for a manifold $X$, and the Dirac element $[d_X]\in K^0(Cl_\tau(X))$ - \cite{Ka88}, section 4, or \cite{Ka16}, section 2. 

\smallskip

\noindent-- Basic definitions for transversally elliptic operators - \cite{Ka16}, section 6.

\smallskip

\noindent-- Bott and Dirac operators on tangent spaces of a manifold $X$; elements $[{\cal B}_\xi], [{\cal B}_{\xi,\Gamma}], [d_\xi]$ - \cite{Ka16}, 2.5, 2.6, 7.2.

\smallskip

\noindent-- Dirac element $[d_{X,\Gamma}]\in K^0(C^*(G,Cl_{\tau\oplus\Gamma}(X)))$ - \cite{Ka16}, 8.8.

\smallskip

\noindent-- Dolbeault elements $[{\cal D}_X], [{\cal D}_{X,\Gamma}]$ and $[{\cal D}^{cl}_{X,\Gamma}]$ for the tangent manifold $TX$ - \cite{Ka16}, 2.8 and 8.17.

\smallskip

\noindent-- Clifford symbol for elliptic and transversally elliptic operators - \cite{Ka16}, 3.8 and 8.11.

\smallskip

\noindent-- Tangent Clifford symbol for transversally elliptic operators - \cite{Ka16}, 8.13. 

 
\section{Operator integration}

We will use operator integration for the construction of pseudo-differential operators. We will need two types of integration described in the two following subsections.

\subsection{Riemann integration}

This kind of operator integration is essentially described in section 3 of \cite{Ka75}. We will add here a few more details. 

Let $X$ be a second countable, locally compact, $\sigma$-compact space, $D$ a {$C^*$}-algebra, and $\phi:C_0(X)\to {\cal M}(D)$ a homomorphism such that $\phi(C_0(X))\cdot D$ is dense in $D$. Then $\phi$ extends to a unital homomorphism $\phi:C_b(X)\to {\cal M}(D)$, where $C_b(X)$ is the {$C^*$}-algebra of all bounded continuous functions on $X$. Given a bounded norm-continuous map $F:X\to {\cal M}(D)$ such that $F(x)$ commutes with $\phi(C_0(X))$ modulo $D$ (or al least each $F(x)$ commutes modulo $D$ with $\phi(C_0(U_x))$ for some open set $U_x\owns x$ for any $x\in X$), we will construct a Riemann type operator integral $\int_X F(x)d\phi\in {\cal M}(D)$ which is defined modulo $D$ and has the following properties (cf. \cite{Ka75}, section 3, theorem 1):

\begin{theo} $1^{\rm o}$. If $||F(x)|| \le c$ for all $x\in X$ then $\int_X F(x)d\phi \le c$. 

$2^{\rm o}$. The integral is additive, multiplicative, and 
$\int_X F^*(x)d\phi=(\int_X F(x)d\phi)^*$.  

$3^{\rm o}$. If $F$ is a scalar function $F(x)=f(x)\cdot 1$, where $f\in C_b(X)$, then $\int_X F(x)d\phi=\phi(f)$.

$4^{\rm o}$. The integral is functorial in $X$: if there is a proper continuous map $h:Y\to X$, and $\tilde F=F\cdot h:Y\to{\cal M}(D)$, $\phi=\psi \cdot h^*: C_0(X)\to C_0(Y)\to {\cal M}(D)$, then $\int_X F(x)d\phi=\int_Y \tilde F(y) d\psi$.
\end{theo}

The following simple lemma (\cite{Ka75}, section 3, lemma 1) provides the necessary estimates for the proof:

 \begin{lem} Let $B$ be a {$C^*$}-algebra, and elements $F_1,...,F_n;\alpha_1,...,\alpha_n\in B$ satisfy the following conditions: $\sum_{i=1}^n \alpha_i^*\alpha_i=1$, and $||F_i||\le c$ for any $i$. Then $||\sum_{i=1}^n \alpha_i^*F_i\alpha_i||\le c$. 
 \end{lem}
 
 \noindent {\em Proof.}  Using a faithful representation of $B$ in a Hilbert space $H$, we get for any $\xi,\eta\in H$: 
 $$|(\sum_i \alpha_i^*F_i\alpha_i(\xi),\eta)|=|\sum_i (F_i\alpha_i(\xi),\alpha_i(\eta))| \le \sum_i ||F_i|| \cdot ||\alpha_i(\xi)|| \cdot ||\alpha_i(\eta)||$$ 
 $$\le c (\sum_i||\alpha_i(\xi)||^2)^{1/2}(\sum_i||\alpha_i(\eta)||^2)^{1/2}=c||\xi|| ||\eta||. \;{\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10 $$

 \noindent\emph{Proof of the theorem.} The basic idea of the integral is the following. Let us assume that $X$ is compact. Let $B$ be the {$C^*$}-subalgebra of ${\cal M}(D)/D$ generated by $\phi(C(X))$ and all elements $F(x),\, x\in X$. Then there is a unital homomorphism: $C(X)\to B$ which maps $C(X)$ to the center of $B$. The map $F:X\to B$ represents an element of $C(X,B)\simeq C(X)\otimes B$. Consider the multiplication homomorphism: $C(X)\otimes B\to B$. The image of $F$ under this homomorphism is by definition $\int_X F(x)d\phi\in B\subset {\cal M}(D)/D$. 
 
 Now let us translate this into the Riemann operator integration context. We will still continue to assume for the moment that $X$ is compact. The integral is constructed in the following way. Taking a locally finite covering $\{U_i\}$ of $X$, points $x_i\in U_i$, and a partition of unity $\sum_i \alpha_i^2(x)=1$ associated with $\{U_i\}$ (all functions $\alpha_i$ are non-negative), we consider the integral sum: $\Sigma(\{U_i\},\{\alpha_i\},\{x_i\})=\sum_i \phi(\alpha_i)F(x_i)\phi(\alpha_i)$. We assume that $||F(x)||\le c$ for all $x\in X$, so by lemma 2.2 the norm of the integral sum is $\le c$. 
 
 We will call $\{U_i\}$ an $\epsilon$-covering if for any $i$ and any $x,y\in U_i$, one has $||F(x)-F(y)||\le\epsilon$. To verify that the integral sums for two $\epsilon$-coverings $\{U_i\}$ and $\{V_j\}$ differ in norm no more than by $2\epsilon$ in ${\cal M}(D)/D$, we form the covering $W_{i,j}=U_i\cap V_j$. Let $\sum_j\beta_j^2=1$ be the partition of unity for $\{V_j\}$, and set $\gamma_{i,j}=\alpha_i\beta_j,\, x_{i,j}\in W_{i,j}$. Then we have the following estimate in ${\cal M}(D)/D$:
 $$||\sum_i F(x_i)\phi(\alpha_i^2)-\sum_{i,j} F(x_{i,j})\phi(\gamma_{i,j}^2)||=||\sum_{i,j}(F(x_i)-F(x_{i,j}))\phi(\gamma_{i,j}^2)||\le \epsilon$$
by lemma 2.2. The integral is the limit in ${\cal M}(D)/D$ of the integral sums for all $\epsilon$-coverings when $\epsilon\to 0$. For a compact $X$, all properties listed in the theorem are easy exercises (see \cite{Ka75}, section 3, theorem 1). 

In the general case of a non-compact $X$, let $\{U_i\}$ be a locally finite covering of $X$ and $\sum_i \alpha_i^2(x)=1$ the corresponding partition of unity. Then, by lemma 2.2, the norms of operators ${\cal F}_m=\sum_{i=1}^m \phi(\alpha_i) F(x_i) \phi(\alpha_i)$ are uniformly bounded in $m$. We claim that the sums ${\cal F}_m$ converge strictly in ${\cal M}(D)$ when $m\to \infty$. Indeed, for any bounded approximate unit $\{u_k\}\subset C_0(X)$ consisting of functions with compact support in $X$, our initial assumption on $\phi$ says that $\{\phi(u_k)\}$ converges strictly in ${\cal M}(D)$. Since ${\cal F}_m$ is uniformly bounded in $m$, it is enough to show that both sums ${\cal F}_m\phi(u_k)$ and ${\cal F}^*_m\phi(u_k)$ converge for any fixed $k$ when $m\to \infty$. This is true because all $u_k$ have compact support, so $\alpha_iu_k=0$ for large $i$.

Now we can take any locally finite $\epsilon$-covering of $X$ and form the integral sum ${\cal F}=\sum_{i=1}^\infty \phi(\alpha_i) F(x_i) \phi(\alpha_i)$. Modulo $D$ any two such sums for two different $\epsilon$-coverings will differ in norm by $\le 2\epsilon$ (because all  corresponding finite sums ${\cal F}_m$ differ by $\le 2\epsilon$, as shown above). So in ${\cal M}(D)/D$ these sums converge in norm when $\epsilon\to 0$. The lifting of the limit to ${\cal M}(D)$ is denoted $\int_X F(x)d\phi$. 

The proof of all properties listed in the theorem for the case of a non-compact $X$ is the same as for a compact $X$. For example, for the multiplicativity property, one needs to estimate the difference modulo $D$ of two sums:
$$\sum_{i,j}\phi(\alpha_i)F_1(x_i)\phi(\alpha_i)\phi(\alpha_j)F_2(x_j)\phi(\alpha_j)-\sum_{i,j}\phi(\alpha_i)F_1(x_i)\phi(\alpha_i)\phi(\alpha_j)F_2(x_i)\phi(\alpha_j).$$
Both sums converge in the strict topology, so one needs only to show that all finite portions of these sums differ modulo $D$ by less than $2\epsilon c$ if $\{U_i\}$ is an $\epsilon$-covering for $F_2$ and $||F_1(x)||\le c$ for all $x\in X$. Obviously, we can leave only those summands for which $U_i\cap U_j\ne\emptyset$. Then modulo $D$, $||F_2(x_i)-F_2(x_j)||\le 2\epsilon$, and the estimate of the difference of those two sums $\sum_{1\le i,j\le n}$ modulo $D$ comes from lemma 2.2. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10
 
 \begin{cor} In the assumptions of the theorem, suppose that $||F(x)||\to 0$ when $x\to\infty$ in $X$. Then the sums ${\cal F}_m$ in the above proof of the theorem converge uniformly in ${\cal M}(D)$.
\end{cor}
  
 \noindent {\em Proof.}  For any $\epsilon>0$, we can choose $m$ and $n$ large enough so that for any $x$ in the supports of all functions $\alpha_i$, for $i$ in the interval $[m,n]$, we have $||F(x)||\le\epsilon$. Then $||{\cal F}_m-{\cal F}_n||\le \epsilon$ by lemma 2.2. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10
  
  
  \subsection{Group averaging}
  
  We keep all assumptions of the previous subsection, but assume in addition that a locally compact, second countable group $G$ acts on $X$ properly, $D$ is a $G$-algebra, and the homomorphism $\phi:C_0(X)\to {\cal M}(D)$ is $G$-equivariant. In integration over $G$, we will use the left Haar measure.
 
 \begin{lem} If $F:X\to {\cal M}(D)/D$ is $G$-equivariant, then $I(F)=\int_X F(x) d\phi$ is $G$-invariant modulo $D$ and $G$-continuous in norm (i.e. the map $G\to {\cal M}(D): g\mapsto g(I(F))$ is norm-continuous). 
 \end{lem}
 
 \noindent {\em Proof.}  Because the $G$-action transforms an $\epsilon$-covering into another $\epsilon$-covering, it is clear that $I(F)$ is $G$-invariant modulo $D$.  The last assertion follows from \cite{Thom_prepr}, 1.1.4. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10
 
In order to make $F$ exactly $G$-invariant we need averaging over $G$. Essentially, we will use the averaging method of \cite{CM:L^2}, proposition 1.4. We will adapt it to the generality that we need. But first let us adapt to our setting the definitions of a compactly supported distributional kernel and a properly supported distributional kernel.

\begin{defi} An operator $T\in {\cal M}(D)$ will be called compactly supported if there exists a compact subset $K\subset X$ such that for any $a\in C_c(X)$ one has $\phi(a)T=T$ and $T\phi(a)=T$ as soon as $a=1$ on $K$. (In particular, if $b\in C_c(X)$ and $b=0$ on $K$, then $\phi(b)T=0$ and $T\phi(b)=0$.)

An operator $T\in {\cal M}(D)$ will be called properly supported if there exists a closed subset $L\subset X\times X$, with both projections $p_1:L\to X$ and $p_2:L\to X$ proper maps, such that for any $a,b\in C_b(X)$ one has $\phi(a)T\phi(b)=0$ as soon as ${\rm supp\,}(a)\times {\rm supp\,}(b)\cap L=\emptyset$. 
\end{defi}

\begin{rem} {\rm (1) Although the operator integral $\int_X F(x) d\phi$ is defined only modulo $D$, when lifted to ${\cal M}(D)$ it can be assumed to have proper support. Indeed, we can require all functions $\sum_i \alpha_i(x)\alpha_i(y)$ (where $(x,y)\in X\times X$) to have support in some subset $L\subset X\times X$ with the properties as in the above definition. When the operator function $F:X\to {\cal M}(D)/D$ is zero outside of a compact set, the integral $\int_X F(x) d\phi$ can be made to have compact support. 

(2) An important example of a compactly supported operator is the following. Suppose $T$ is properly supported and $a\in C_c(X)$. Then both $\phi(a)T$ and $T\phi(a)$ are compactly supported. Indeed, if ${\rm supp\,}(T)=L\subset X\times X$ and ${\rm supp\,}(a)=K\subset X$, then by the previous definition $L_K=p_1^{-1}(K)\cap L$ is compact in $X\times X$, so $p_2(L_K)\subset X$ is also compact. Let $b\in C_c(X)$ be a function which is equal $1$ on $p_2(L_K)$. Then $\phi(a)T\phi(b)=\phi(a)T$. Indeed, this means simply that $\phi(a)T(1-\phi(b))=0$, which easily follows from the definition.}
\end{rem}

 The analog of \cite{CM:L^2}, proposition 1.4, will be the following:
 
 \begin{prop} Let $T\in {\cal M}(D)$ be an operator with support in a compact set $L\subset X$. Then one can define the average of $T$ over $G$, denoted $Av_G(T)$ or $\int_G g(T)dg$, as the limit of integrals $\int_C g(T)dg$, in the strict topology of ${\cal M}(D)$, over the increasing net of all compact subsets $C$ of $G$, where $dg$ denotes the Haar measure of $G$. Moreover, $||Av_G(T)||\le c ||T||$, where $c$ depends only on $L$.
 \end{prop}
 
 For the proof we need the following lemma:
 
 \begin{lem} If an operator $T\in {\cal M}(D)$ has the property that $T^*g(T)=0$ and $Tg(T^*)=0$ for any $g$ outside of a compact set $K\subset G$, then for any compact subset $C\subset G$, $(\int_C g(T)dg)^*(\int_C g(T)dg)\le |K|^2||T||^2$, where $|K|$ is the Haar measure of $K$.
 \end{lem}
 
 \noindent {\em Proof.}  Consider $[(\int_C g(T)dg)^*(\int_C g(T)dg)]^n$ as the $2n$-fold integral 
 $$\int_C...\int_C g_1(T^*)g_2(T)...g_{2n-1}(T^*)g_{2n}(T)dg_1...dg_{2n}.$$
 Because of our assumption on $T$, this multiple integral actually goes over the subset of $C\times...\times C$ such that $g_i^{-1}g_{i+1}\in K$ for all $i$. So we can rewrite it as a repeated integral:
 $$\int_C\int_K...\int_K g_1(T^*h_2(Th_3(T^*h_4(T...h_{2n-1}(T^*h_{2n}(T)...)dg_1dh_2...dh_{2n},$$
where $h_{i+1}=g_i^{-1}g_{i+1}$. The latter integral is estimated as $\le |C|\cdot |K|^{2n-1}||T||^{2n}$, therefore $[(\int_C g(T)dg)^*(\int_C g(T)dg)]^n\le |C|\cdot |K|^{2n-1}||T||^{2n}$. Taking the $n$-th root of both sides and letting $n\to\infty$, we get the result. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\medskip

\noindent\emph{Proof of the proposition.} If $a\in C_c(X)$ is a function equal to $1$ on $L$, then $T\phi(a)=T$ and $T^*\phi(a)=T^*$. Because the $G$-action on $X$ is proper, there is a compact set $K\subset G$ such that ${\rm supp\,}(a)$ does not intersect with any $g(L)$ for any $g\in G$ outside $K$. So for $g$ outside $K$ we have $T^*g(T)=T^*\phi(a)g(T)=0$, and similarly for $Tg(T^*)$. In view of the previous lemma, $||\int_C g(T)dg||\le |K| \cdot ||T||$ for any compact subset $C\subset G$, and similarly for $T^*$. 

Let us denote $\int_C g(T)dg$ by $I(C)$. We need to show that for any $d\in D$, the integrals $d\cdot I(C)$ and $d\cdot I(C)^*$ converge over the net of all compact $C$. Clearly, this is true if we replace $d\in D$ with $\phi(a)$, where $a\in C_c(X)$, because if we define $C_{max}(a,T)$ as the maximal compact subset of $G$ such that ${\rm supp\,}(a)\cap g(L)\ne \emptyset$, then both integrals will not depend on $C$ as soon as $C_{max}(a,T)\subset C$. 

Given $d\in D$ and $\epsilon>0$, we can find $a\in C_c(X)$ such that $||d-d\phi(a)||\le \epsilon$. By the previous lemma, both $I(C)$ and $I(C)^*$ are bounded in $C$, more precisely, $||dI(C)-\phi(a)dI(C)||\le \epsilon |K|\cdot ||T||$, and similarly for $I(C)^*$. This means that $dI(C)$ varies with $C$ no more than by $\epsilon |K|\cdot ||T||$ when $C_{max}(a,T)\subset C$. This proves the convergence. The norm estimate is also clear. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

 
 \section{Coarse PDO calculus}

Index theory deals with what is usually called `elliptic' operators. These are some kind of bounded (`zero order') operators, invertible modulo some kind of `negative order' operators. In the simplest case, one has a compact manifold $X$ and a representation of $C(X)$ in a Hilbert space $H=L^2(X)$. `Zero order' operators are bounded operators on $H$ which commute with the action of $C(X)$ modulo ${\cal K}(H)$, `negative order' operators are just compact operators. If $X$ is locally compact, the requirement for `zero order' operators is the same: commutation with the action of $C_0(X)$ modulo ${\cal K}(H)$. `Negative order' operators $T$ are those which satisfy the condition: $T\cdot C_0(X)\subset {\cal K}(H)$. Index theory seeks invariants of operators of `zero order' modulo `negative order'. There is an obvious similarity with the basic $KK$-theory (cf. \cite{Ka88}, definition 2.1).

We will use this scheme (`zero order' modulo `negative order' operators) in our coarse PDO calculus. This calculus will correspond to the H\" ormander $\rho=1,\delta=0$ PDO calculus. In fact, H\" ormander discussed such kind of `simple' PDO calculus in \cite{Hor} at the end of subsection 2.1, but based on his definition of PDOs. Our approach is different and aimed at simplifying as much as possible the definition of a PDO for zero order operators. 

There are positive and negative sides of this approach. On the positive side, many proofs get simpler. On the negative side, this approach is not very convenient in treating differential operators and their parametrices. But when necessary, one can always shift back to H\" ormander's approach which is perfectly compatible with the coarse one that we propose.

Our coarse PDO theory uses the operator integration of section 2. In all constructions, the Riemannian metric of the manifold and a Hermitian metric on vector bundles will play an important role. 

We will follow the notation of \cite{Ka16}. $X$ will be a complete Riemannian manifold (equipped with a proper isometric action of a locally compact group $G$ in later sections of the present paper). We denote by $T(X)$ the tangent bundle of $X$, by $T^*(X)$ the cotangent bundle and by $p:T^*(X)\to X$ the projection. We will often identify $T(X)$ and $T^*(X)$ via the Riemannian metric of $X$. The tangent manifold will be denoted $TX$, and the projection $TX\to X$ also by $p$.

\medskip

We start with the following theorem, which is definitely known, but I don't have a reference. 

\begin{theo} Let $f=f(\xi)$ be a bounded, differentiable function on ${\bf R}^n$ such that all its first derivatives vanish at infinity. Denote by $\Phi$ the operator of Fourier transform on $L^2({\bf R}^n)$, by $f$ the operator of multiplication by the function $f$ on $L^2({\bf R}^n)$, and by $F$ the operator $\Phi^{-1}f\Phi$. Then for any function $a=a(x)\in C_0({\bf R}^n)$ considered as a multiplication operator on $L^2({\bf R}^n)$, the commutator $[F,a]$ is a compact operator on $L^2({\bf R}^n)$.

More generally, the assertion remains true in the situation with a compact parameter space $Z$. More precisely, if $L^2({\bf R}^n)$ is replaced with $L^2({\bf R}^n)\otimes C(Z)$ and $f(\xi)$ with $f(\xi,z)\in C_b({\bf R}^n\times Z)$, continuous in $z$ uniformly in $\xi$, satisfying the same assumption on its first derivatives in $\xi$ (uniformly in $z\in Z$), then for any $a\in C_0({\bf R}^n\times Z)$, the commutator $[F,a]$ belongs to ${\cal K}(L^2({\bf R}^n))\otimes C(Z)$.
\end{theo}

\noindent {\em Proof.}  We will prove the Fourier-dual assertion. Recall that $\Phi^{-1}C_0({\bf R}^n)\Phi=C^*({\bf R}^n)$, the {$C^*$}-algebra of the abelian group ${\bf R}^n$. The algebra $C^*({\bf R}^n)$ contains the dense subalgebra $C_c({\bf R}^n)$ (compactly supported continuous functions) with convolution as multiplication. We need to prove that for any $b\in C_c({\bf R}^n)$, the commutator $[b,f]\in {\cal K}(L^2({\bf R}^n))$.

The assumption on the first derivatives of $f$ implies that $|f(y)-f(x)|\le \int_0^1 |\partial/\partial t f(x+t(y-x))|dt\to 0$ when $||x-y||$ is bounded. The commutator $[b,f]$ is an integral operator with the kernel $k(x,y)=b(x-y)(f(x)-f(y))$. Since $b$ has compact support, we obviously get $\int |k(x,y)|dx\to 0$ when $y\to\infty$ and $\int |k(x,y)|dy\to 0$ when $x\to\infty$. The Schur lemma \cite{Hor_book}, 18.1.12, easily implies that the integral operator with the kernel $k$ is compact. 

The proof of the generalized version of the statement (with the parameter space $Z$) is the same. In fact, it is enough to work with $f(\xi,z)=f_1(\xi)f_2(z)$ and $a(x,z)=a_1(x)a_2(z)$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\begin{cor} The assertion of the theorem remains true in the following weaker assumptions: $f$ is a bounded measurable function on ${\bf R}^n$ which is differentiable outside of a compact subset of ${\bf R}^n$, and all its first derivatives vanish at infinity. Moreover, in the same assumptions, the norm of the operator $F\,{\rm mod\,} {\cal K}(L^2({\bf R}^n))$ does not exceed $\limsup_{\xi\to\infty}|f(\xi)|$.
\end{cor}

\noindent {\em Proof.}  We can write $f=f_0+f_1$ where $f_0$ is bounded,  measurable and has compact support, and $f_1$ satisfies the assumptions of the theorem. For any $a\in C_0({\bf R}^n)$, the products $\Phi^{-1}f_0\Phi\cdot a$ and $a\cdot\Phi^{-1}f_0\Phi$ belong to ${\cal K}(L^2({\bf R}^n))$ by the Rellich lemma. So the assertion of the theorem remains true. For the last statement, we can take $f_0$ with as large compact support as we want. The norm of $F\,{\rm mod\,} {\cal K}$ depends only on $\sup |f_1|$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\medskip

\begin{prop} Keeping the assumptions of theorem 3.1 concerning the function $f$ and the operator $F$, let $V$ be a finite-dimensional complex Hermitian vector space, $h$ and $k$ - continuous maps ${\bf R}^n\to {\cal L}(V)$, such that $h=h(x)$ is linear in $x$ and $(h(x))^*=h(x)$ for any $x$, $k=k(x)$ is bounded, and both $||(h^2+1)^{-1}k||$ and $||(h^2+1)^{-1}hk||$ vanish at infinity in $x$. Then we have: $[F,h(h^2+1)^{-1/2}]k\in {\cal K}(L^2({\bf R}^n)\otimes V)$. 
\end{prop}

\noindent {\em Proof.}  We will use the integral presentation: 
$$h(h^2+1)^{-1/2}=2/\pi \int_0^\infty h(1+h^2+\lambda^2)^{-1}d\lambda.$$
The commutator $[F,h(h^2+1)^{-1/2}]$ equals 
$$2/\pi \int_0^\infty (1+h^2+\lambda^2)^{-1}\cdot ((1+\lambda^2)[F,h]+h[F,h]h)\cdot(1+h^2+\lambda^2)^{-1}d\lambda.$$
Since $h$ is a linear function, the commutator $[F,h]$ belongs to $C^*({\bf R}^n)\otimes {\cal L}(V)$. (Indeed, the Fourier-dual assertion says that the commutator of $f$ with the first order differential operator $\hat h$ equals to the multiplication by a function vanishing at infinity.) Therefore the latter integral expression converges in norm, and because $(h^2+\lambda^2+1)^{-1}k$ and $(h^2+\lambda^2+1)^{-1}hk$ both belong to $C_0({\bf R}^n)\otimes{\cal L}(V)$, the assertion follows from the Rellich lemma. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\medskip

Here are a couple more lemmas to justify the PDO calculus which follows.

\begin{lem} Let $\sigma=\sigma(\xi)$ be a bounded, differentiable function on ${\bf R}^n$ such that all its first derivatives vanish at infinity. Denote by $\hat\sigma\in {\cal M}(C^*({\bf R}^n))$ its Fourier transform. Let also $a\in C_0({\bf R}^n-\{0\})$. Consider both $\hat\sigma$ and $a$ as operators on $L^2({\bf R}^n)$ (as operators of convolution and multiplication respectively). Then $\hat\sigma\cdot a\in {\cal K}(L^2({\bf R}^n))$. In other words, if $a=a'\cdot a''$, where $a', a''\in C_b({\bf R}^n)$, $a'$ vanishes at infinity of ${\bf R}^n$ and $a''$ vanishes at $0$, then the product of operators $\hat\sigma a'$ and $a''$ belongs to ${\cal K}(L^2({\bf R}^n))$.
\end{lem}

\noindent {\em Proof.}  Let $\{x_k,\, k=1,..,n\}$ be the Euclidean coordinates in ${\bf R}^n$ and $\{\xi_k\}$ the dual coordinates in $({\bf R}^n)^*$. The algebra $C_0({\bf R}^n-\{0\})$ has a dense subalgebra $B$ algebraically generated by the functions: $\{\alpha x_k\exp(-\beta ||x||^2/2, \,\alpha\in {\bf C},\beta>0\}$. Writing $\beta x_k\exp(-\beta ||x||^2/2)$ as $-\partial/\partial x_k(\exp(-\beta ||x||^2/2))$ shows that the Fourier transform of $B$ is generated by the functions $\alpha \xi_k\exp(-\beta ||\xi||^2/2)$, or by the functions $\alpha\partial/\partial \xi_k(\exp(-\beta ||\xi||^2/2))$. It is enough to prove our assertion for $a\in B$, and we can even assume that $\hat a=\partial/\partial\xi_k(\alpha(\exp(-\beta||\xi||^2/2)))$.

The Fourier transform of  $\hat\sigma\cdot a$ is $\sigma * \hat a$ (where $*$ means convolution). 
The latter is equal (by our assumption about $\hat a$) to $\sigma * (\partial/\partial\xi_k(\alpha(\exp(-\beta||\xi||^2/2))))=(\partial/\partial \xi_k(\sigma))*(\alpha\exp(-\beta||\xi||^2/2))$. Because $\partial/\partial \xi_k(\sigma)\in C_0({\bf R}^n)$ by the assumption of the lemma, after making the inverse Fourier transform of our convolution product, we get an element of $C^*({\bf R}^n)\cdot C_0({\bf R}^n)\subset {\cal K}(L^2({\bf R}^n))$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\begin{rem} {\rm The assertion of the lemma means that when we localize the operator $\hat\sigma$ to any neighborhood $U$ of $0$ (by multiplying it with $a'$, where ${\rm supp\,}(a')\subset U$), the non-trivial part of the operator $\hat\sigma$ (i.e. modulo ${\cal K}$) will be contained in an arbitrarily small neighborhood of $0$. This is a well known property in the PDO calculus.}
\end{rem}

\begin{lem} Let $\sigma(x,\xi)$ be a function on $K\times {\bf R}^n$, where $K$ is a compact subset of ${\bf R}^n$, and let $\{\psi_x\}:(x,\xi)\mapsto (x,\psi_x(\xi))$ be a norm-continuous family of invertible linear maps of $K\times {\bf R}^n$ into itself.

$1^{\rm o}$ Assume that $\sigma(x,\xi)$ is continuous in $x$ uniformly in $\xi$. 

$2^{\rm o}$ Also assume that $\sigma(x,\xi)$ is differentiable in $\xi$, and for the exterior derivative $d_\xi$, there is an estimate: $||d_\xi \sigma(x,\xi)||\le C\cdot (1+||\xi||)^{-1}$ with the constant $C$ which does not depend on $(x,\xi)$. 

Then $\sigma(x,\psi_x(\xi))$ satisfies the same two conditions as $\sigma(x,\xi)$ (with a different constant $C$). Moreover, if the assumptions concerning $\sigma$ hold only outside of a compact subset in $\xi\in{\bf R}^n$, the assertion remains true outside of a (possibly larger) compact subset in $\xi\in{\bf R}^n$.
\end{lem}

\noindent {\em Proof.}  The assertion about the second condition is clear. To show that the first condition is also preserved, let points $x,y\in K$ be so close to each other that $||\sigma(x,\xi)-\sigma(y,\xi)||\le \delta$ for any $\xi$ (by the assumption of continuity of $\sigma$ in $x$ uniformly in $\xi$). We can write $\sigma(x,\psi_x(\xi))-\sigma(y,\psi_y(\xi))$ as a sum of two expressions: $(\sigma(x,\psi_x(\xi))-\sigma(x,\psi_y(\xi)))+(\sigma(x,\psi_y(\xi))-\sigma(y,\psi_y(\xi)))$. The norm of the second expression is estimated by $\delta$. 

The norm of the first expression is estimated as $\le ||d_\eta(\sigma(x,\eta))||\cdot ||\psi_x(\xi)-\psi_y(\xi)||$, where $\eta$ is some point on the segment joining $\psi_x(\xi)$ and $\psi_y(\xi)$, i.e. $\eta=t\psi_x(\xi)+(1-t)\psi_y(\xi)$ with $0\le t \le 1$. The first multiple is estimated as $\le C(1+||\eta||)^{-1}$. The second multiple is estimated as $\le ||\psi_x-\psi_y||\cdot ||\xi||$. Because $\psi_x$ is norm-continuous in $x$ and invertible for any $x\in K$, the map $\xi\mapsto \eta=t\psi_x(\xi)+(1-t)\psi_y(\xi)=t(\psi_x(\xi)-\psi_y(\xi))+\psi_y(\xi)$ is still an invertible linear map which is close to the map $\xi\mapsto \psi_y(\xi)$ when $x$ and $y$ are close enough. Therefore $||d_\eta(\sigma(x,\eta))||\cdot ||\psi_x(\xi)-\psi_y(\xi)||\le C'(1+||\psi_y(\xi)||^{-1})\cdot ||\psi_x-\psi_y||\cdot ||\xi||$, thus proving the uniform continuity of $\sigma(x,\psi_x(\xi))$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\medskip

Now we are ready to give the definition of symbols of `order $0$' and `negative order'. 
 
\begin{defi} Let $E$ be a (complex) vector bundle over $X$. A symbol $\sigma(x,\xi)$ of order $0$ is a bounded measurable section of the bundle $p^*(E)$ over $TX$ satisfying the following  conditions:

$1^{\rm o}$ For any compact subset in $x\in X$, $\sigma(x,\xi)$ is continuous in $x$ uniformly in $\xi$ outside of a compact subset in $\xi$. 

$2^{\rm o}$ For any compact subset in $x\in X$, $\sigma(x,\xi)$ is differentiable in $\xi$ outside of a compact subset in $\xi$, and for the exterior derivative $d_\xi$, there is an estimate for any compact subset $K\subset X$: $||d_\xi \sigma(x,\xi)||\le C\cdot (1+||\xi||)^{-1}$ with the constant $C$ which depends only on $K$ and $\sigma$ and not on $(x,\xi)$. 

We will say that a symbol $\sigma$ is of `negative order' if $||\sigma(x,\xi)||$ converges to $0$ uniformly in $x\in X$ on compact subsets of $X$ when $\xi\to\infty$. We will say that $\sigma$ is of `strongly negative order' if, additionally, $||\sigma(x,\xi)||$ converges to $0$ uniformly in $\xi$ when $x\to \infty$ in $X$. 
\end{defi}

\begin{rem} {\rm Condition $1^{\rm o}$ of definition 3.7 does not depend of the choice of local coordinates on the tangent bundle $T(X)$ in view of lemma 3.6. 

In fact, condition $2^{\rm o}$ of definition 3.7 was stated with this strong estimate on the first derivative of $\sigma$ because this estimate was required in the statement of lemma 3.6. However, there may be situations when, for example, there is a preferred choice for the trivialization of $T(X)$. In this case the precise estimate in condition $2^{\rm o}$ may be omitted. The minimal necessary requirement in condition $2^{\rm o}$ would be the vanishing of $d_\xi(\sigma)$ at infinity in $\xi$ uniformly on compact subsets in $x$.

Also note that the class of symbols of definition 3.7 includes the `classical' symbols (i.e. symbols homogeneous of order $0$ in the $\xi$ variable).} 
\end{rem} 

In our present approach, an operator corresponding to the symbol $\sigma$ will be constructed using the operator integration described in subsection 2.1. Let us fix a complex vector bundle $E$ over $X$ (equipped with a Hermitian metric). We will define operators acting on sections of $E$.

Let $x\in X$ and $U_x\subset X$ be a small open ball of radius $r_x$ centered at $x$. Using the Riemannian exponential map, consider the open Euclidean ball $\tilde U_x$ of the same radius around the point $0\in T_x(X)$ as a Euclidean coordinate neighborhood for $U_x$. 

For any fixed $x\in X$, the symbol $\tilde \sigma_x(\xi)=\sigma(x,\xi):E_x\to E_x$ is a bounded function of $\xi\in T^*_x(X)$. If we denote the Fourier transform $L^2(T_x(X))\otimes E_x \to L^2(T^*_x(X))\otimes E_x$ by $\Phi$, then $\tilde F_x=\Phi^{-1}\tilde \sigma_x\Phi$ is a bounded operator on $L^2(T_x(X))\otimes E_x$. This operator is just the operator with the symbol $\tilde\sigma_x(\xi)$. (Here $x$ is fixed, only $\xi$ varies.) 

Next, choose a function $\nu\in C_0^\infty([0,1))$ such that $0\le \nu \le 1$, $\nu(t)=1$ for $t\le 1/2$, and $\nu(t)=0$ for $t\ge 2/3$. By theorem 3.1, $\tilde F_x$ commutes modulo compact operators with multiplication by functions from $C_c(T_x(X))$. In particular, it commutes modulo compact operators with the function $\nu_x(v)=\nu(||v||/r_x)$ (here $t=||v||/r_x$). 

It is well known that when all sectional curvatures at any $y\in U_x$ are contained between $-c$ and $c>0$, then the exponential map $\tilde U_x\to U_x$ is $\varepsilon$-close to an isometry in a smaller ball of radius $\varepsilon/c$ around $x$ (see e.g. \cite{Ka88}, proposition 4.3). The Riemannian measure on $U_x$ is defined via the Riemannian metric $(g_{ij})$ on $U_x$ using the function $(\det(g_{ij}))^{1/2}$. When $\varepsilon$ is small, $(g_{ij})$ is close to the identity matrix. In this case, the map $L^2(\tilde U_x)\to L^2(U_x)$ given by the exponential map $\tilde U_x\to U_x$ is close to an isometry. (In fact, this map of $L^2$-spaces is given by multiplication by the function $(\det(g_{ij}))^{1/4}$ which is close to $1$.)

Now we need a trivialization of $E$ over $U_x$. Of course, this trivialization will depend on $x$, so we need a continuous family of trivializations $\{E|_{U_x}\}$ over $X$. Using this and the almost isometry between $L^2(\tilde U_x)$ and $L^2(U_x)$ described above, we can transfer the operator $\nu_x\tilde F_x\nu_x$ to $L^2(E|_{U_x})$. Denote the resulting operator on $L^2(E|_{U_x})\subset L^2(E)$ by $F(x)$. 

Note that if the original symbol $\sigma$ is a symbol in the H\" ormander $\rho=1,\, \delta=0$ class, then $F(x)$ belongs to the corresponding class of PDOs in the H\" ormander theory and its symbol at the point $x$ is $\sigma(x,\xi)$ modulo symbols of lower order. This is essentially the homeomorphism invariance of the H\" ormander $\rho=1,\, \delta=0$ class (see \cite{Hor}, end of subsection 2.1).

\begin{rem} {\rm Lemma 3.4 shows that a different choice of the function $\nu$ would result only in a compact perturbation of $F(x)$.}
\end{rem}

Theorem 3.1 ensures that the operator $F(x)$ commutes modulo compact operators with the multiplication by functions from $C_0(X)$. (Each $F(x)$ commutes with $C_0(U_x)$, and because the cutting function $\nu_x$ has support inside $U_x$, the operator $F(x)$ commutes with $C_0(X)$.) Moreover, $F(x)$ modulo compact operators is a norm-continuous function of $x\in X$. Indeed, by our assumptions, $\sigma(x,\xi)$ is continuous in $x$ uniformly in $\xi$. So the Euclidean operators $\tilde F_x(\xi)$ and $\tilde F_y(\xi)$ are norm-close to each other if $x$ and $y$ are close because they are just Fourier transforms of two uniformly close functions. Therefore the Euclidean operators $\nu_x\tilde F_x\nu_x$ and $\nu_y\tilde F_y\nu_y$ are also norm-close to each other. If $x$ and $y$ are close enough, both functions, $\nu_x$ and $\nu_y$, have their supports in $U_x\cap U_y$. The transition function between $E|_{U_x}$ and $E|_{U_y}$ will be close to the identity. Therefore, modulo compact operators, $F(x)$ and $F(y)$ will be norm-close to each other in $L^2(U_x\cap U_y)$.

To set up the operator integration, let $D={\cal K}(L^2(E))$ and $\phi:C_0(X)\to {\cal L}(L^2(E))$ be the multiplication by functions. The function $F$ constructed above will be considered as the function $X\to {\cal M}(D)/D$. The lifting of the operator integral $\int_X F(x)d\phi$ to ${\cal L}(L^2(E))$ will be denoted ${\cal F}$. 

\begin{defi} We define `negative order' operators as those which become compact after multiplication by an element of $C_0(X)$.
\end{defi}

In the case when there is a continuous, isometric, and proper action of a locally compact group $G$ on $X$ (and on the vector bundle $E$), and the symbol $\sigma$ is $G$-invariant, we can average ${\cal F}$ over $G$. For this, we take a cut-off function $\mathfrak{c}$ on $X$, i.e. a non-negative function $\mathfrak{c}\in C_b(X)$ whose support has compact intersection with any $G$-compact set in $X$ and which satisfies the property $\int_G g(\mathfrak{c})dg=1$. Let $\sum_i \alpha_i=1$ be a partition of unity on $X/G$ so that all $\alpha_i\ge 0$, all supports ${\rm supp\,}\alpha_i$ are compact in $X/G$ and form a locally finite covering of $X/G$. We will consider all functions $\alpha_i$ as $G$-invariant functions on $X$. 

Using remark 2.6 and proposition 2.7, we take $Av_G (\alpha_i\mathfrak{c}{\cal F})$ for every $i$. The operators $\alpha_i\mathfrak{c}{\cal F}$ have compact support because all $\alpha_i\mathfrak{c}$ have compact support and ${\cal F}$ has proper support - see remark 2.6. After that we redefine ${\cal F}$ as $\sum_i Av_G (\alpha_i\mathfrak{c}{\cal F})$. By lemma 2.4, the averaged operator will differ from the original ${\cal F}$ by an operator of `negative order'. The operator ${\cal F}$ will still be properly supported (see remark 2.6).

\begin{theo} The correspondence between the symbol $\sigma$ and the operator ${\cal F}$ constructed out of it has the following properties (modulo negative order symbols and negative order operators):

$1^{\rm o}$ Composition of symbols $\mapsto$ composition of operators.

$2^{\rm o}$ $\sigma^* \mapsto {\cal F}^*$.

$3^{\rm o}$ The operator corresponding to a negative order symbol has negative order, and the operator corresponding to a strongly negative order symbol is compact.
\end{theo}

\noindent {\em Proof.}  All assertions follow directly from the construction and theorem 2.1. To obtain the last assertion concerning strongly negative order symbols, one can either use corollary 2.3, or one may approximate the symbol by symbols with compact support in $x\in X$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\begin{theo} Let $\sigma(x,D)$ be a bounded, properly supported pseudo-differen\-tial operator of order $0$ with  the symbol $\sigma$ in the H\" ormander $\rho=1,\,\delta=0$ calculus, and let ${\cal F}$ be the operator constructed out of $\sigma$ above. Then, the difference $\sigma(x,D)-{\cal F}$ is an operator of `negative order'. 

In particular, the operator ${\cal F}$ has symbol and this symbol is well defined modulo symbols of negative order.
\end{theo}

\noindent {\em Proof.}  Because both $\sigma(x,D)$ and ${\cal F}$ are bounded we can consider only $f\in C_c(X)$, so we need to prove that $\sigma(x,D)$ and ${\cal F}$ coincide on any compact piece of $X$. Let $\{U_i\}$ be a locally finite covering of $X$ consisting of Riemannian balls $U_{x_i}$ and $\sum_i \alpha_i^2=1$ the corresponding partition of unity. Let us present $\sigma(x,D)$ as the sum: $\sum_i \sigma(x,D)\alpha^2_i$. Modulo compact operators, we can change this sum slightly by replacing it with $\sum_i \tilde\nu_i\sigma(x,D)\tilde\nu_i\alpha^2_i$, where the function $\tilde\nu_i$ is the function $\nu_x$ used in the construction above for the ball $U_i$.

Let us compare the latter sum with the integral sum $\sum_i F(x_i)\alpha^2_i$. For any $i$, both $F(x_i)$ and $\tilde\nu_i\sigma(x,D)\tilde\nu_i$ are PDOs of order $0$ with compactly supported distributional kernel. The difference between their symbols goes to $0$ uniformly in $i$ when the radii of the balls $U_i$ go to $0$ (on our compact piece of $X$). Now the assertion follows from the usual norm estimate results: lemma 2.2 and \cite{Hor}, corollary 2.2.3. 

The definition of a symbol for a properly supported PDO of the H\" ormander class is well known, and this symbol modulo symbols of negative order (called `principal symbol') depends only on the operator modulo operators of negative order (see \cite{Hor}, end of subsection 2.1).
{\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\medskip

Finally, a remark about notation. In the previous construction we have actually used not the symbol of an operator itself but its Fourier transform. If considered point-wise, the usual symbol lives in $C_b(T^*_x(X))={\cal M}(C_0(T^*_x(X)))$, $x\in X$. Its Fourier transform lives in its Fourier dual.

\begin{defi} We will denote the tangent bundle $T(X)$ by $\tau$ and the tangent space $T_x(X)$ at $x\in X$ by $\tau_x$. The algebra $C^*(\tau_x)$ is the {$C^*$}-algebra of the abelian group $\tau_x$ with its Euclidean topology. The algebra of continuous sections (vanishing at infinity) of the field $\{C^*(\tau_x),x\in X\}$ will be denoted $C^*_\tau(X)$. We will consider the Fourier transform $\hat\sigma_x$ of the symbol $\sigma_x$ as an element of the multiplier algebra ${\cal M}(C^*_\tau(X))$. We will call it \emph{convolutional symbol} corresponding to the usual symbol. 
\end{defi}

\medskip

We end up this section with a simple technical lemma which will be needed later in the proof of our index theorems. 

Let $Y\to X$ be a smooth, locally trivial fiber bundle, both $X$ and $Y$ are complete Riemannian manifolds. We will denote the fiber over a point $x\in X$ by $Y_x$. Assume that the transition functions are isometries in the Riemannian metrics on all $Y_x$. Let ${\cal E}$ be the Hilbert module over $C_0(X)$ defined by the continuous field $L^2(Y_x),\, x\in X$. Consider the Hilbert space $H={\cal E}\otimes_{C_0(X)} L^2(X)$. For any $e\in {\cal E}$, define the operator $\theta_e\in {\cal L}(L^2(X),H)$ by $\theta_e(l)=e\otimes l\in H$ for any $l\in L^2(X)$. 

Now a small reminder: Suppose we have a bounded operator $F\in {\cal L}(L^2(X))$. Then a \emph{$K$-theoretic connection} (cf. \cite{CS}, A1; \cite{Ka88}, 2.6) for the operator $F$ is an operator $\tilde F\in {\cal L}(H)$ such that for any $e\in {\cal E}$, the operators $\theta_e F-\tilde F \theta_e$ and $\theta_e F^*-\tilde F^* \theta_e$ are compact. (In fact, when there are some additional graded vector bundles over $X$ and $Y$ and we consider graded $L^2$-spaces, then the multiple $(-1)^{deg(e)\cdot deg(F)}$ has to be introduced in front of $\tilde F$ and $\tilde F^*$, so that the commutators become graded).

\begin{lem} In the notation above, suppose that $F$ is a PDO with the symbol $\sigma$. Let us not distinguish between tangent and cotangent bundles (using the Riemannian metrics) and lift $\sigma$ from $TX$ to $TY$ via the projection $Y\to X$. Denote the lifted symbol by $\tilde \sigma$. Then the PDO $\tilde F$ on the space $H$ with the symbol $\tilde \sigma$ (constructed by means of our coarse PDO calculus) is a $K$-theoretic connection for $F$.
\end{lem}

\noindent {\em Proof.}  If $Y$ is a direct product $X\times Y_x$, the assertion is obviously true. But the lifted symbol commutes with the transition functions. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10


\section{{$C^*$}-algebras associated with a group action}

Let $X$ be a locally compact, $\sigma$-compact space equipped with a continuous proper action of a locally compact group $G$. The left-invariant Haar measure on $G$ will be denoted $dg$, and the modular function of $G$ by $\mu$. The stability subgroup $M_x$ of any point $x\in X$ is compact because the action is proper. We will assume that the Haar measure of any compact group is normalized to the total mass $1$. 

For any Hilbert $C_0(X)$-module $E$, the fiber of $E$ over $x\in X$ will be denoted $E_x$. The space of compactly supported elements of $E$, i.e. $E\cdot C_c(X)$, will be denoted $E_c$ (or $C_c(E)$ if we consider $E$ as a continuous field of vector spaces).


\subsection{$C^*(G,B)$ for a $C_0(X)$-algebra $B$}

For a proper $G$-algebra $B$, we will use the notation $B^G$ for a certain subalgebra of ${\cal M}(B)$ consisting of $G$-invariant elements - see \cite{Ka88}, 3.2.
  
We start with the construction which originates in the work of M. Rieffel. Here we follow \cite{KS03}, the beginning of section 5 and the proof of theorem 5.4. 

\begin{defi} Let $B$ be a $G-C_0(X)$-algebra, where $X$ is a proper $G$-space, and $E$ a Hilbert $G-B$-module. Denoting by $C_c(X)$ the set of compactly supported functions on $X$, let $E_c=C_c(X)\cdot E$. We define a pre-Hilbert module structure over $C_c(G,B)$ on $E_c$ by 
$$e\cdot b=\int_G g(e)\cdot g(b(g^{-1}))\cdot \mu(g)^{-1/2}dg\in E_c,$$
$$(e_1,e_2)(g)=\mu(g)^{-1/2}(e_1,g(e_2))_E \in C_c(G,B),$$
where $e,e_1,e_2\in E_c,\; b\in C_c(G,B)$. We will denote by ${\cal E}$ the Hilbert $C^*(G,B)$-module which is the completion of $E_c$ in the norm defined by the above $C_c(G,B)$ inner product on $E_c$.
\end{defi}

\begin{rem} {\rm The positivity of the inner product comes from the embedding $i:{\cal E}\subset C^*(G,E)$ (onto a direct summand) given by $i(e)(g)=\mu(g)^{-1/2}c^{1/2}g(e)$, where $c$ is a cut-off function on $X$ (see \cite {Ka16}, section 5). The Hilbert module $C^*(G,E)$ is defined in \cite{Ka88}, 3.8. It is isomorphic to $E\otimes_B C^*(G,B)$.}  
\end{rem}

\begin{lem} ${\cal K}({\cal E})\simeq {\cal K}(E)^G$. In particular, when $E=B$, we have ${\cal K}({\cal E})=B^G$, and when $E=B=C_0(X)$, we get ${\cal K}({\cal E})=C_0(X/G)$. 
\end{lem}

\noindent {\em Proof.}  The `rank 1' operators $\theta_{e_1,e_2}$ in ${\cal K}({\cal E})$ (i.e. $\theta_{e_1,e_2}(e)=e_1(e_2,e)$) are easily seen to be $\theta_{e_1,e_2}(e)=\int_G \theta_{g(e_1),g(e_2)}(e)dg,$ 
where $e_1,e_2,e\in E_c$, and on the right side, the notation $\theta$ is used for the `rank 1' operators in ${\cal K}(E)$. This proves the assertion. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\begin{lem} When $E=L^2(G,B)$, then ${\cal E}\simeq C^*(G,B)$ as a Hilbert module over $C^*(G,B)$. 
\end{lem}

\noindent {\em Proof.}  Consider $L^2(G,B)$ with the right $G$-action: $g(e)(t)=\mu(g)^{1/2}g(e(tg))$ instead of the usual left one. (These two $G$-actions correspond to each other under the automorphism $e(g)\mapsto \mu(g)^{-1/2} e(g^{-1})$ of $L^2(G,B)$.) Let us use elements $e=e(g)\in E_c$ with compact support in $g\in G$ (i.e. elements from $C_c(X)\cdot C_c(G,B)$). Then it is easy to check that the isomorphism ${\cal E}\simeq C^*(G,B)$ is given by the formula: $e(g)\mapsto g(e(g))$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\medskip

Lemmas 4.3 and 4.4 imply the following result, which is a generalization of the well known similar fact for compact groups (cf. e.g. \cite{GHT}, 11.2):

\begin{theo} $C^*(G,B)\simeq {\cal K}(L^2(G,B))^G$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10
\end{theo}

Next, we will prove a version of the structure theorem of P. Green (\cite{Gr80}, 2.13). (We need only a weak version of it and we want to avoid using a measurable cross-section in the construction.) Let us assume that $X$ is a fibered product $X=G\times_M Z$, where $M$ is a compact subgroup of $G$ and $Z$ an $M$-space. There is a natural projection $\pi:X\to G/M$ in this case, and $\pi(Z)=(M)\in G/M$. Let $B_Z$ be the part of $B$ over $Z$, i.e. $B_Z=B/(B\cdot C_0(X-Z))$. (In the Riemannian manifold case which will be considered later, $Z$ will be a small Euclidean disk orthogonal to the orbit $O_x$ at the point $x\in X$, and $M$ will be the stability subgroup at $x$.) 

It is clear that $B$ is the algebra defined by the continuous field of {$C^*$}-algebras $\{B_t, t\in G/M\}$, where $B_t=t(B_Z)$. This means that $B$ is isomorphic to the algebra of continuous sections (vanishing at infinity) of the fiber bundle of algebras $G\times_M B_Z \to G/M$ associated with the principal bundle $G\to G/M$. We will use the notation $(C_0(G)\otimes B_Z)^M$ for this algebra.

\begin{prop}  In the above assumptions and notation, if $X=G\times_M Z$, then
$C^*(G,B)\simeq (B_Z\otimes {\cal K}(L^2(G)))^M$, where $M$ acts on $B_Z\otimes{\cal K}(L^2(G))$ diagonally (with the action on $L^2(G)$ by right translations). 
\end{prop} 

\noindent {\em Proof.}  By theorem 4.5, $C^*(G,B)\simeq (B\otimes {\cal K}(L^2(G)))^G$. This can be rewritten as $((C_0(G)\otimes B_Z)^M\otimes {\cal K}(L^2(G)))^G$. Here $G$ acts on $C_0(G)$ by left translations and $M$ by right translations, and these two actions commute. The action of $G$ on $L^2(G)$ is by right translations, and the action of $M$ on $L^2(G)$ is trivial. Therefore $C^*(G,B)$ is isomorphic to $(C_0(G)\otimes B_Z\otimes {\cal K}(L^2(G)))^{M\times G}$. 

The isomorphism stated in the proposition comes from the obvious isomorphism: $(C_0(G)\otimes A)^G\simeq A$ for $A=B_Z\otimes {\cal K}(L^2(G))$. Note that the inverse of the latter isomorphism maps $a\in A$ to $g\mapsto \{g(a)\}$ in $C_b(G,A)$. Because $M$ acts on $C_b(G)$ by right translations, we get the action of $M$ on $C_b(G,A)$: $\{g(a)\}\mapsto \{gm(a)\}$, which corresponds to the diagonal action $a\mapsto m(a)$ of $M\subset M\times G$ on $A$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10


\subsection{Leaf-wise $L^2$-spaces and integral operators}

\begin{defi} Let $E$ be a $G-C_0(X)$-Hilbert module. We define the leaf-wise Hilbert module $L^2_{lf}(E)$ over $C_0(X/G)$ as follows. For $e_1,e_2\in C_c(E)$ the inner product $(e_1,e_2)_{\cal E}\in C_c(X/G)$ is the function $\int_G g(e_1,e_2)_Edg$. There is also an obvious multiplication by elements of $C_c(X/G)$ on $C_c(E)$. The completion of $C_c(E)$ in the norm associated with this inner product gives the Hilbert module $L^2_{lf}(E)$ over $C_0(X/G)$. 

More generally, in the situation when $B$ is a $G-C_0(X)$-algebra and $E$ is a $G-B$-Hilbert module, repeating the above definition, we get a Hilbert module $L^2_{lf}(E)$ over $B^G$. Indeed, $\int_G g(e_1,e_2)_Edg\in B^G$ by definition 3.2 \cite{Ka88}.
\end{defi}

Consider the map $f:G\times X\to X$ defining the $G$-action on $X$. For each $x\in X$, the orbit ${\cal O}_x$ through $x$ is the image of $G\times \{x\}$. We have: $L^2({\cal O}_x)\simeq L^2(G/M_x)\subset L^2(G)$. The image consists of all $L^2$-functions on $G$ right-invariant under $M_x$. So the $L^2$-space of each orbit is embedded into $L^2(G)$. This embedding  $\iota_x:L^2({\cal O}_x)\subset L^2(G)$ depends on the point $x$ of the orbit.  If $O_y$ is another orbit and $M_y\subset M_x$ then there is an embedding $\iota_{x,y}=\iota_y^*\iota_x:L^2(O_x)\subset L^2(O_y)$. 

\medskip

The following proposition is a part of \cite{Gr80}, 2.13.
 
 \begin{prop} Let $E$ be a $G-B$-Hilbert module, where $B$ is a $G-C_0(X)$-algebra. The natural representation $C^*(G,{\cal K}(E))\to {\cal L}(L^2_{lf}(E))$ admits a natural cross-section homomorphism: ${\cal K}(L^2_{lf}(E))\subset C^*(G,{\cal K}(E))$. This means that ${\cal K}(L^2_{lf}(E))$ is a direct summand in $C^*(G,{\cal K}(E))$.
\end{prop}

\noindent {\em Proof.}  We denote the natural covariant representation of the pair $(G, {\cal K}(E))$ in ${\cal L}(L^2_{lf}(E))$ by $(u,\phi)$, where $u$ is the $G$-action. Let us denote by $\psi$ the corresponding representation $C^*(G,{\cal K}(E))\to {\cal L}(L^2_{lf}(E))$: $k(g)\mapsto \int_G \phi(k(g))u(g)dg$. On each orbit $O_x=G/M_x$, we have: $C^*(G, {\cal K}(E|_{O_x}))\simeq {\cal K}(L^2(G)\otimes E_x)^{M_x}$ (proposition 4.6), and the map $\psi$ is given by the restriction of an element of ${\cal K}(L^2(G)\otimes E_x)^{M_x}$ to the subspace $(L^2(G)\otimes E_x)^{M_x}\simeq {L^2_{lf}}(E)|_{O_x}$. 

To construct the cross-section, on each orbit $O_x$, consider the obvious embedding $(L^2(G)\otimes E_x)^{M_x}\subset L^2(G)\otimes E_x$, which gives an embedding ${\cal K}((L^2(G)\otimes E_x)^{M_x})\subset {\cal K}(L^2(G)\otimes E_x)^{M_x}$. These embeddings produce the global embedding $i:{\cal K}(L^2_{lf}(E))\subset C^*(G,{\cal K}(E)$. The relation $\psi\cdot i={\rm id}$ is obvious. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\medskip

We will define now \emph{leaf-wise integral operators}. 

\begin{defi} Let $E$ be a $G-B$-Hilbert module, where $B$ is a $G-C_0(X)$-algebra. An integral leaf-wise operator with compact support on ${L^2_{lf}}(E)$ is any element of ${\cal K}({L^2_{lf}}(E))$ which has compact support in $X$. The set of such operators will be denoted ${\cal K}_c({L^2_{lf}}(E))$.

The algebra of $G$-invariant integral leaf-wise operators  is ${\cal K}({L^2_{lf}}(E))^G$. It is a subalgebra of ${\cal L}({L^2_{lf}}(E))$. A $G$-invariant integral leaf-wise operator with $G$-compact support is any element of ${\cal L}({L^2_{lf}}(E))$ which can be presented as $\int_G g(k)dg$ where $k\in {\cal K}_c({L^2_{lf}}(E))$. The set of such operators is dense in ${\cal K}({L^2_{lf}}(E))^G$. 
\end{defi}

\begin{prop} For a one-orbit space, if $X=G$ and $B=E=C_0(G)$, then ${\cal K}({L^2_{lf}}(E))^G$ is isomorphic to $C^*(G)$; and if $X=G/M$ and $E=C_0(G\times_M V)$ (where $V$ is the fiber of $E$ at the point $(M)\in G/M$), then ${\cal K}({L^2_{lf}}(E))^G$ is isomorphic to a subalgebra of ${\cal K}(V)\otimes C^*(G)$ satisfying certain right $M$-invariance conditions.
\end{prop}

\noindent {\em Proof.}   When $B$ is an arbitrary $G-C_0(G)$-algebra, the $G$-invariant properly supported kernels $k(g_1,g_2)$ with values in $B$ satisfy the usual condition: $k(gg_1,gg_2)=g(k(g_1,g_2))$, which implies that $k(g_1,g_2)=g_1(f(g_1^{-1}g_2))$, where $f$ is compactly supported. For $B=C_0(G)$, this gives $C^*(G)$. 

In the case of $X=G/M$, there are additional right $M$-invariance conditions for the kernel $k:G\times G\to {\cal K}(V)$, namely, $m_1k(g_1m_1,g_2m_2)m_2^{-1}=k(g_1,g_2)$, where $m_1,m_2\in M$.
 {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10 
 
 \begin{prop} The natural embedding ${\cal K}(L^2_{lf}(E))\subset C^*(G,{\cal K}(E))$ of proposition 4.8 induces the map ${\cal I}:{\cal K}({L^2_{lf}}(E))^G\to {\cal M}(C^*(G,{\cal K}(E)))$ which extends to the multipliers ${\cal M}({\cal K}({L^2_{lf}}(E))^G)$. This map has the following property: $C_0(X)\cdot {\rm Im}\,({\cal I})\subset C^*(G,{\cal K}(E))$.
 \end{prop}
 
 \noindent {\em Proof.}  The map ${\cal K}({L^2_{lf}}(E))^G\to {\cal L}({L^2_{lf}}(E))$ extends to ${\cal M}({\cal K}({L^2_{lf}}(E))^G)$ because there is an approximate unit in ${\cal K}({L^2_{lf}}(E))^G$ which maps to the sequence converging to $1\in {\cal L}({L^2_{lf}}(E))$ in the strict topology.
 
 On the other hand, because  ${\cal K}(L^2_{lf}(E))$ is a direct summand in $C^*(G,{\cal K}(E))$, the algebra  ${\cal L}(L^2_{lf}(E))$ is a direct summand in ${\cal M}(C^*(G,{\cal K}(E)))$. The composition of these two maps gives the required map ${\cal I}$.
 
 The last assertion follows from the definition of ${\cal K}({L^2_{lf}}(E))^G$, namely, $C_0(X)\cdot {\cal K}({L^2_{lf}}(E))^G\subset {\cal K}({L^2_{lf}}(E))$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10
 
 \begin{rem} {\rm For each orbit of $X$, $C^*(G,{\cal K}(E))={\cal K}(V\otimes L^2(G))^M$ (in the notation of proposition 4.10), so this algebra is generated by compactly supported integral operators on $G$ with the same right $M$-invariance conditions as for ${\cal K}(L^2_{lf}(E))^G$ (but no left $G$-invariance condition). Obviously, ${\cal K}({L^2_{lf}}(E))^G$, for each orbit of $X$, naturally maps to ${\cal M}({\cal K}(V\otimes L^2(G))^M)$.}
\end{rem}

 
\subsection{Admissible Hilbert modules and {$C^*$}-algebras}

\emph{From now on we will assume that $X$ is a complete Riemannian manifold equipped with the proper, isometric action of the Lie group $G$.}

\medskip

Any orbit $O_x=G/M_x\subset X$ has a closed tubular neighborhood of the type $U=G\times_{M_x} Z_x$, where the slice $Z_x$ is a closed disk orthogonal to $O_x$ at $x\in X$. The action of $M_x$ on $Z_x$ is orthogonal, therefore all concentric spheres in $Z_x$ have the same $M_x$-orbit structure. The natural projection $U\to O_x$ will be denoted $\pi$. This map induces the map $\pi^*:C_0(O_x)\to C_0(U)$ which allows to lift Hilbert modules from $O_x$ to $U$. Namely, the lifting $\pi^*(E)$ of a Hilbert $C_0(O_x)$-module $E$ is defined as $E\otimes_{C_0(O_x)} C_0(U)$, which is a Hilbert $C_0(U)$-module. There is also the map: $\pi^*:{\cal L}(E)\to {\cal L}(E\otimes_{C_0(O_x)} C_0(U))$ given by $k\mapsto k\otimes 1$.
 
We will consider $U$ as a `$G$-cone' with the `$G$-vertex' $G/M_x$ (corresponding to the orbit $O_x$) and the `$G$-base' $G\times_{M_x} \partial Z_x$ (where $\partial Z_x$ is the boundary sphere of $Z_x$). For any concentric sphere $S_r\subset Z_x$ of radius $r$, we will call $Q_r=G\times_{M_x} S_r$ the `$G$-level' of radius $r$. 

\medskip

Now we introduce a certain class of Hilbert $G-C_0(X)$-modules which will be used in the definition of leaf-wise operators. It is clear that the restriction of any $G-C_0(X)$-Hilbert module $E$ to an orbit is isomorphic to the module of continuous sections of a certain vector bundle over this orbit (see the discussion before proposition 4.6). First of all, we want all these vector bundles to be finite-dimensional with dimension bounded over $X$. Secondly, although the dimensions of these orbital bundles are allowed to jump, the global structure of the orbital vector bundles for $E$ should resemble the structure of tangent bundles to the orbits.

\begin{defi} We define the class of admissible $G-C_0(X)$-Hilbert modules $E$ inductively on the dimension of $X$. The main requirement is that $E$ is a Hilbert submodule (not necessarily a direct summand) of the Hilbert module ${\cal V}$ of continuous sections (vanishing at infinity of $X$) of some finite-dimensional $G$-vector bundle $V$ over $X$. Secondly, for any orbit $O_x$, there should exist a tubular neighborhood $U$ as above such that:

$1^{\rm o}$ There exists a natural isometric embedding $\pi^*(E|_{O_x})\subset E|_U$ compatible with the embedding $E\subset {\cal V}$, i.e. the natural maps $\pi^*$ for $E$ and ${\cal V}$ must be compatible. 

$2^{\rm o}$ The restriction of $E$ to any $G$-level of $U$ satisfies the inductive requirements for manifolds of smaller dimension, and all restrictions of $E$ to all such $G$-levels (except the $G$-vertex) must be isomorphic (upon the identification of the corresponding concentric spheres of $Z_x$).
\end{defi}

Note that a Hilbert module of continuous sections (vanishing at infinity) of any $G$-vector bundle over $X$ is certainly admissible.

\medskip

A similar definition can be given for a certain class of $C_0(X)$-algebras. This class will be a little more general than the class of algebras ${\cal K}(E)$ for $E$ admissible. 

\begin{defi} A $C_0(X)$-algebra $B$ will be called $C_0(X)$-\emph{unital} if there exists a subalgebra in $B$ isomorphic to $C_0(X)$, and the structure of the $C_0(X)$-algebra on $B$ is given by multiplication with elements of this subalgebra. (This means that all fibers $B_x,\, x\in X$, are unital.)

We will call a $C_0(X)$-algebra $B$ admissible if it is a $C_0(X)$-unital subalgebra of ${\cal K}({\cal V})$ for a $C_0(X)$-Hilbert module ${\cal V}$ of continuous section of a finite-dimensional vector bundle $V$ over $X$, and the conditions $1^{\rm o}-2^{\rm o}$ of definition 4.6 are satisfied for $B$ in place of $E$. 
\end{defi}

\subsection{Orbit-continuity}

\emph{Now we will define an important notion of orbit-continuity. We will do it simultaneously for all three types of algebras we considered in this section: $C^*(G,{\cal K}(E))$, ${\cal K}({L^2_{lf}}(E))$, and ${\cal K}({L^2_{lf}}(E))^G$. To avoid repetitions, in any definition or statement of this subsection, we will assume that one of the above three types of algebras is selected, and will refer to such algebra as ${\cal A}(E)$. Similar definitions and statements for admissible {$C^*$}-algebras (of definition 4.14) are left as an exercise.} 

\medskip

Let $E$ be an admissible $C_0(U)$-Hilbert module. Consider an element $F$ in the algebra ${\cal A}(E)$. Using the restriction map from $U$ to $Q_r$ for each $G$-level $Q_r$ (including the orbit $O_x$ for $r=0$), we get a family of elements $F_r$ depending on the radius $r$. 

We can compare the elements of this family using the fact that any two levels for non-zero $r$ are isomorphic, and the zero level can be mapped (lifted) to $U$ via $\pi^*$ and then restricted to any other level. 

\begin{prop} If $F$ is an element of ${\cal A}(E)$, the family $\{F_r,\,r\ge 0\}$ is norm-continuous in $r$. Vice versa, a norm-continuous family $\{F_r,\,r\ge 0\}$ in $\{{\cal A}(E|_{Q_r})\}$ defines an element of the algebra ${\cal A}(E)$.
\end{prop}

\noindent {\em Proof.}  For $C^*(G,{\cal K}(E))$, this follows from proposition 4.6, for the other two types - from the definitions. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\begin{cor} Multipliers of ${\cal A}(E)$ are characterized by the continuity of their family of restrictions to the levels $\{Q_r,\, r\ge 0\}$ in the strict topology. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10
\end{cor}

In the next definition we say that a sequence of orbits $O_{x_k}$ converges to the orbit $O_x$ if all points $\{x_k\}$ belong to $Z_x\subset U$ and $x_k\to x$.

\begin{defi} Let $E$ be an admissible $G-C_0(X)$ Hilbert module. If we are given a function $O\to F_O\in {\cal A}(E|_O)$, we will say that this function is orbit-continuous at the given orbit $O_x$ if whenever a sequence of orbits $O_{x_k}$ converges to the orbit $O_x$, we have: $||F_{O_{x_k}}-\pi_k^*(F_{O_x})||\to 0$, where $\pi_k^*$ means the lifting to a small tubular neighborhood $U$ of the orbit $O_x$ containing the orbit $O_{x_k}$, composed with the restriction to the orbit $O_{x_k}$. 
\end{defi}

The obvious reformulation of this definition for the case of orbit-continuity of multipliers in the strict topology is  an exercise. 

\begin{prop} Let $E$ be an admissible $G-C_0(X)$-Hilbert module. Then elements of the algebra ${\cal A}(E))$ are orbit-continuous in norm at any orbit $O$. Conversely, orbit-continuity in norm at each orbit  $O$ of the family of elements $\{F_O\in {\cal A}(E|_O),\,O\in X/G\}$ is sufficient for $F$ to be an element of ${\cal A}(E)$. The similar statement is true for multiplier algebras with the strict topology instead of the norm topology.
\end{prop}

\noindent {\em Proof.}  This follows from 4.15, 4.16. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\begin{ex} {\rm Let us consider the case of the group $G=S^1$ acting on the disk $X=D^2$ by rotations, and the Hilbert module $E=C(X)$.  For each circle $S_r$ of non-zero radius $r$, $C^*(G,C(S_r))\simeq{\cal K}(L^2(S^1))$, and for $r=0$, $C^*(G,C(S_0))=C^*(S^1)$, and ${\cal K}(L^2(S_0))={\bf C}=\{{\rm multiples\; of}\; 1\in C^*(G)\}$. When $r\to 0$, an element of $C^*(G,C(S_r))$ converges in norm to an element of $C^*(G)$, and an element of ${\cal K}(L^2(S_r))$ converges in norm to a multiple of $1\in C^*(G)$.

Let ${\cal D}$ be the differential operator on $S^1$ given by ${\cal D}=-i\,\partial/\partial\theta$, where $\theta$ is the angle. Put $F={\cal D}({\cal D}^2+1)^{-1/2}$. This is a bounded operator in $L^2(S^1)$ and a multiplier of $C^*(S^1)$. It defines a bounded operator $F_r$ on each $C^*(G,C(S_r))$ for $r\ge 0$ and on each $L^2(S_r)$ for $r > 0$. We define the operator $F_0$ for $r=0$ as $0$ in the case of $L^2(S_0)$. The family $\{F_r\}$ is strictly continuous in $r\ge 0$ in both cases: $\{C^*(G,C(S_r))\}$ and $\{{\cal K}(L^2(S_r))\}$, and gives an element of ${\cal M}(C^*(G,{\cal K}(E)))$ and ${\cal L}({L^2_{lf}}(E))$ respectively.  

The operator $1-F^2=({\cal D}^2+1)^{-1}\in C^*(S^1)$ is the convolution by the function $\sum_n (1+n^2)^{-1}e^{in\theta}$ on each orbit $S_r,\, r>0$. In the case of $C^*(G,{\cal K}(E))$, the family of these operators is norm-continuous in $r$ for all $r\ge 0$ and defines an element of $C^*(G,{\cal K}(E))$. 

However, the corresponding family $\{1-F^2_r\}$ for ${L^2_{lf}}(E)$ is not norm-conti\-nuous at $r=0$ and does not define an element of ${\cal K}({L^2_{lf}}(E))$. One can force $1-F^2$ to be an element of ${\cal K}({L^2_{lf}}(E))$ by changing the definition of $F_r$ into $F_r={\cal D}({\cal D}^2+r^2)^{-1/2}$ on the circle of radius $r>0$ (and leaving $F_0=0$).}
\end{ex}


\section{Leaf-wise PDOs}

In this section, we give a definition of symbol algebras for leaf-wise PDOs, followed by the construction of leaf-wise operators. An important leaf-wise Dirac operator will be discussed in detail.

The notation and assumptions concerning the manifold $X$ will be the same as stated in section 3: $X$ will be a complete Riemannian manifold equipped with a proper isometric action of a Lie group $G$, $T(X)$ and $T^*(X)$ the tangent and cotangent bundles of $X$ respectively (isomorphic via the Riemannian metric). The tangent manifold will be denoted $TX$, and the projection $TX\to X$ by $p$. 

\subsection{Notation and preliminaries.}

Let us first recall some notation from \cite{Ka16}, sections 6 and 7.

Let $x\in X$ and $f_x:G\to X$ be the map defined by $g\mapsto g(x)$. We will denote by $\mathfrak{g}$ the Lie algebra of the group $G$ (left-invariant vector fields on $G$). Let $f'_x: \mathfrak{g}\to T_x(X)$ be the tangent map (first derivative of $f_x$) at the identity of $G$, and $f'^*_x:T_x^*(X)\to \mathfrak{g}^*$ the dual map. It is easy to see that for any $x\in X,\; g\in G,\; v\in \mathfrak{g}$, one has: $g(f'_x(v))=f'_{g(x)}(Ad(g)(v))$. 

Let us consider the trivial vector bundle $\mathfrak{g}_X=X\times \mathfrak{g}$ over $X$ with the $G$-action given by $(x,v)\mapsto (g(x),Ad(g)(v))$. Because the $G$-action on $X$ is proper, there exists a $G$-invariant Riemannian metric on $\mathfrak{g}_X$. Equivalently, one can say that there exists a smooth map from $X$ to the space of Euclidean norms on $\mathfrak{g}$: $x\mapsto ||\cdot ||_x$,  such that for any $x\in X,\; g\in G,\; v\in \mathfrak{g}$, one has: $||Ad(g)(v)||_{g(x)}=||v||_x$. From the above formulas, it is clear that the map $f':\mathfrak{g}_X\to T(X)$, defined by $(x,v)\mapsto f'_x(v)$ at any $x\in X$, is $G$-equivariant. Note that by multiplying our Riemannian metric on $\mathfrak{g}_X$ by a certain strictly positive $G$-invariant function, we can also arrange that the following condition is satisfied: for any $v\in \mathfrak{g},\; ||f'_x(v)||\le ||v||_x$. We will assume that this is the case, and so we have $||f'_x||\le 1$ for any $x\in X$.

Let us identify $\mathfrak{g}_X$ with its dual bundle via the Riemannian metric. The fiber of $\mathfrak{g}_X$ at the point $x$ will be denoted $\mathfrak{g}_x$. Consider a continuous field of subspaces of the tangent bundle $T(X)$ defined as $\{{\rm Im}\, f'_x\subset T_x(X),\;x\in X\}$. Call this field $\Gamma$. This is the field of tangent spaces to the orbits of $G$. Another field is defined as $\{{\rm Im}\, f'^*_x\subset \mathfrak{g}^*_x,\; x\in X\}$. These two fields are isometrically isomorphic. The isomorphism is given by the isometry $(f'f'^*)^{-1/2}f':{\rm Im}\, f'^*\to {\rm Im}\, f'$. (\emph{Note that we do not assume that $f'$ is invertible.}) The continuous sections of these two fields are $\{f'(v),\; v\in C_0(\mathfrak{g}_X)\}$ and $\{f'^*(v),\; v\in C_0(T(X))\}$. \emph{We will denote by $\Gamma$ any of these two fields of vector spaces and by $\Gamma_x$ the fiber of $\Gamma$ at $x\in X$.} 

We define a $G$-invariant map $\{\varphi_x=(f'_xf'^*_x)^{1/2},\;x\in X:T^*(X)\to T^*(X)\}$. (\emph{Note the change in notation from \cite{Ka16}, section 6, where the same notation $\varphi_x$ was used for the map $f'_xf'^*_x$.}) We also define a $G$-invariant quadratic form $q$ on covectors $\xi\in T_x^*(X)$: $q_x(\xi)=(f'_xf'^*_x(\xi),\xi)=||f'^*_x(\xi)||_x^2=||\varphi_x(\xi)||^2$. Our previous assumption $||f'_x||\le 1$ implies  that $q_x(\xi)\le ||\xi||^2$ for any covector $\xi$. Being restricted to any orbit, the form $q_x(\xi)$ can serve as a Riemannian metric on this orbit. This is compatible with the natural topology on each orbit and on the orbit space globally. 

Note that a covector $\xi\in T_x^*(X)$ is orthogonal to the orbit passing through $x$ if and only if $q_x(\xi)=0$. The subspace $\{(x,\xi), q_x(\xi)=0\}\subset TX$ is usually denoted $T_GX$. 

The Lie algebra of the stability subgroup $M_x$ will be denoted $\mathfrak{m}_x$, and its orthogonal complement by $\mathfrak{m}_x^\perp\subset \mathfrak{g}_x^*$. Note that $\mathfrak{g}_x/\mathfrak{m}_x$ is isomorphic (via $f'_x$) to the tangent space $T_x(O_x)$, and $\mathfrak{m}_x^\perp=(\mathfrak{g}_x/\mathfrak{m}_x)^*$ is isomorphic (via $f'^*_x$) to the cotangent space $T^*_x(O_x)$.

If there is an orbit $O_y$ inside the tubular neighborhood $U$ of the orbit $O_x$ (as in subsection 4.2), and $y\in Z_x$, then the projection $\pi:U\to O_x$ defines the map $O_y\to O_x$, such that $\pi_*:T(O_y)\to T(O_x)$ is surjective, and the dual map $\pi^*:T^*(O_x)\to T^*(O_y)$ is injective. 
Obviously, when these tangent and cotangent spaces are identified with subspaces or quotient spaces of $\mathfrak{g}$, these maps become tautological maps (like $T(G/M_y)\to T(G/M_x)$ when $M_y\subset M_x$).  

\medskip

Finally, concerning the orbit structure, we refer to \cite{Br}, ch. 4, theorems 3.1 and 3.2. Although \cite{Br} deals only with compact group actions, these results are valid with the same proof for proper smooth actions. \emph{We assume in this discussion that $X/G$ is connected.} (In general, one can consider each connected component of $X/G$ separately.)

Orbits of maximal dimension are called \emph{principal orbits}. All principal orbits have the same orbit type (call it $G/M$). The main result is that the stability subgroup $M_x$ of any point $x\in X$ contains a subgroup conjugate to $M$. Therefore, if $O$ is a principal orbit and $O'$ any other orbit, there exists an equivariant map $O\to O'$. The union $X_{\rm max}$ of all principal orbits is open and dense in $X$. Furthermore, if $U=G\times_M Z$ is a tubular neighborhood of a principal orbit $O\simeq G/M$, then $M$ acts trivially on $Z$, therefore $U=G/M\times Z$. For any other orbit $O_x$ and its tubular neighborhood $U=G\times_{M_x} Z_x$, the principal orbits of the $M_x$-action on $Z_x$ have the orbit type $M_x/M$, and the space of principal orbits of $U$ is $U_{\rm max}=G\times_{M_x} (Z_x)_{\rm max}$.


\subsection{Symbol algebras}

For any $x\in X$, both vector spaces $\mathfrak{g}_x$ and $\tau_x=T_x(X)$ can be considered as abelian groups, so there are the corresponding convolutional {$C^*$}-algebras: $C^*(\mathfrak{g}_x)$ and $C^*(\tau_x)$. They are isomorphic via the Fourier transform to $C_0(\mathfrak{g}_x^*)$ and $C_0(\tau_x^*)$ respectively. The map $(f'^*_x)^*:C_b(\mathfrak{g}_x^*)\to C_b(\tau_x^*)$ corresponds to the natural map $f'_x:{\cal M}(C^*(\mathfrak{g}_x))\to {\cal M}(C^*(\tau_x))$ via the Fourier transform.

Let us denote the algebras of continuous sections (vanishing at infinity in $X$) of the fields $\{C^*(\mathfrak{g}_x),\, x\in X\}$ and $\{C^*(\tau_x),\, x\in X\}$ by $C^*_\mathfrak{g}(X)$ and $C^*_\tau(X)$ respectively. We will denote by $\mathfrak{G} X$ the total space of the fiber bundle $\mathfrak{g}_X\simeq X\times \mathfrak{g}$ and by $q$ its projection to $X$. Then $\{C_0(\mathfrak{g}_x^*),\, x\in X\}$ are exactly the fibers of $C_0(\mathfrak{G} X)$ over $X$. Similarly, the fibers of the algebra $C_0(TX)$ over $X$ form a continuous field of algebras $\{C_0(\tau^*_x),\, x\in X\}$. Therefore $C_0(\mathfrak{G} X)\simeq C^*_\mathfrak{g}(X)$ and $C_0(TX)\simeq C^*_\tau(X)$ via the Fourier transform. 

The important conclusion is that there are natural maps $(f'^*)^*:C_b(\mathfrak{G} X)\to C_b(TX)$ and $f'_x:{\cal M}(C^*_\mathfrak{g}(X))\to {\cal M}(C^*_\tau(X))$ related by the Fourier transform. 

\medskip

Now we can define two symbol algebras for leaf-wise operators: the `Lie symbol algebra' and the `space symbol algebra'. These will be the algebras of operators of `negative order'.  

\begin{defi} The map $f'^*:TX\to \mathfrak{G} X$ induces the maps $C_b(\mathfrak{G} X)\to C_b(TX)$ and ${\cal M}(C^*_\mathfrak{g}(X))\to {\cal M}(C^*_\tau(X))$. We will call $C_0(\mathfrak{G} X)$ the `Lie symbol algebra' for the `negative order' leaf-wise operators, and we define the `space symbol algebra' for the `negative order' leaf-wise operators, $\mathfrak{S}_{lf}(X)$, as the image of $C_0(\mathfrak{G} X)$ in $C_b(TX)$. This is a $C_0(X)$-algebra. The convolutional space symbol algebra (Fourier dual and isomorphic to $\mathfrak{S}_{lf}(X)$) is defined as the image of $C^*_\mathfrak{g}(X)$ in  ${\cal M}(C^*_\tau(X))$. It will be denoted $C^*_\Gamma(X)$. 

For an admissible Hilbert module $E$ over $C_0(X)$, we consider ${\cal K}(E\otimes_{C_0(X)} C_0(\mathfrak{G} X))$ and $\mathfrak{S}_{lf}(E)={\cal K}(E\otimes_{C_0(X)} \mathfrak{S}_{lf}(X))$ as the corresponding symbol algebras associated with $E$. Similarly, we can use an admissible algebra $B$ in place of ${\cal K}(E)$, by considering $B\otimes_{C_0(X)} C_0(\mathfrak{G} X)$ and $\mathfrak{S}_{lf}(B)=B\otimes_{C_0(X)} \mathfrak{S}_{lf}(X)$ as the corresponding Lie and space symbol algebras.
\end{defi}

At this point, it will be useful to make some additional clarification concerning the algebra $\mathfrak{S}_{lf}(X)$. Note that the map $f'_x:\mathfrak{g}_x\to\tau_x$ factors as follows: $\mathfrak{g}_x\to\mathfrak{g}_x/\mathfrak{m}_x\simeq T_x(O_x)\subset\tau_x$. The dual map $f'^*_x$ factors as $\tau^*_x\to T^*_x(O_x)\simeq (\mathfrak{g}_x/\mathfrak{m}_x)^*\simeq \mathfrak{m}_x^\perp \subset \mathfrak{g}_x^*$. 

\begin{defi} We will consider $\{C_0(\mathfrak{m}_x^\perp),\,x\in X\}$ as a continuous field of algebras. The continuous sections of this field are, by definition, the restrictions of continuous functions belonging to $C_0(\mathfrak{G} X)$.
We introduce the algebra $C_0((\mathfrak{G}/\mathfrak{M}) X)$ as the algebra of continuous sections (vanishing at infinity of $X$) of the continuous field $\{C_0(\mathfrak{m}_x^\perp),\,x\in X\}$. 
\end{defi}

In definition 5.1, we can replace the algebra $C_0(\mathfrak{G} X)$ with $C_0((\mathfrak{G}/\mathfrak{M}) X)$. We will see in subsection 5.4, that the construction of the operator out of the symbol depends only on the symbol defined as an element of $C_0((\mathfrak{G}/\mathfrak{M}) X)$. 

Because the map $C_0(T^*_x(O_x))\to C_b(\tau^*_x)$ induced by the restriction $\tau^*_x\to T^*_x(O_x)$ is injective, we get the following simple fact:

\begin{prop} The algebra $\mathfrak{S}_{lf}(X)$ is isomorphic to $C_0((\mathfrak{G}/\mathfrak{M}) X)$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10
\end{prop}

\begin{ass} Before discussing actual leaf-wise symbols and operators, here are the two options in which these objects will be defined:

$1^{\rm o}$ The $G$-equivariant option: symbols and operators are $G$-invariant. 

$2^{\rm o}$ The stabilizer-invariant option: In the notation of subsection 4.3, for any $x\in X$, the symbol restricted to the slice $Z_x$ is invariant with respect to the action of the stability subgroup $M_x$.
\end{ass}

Let us fix an admissible $G-C_0(X)$-Hilbert module $E$ (definition 4.13). We will use the product structure of the bundle $\mathfrak{G} X=X\times \mathfrak{g}$.

\begin{defi} We will say that a function $\mathfrak{b}(x,\eta)\in {\cal L}(E\otimes_{C_0(X)} C_0(\mathfrak{G} X))$ is orbit-continuous if it satisfies two conditions:

$1^{\rm o}$ $\mathfrak{b}$ is $G$-continuous, i.e. norm-continuous with respect to the group action;

$2^{\rm o}$ for any orbit $O\subset X$ and $\epsilon>0$, there is a tubular neighborhood $U$ of the orbit $O$ such that for any $x\in O$, $||\mathfrak{b}(y,\eta)-\pi^*(\mathfrak{b}(x,\eta))||< \epsilon$ for all $y\in Z_x$ and $\eta\in \mathfrak{m}_x^\perp$, where $\pi^*$  is the lifting from $O_x$ to $U$ composed with the restriction to $O_y$.
 \end{defi}

Since the map $C_0(\mathfrak{G} X)\to \mathfrak{S}_{lf}(X)$ is surjective, it extends to the map of the multiplier algebras ${\cal M}(C_0(\mathfrak{G} X))\to {\cal M}(\mathfrak{S}_{lf}(X))$. 

\begin{defi} An element $\sigma(x,\xi)\in {\cal L}(\mathfrak{S}_{lf}(E))$ (where $\xi\in \tau_x$) will be called a `space leaf-wise symbol' of order $0$ if it is the image of an element (called `Lie leaf-wise symbol') $\mathfrak{b}(x,\eta)\in {\cal L}(E\otimes_{C_0(X)} C_0(\mathfrak{G} X))$ (where $\eta\in \mathfrak{g}_x$) satisfying the following assumptions:

$1^{\rm o}$ $\mathfrak{b}(x,\eta)$ is orbit-continuous. 

$2^{\rm o}$ $\mathfrak{b}(x,\eta)$ is differentiable in $\eta$, and its exterior derivative $d_\eta(\mathfrak{b})$ vanishes at infinity in $\eta$ uniformly on compact subsets of $X$.

A space symbol $\sigma$ is of `negative order' if it becomes an element of ${\cal K}(\mathfrak{S}_{lf}(E))$ after being multiplied by any element of $C_0(X)$.
\end{defi}

\begin{rem} {\rm The space symbol is the actual symbol of the operator on $X$. The Lie symbol is the symbol of the operator lifted to $G/M_x\simeq O_x$, i.e. with the Riemannian metric of $G/M_x$ on the orbit. This latter symbol will be used for the construction of the operator out of symbol.}
\end{rem}

\begin{rem} {\rm $1^{\rm o}$. The strong estimate on $\mathfrak{b}(x,\eta)$ required in definition 3.7, $2^{\rm o}$, is not needed in the item $2^{\rm o}$ here because of an almost canonical choice of coordinate systems in the tangent spaces along the orbit (and the stabilizer-invariance of the symbols).

$2^{\rm o}$. Definition 5.6 was given with the assumption that all our symbols are continuous functions. Recall from definition 3.7 that one can also use symbols which are not everywhere continuous, e.g. `classical' symbols. In order to take this possibility into account, definition 5.6 should be modified. This refers both to $\sigma(x,\xi)$ and $\mathfrak{b}(x,\eta)$. The terminology that has to be used is given below (cf. also definition 3.7). The words `outside of a compact set in $\eta$' should be introduced in both conditions $1^{\rm o}$ and $2^{\rm o}$.}
\end{rem}

\begin{term}We will say that a function $a(x,\eta)$ satisfies a certain condition outside of a compact set in $\eta$ if for any compact set $L'$ in the variable $x$ there is a compact set $L''$ in the variable $\eta$ such that the said condition holds for all $(x,\eta)$ with $x\in L',\eta\notin L''$.
\end{term}


\subsection{Clifford algebras and the leaf-wise Dirac operator}

Before we give a general construction of leaf-wise operators (in subsection 5.4), we want to discuss an important leaf-wise Dirac operator. First recall some notation from \cite{Ka16}.

\medskip

Let ${\cal V}$ be a real vector bundle over $X$ equipped with a $G$-invariant Riemannian metric and ${\it Cliff}({\cal V},Q)$ the Clifford algebra bundle associated with the quadratic form $Q(v)=||v||^2$ on ${\cal V}$. We denote by $Cl_{\cal V}(X)$ the complexification of the algebra of continuous sections of ${\it Cliff}({\cal V},Q)$ over $X$, vanishing at infinity of $X$. With the sup-norm on sections, this is a {$C^*$}-algebra. It will be denoted $Cl_{\cal V}(X)$. When ${\cal V}=\tau=T^*(X)$, this algebra will be denoted $Cl_\tau(X)$, and when ${\cal V}=\mathfrak{g}_X$, this algebra will be denoted $Cl_\mathfrak{g}(X)$. Recall that according to \cite{Ka16}, theorem 2.7, the algebras $Cl_\tau(X)$ and $C_0(TX)$ are $KK$-equivalent.

Let $f':\mathfrak{g}_X\to \tau=T(X)$ be the tangent map defined in subsection 5.1 and $f'^*:\tau^*\to \mathfrak{g}_X^*$ the dual map. The {$C^*$}-subalgebra of $Cl_\tau(X)$ generated by scalar functions $C_0(X)\subset Cl_\tau(X)$ together with the subspace $\{f'(v)\}$ for all continuous sections $v$ of $\mathfrak{g}_X$ vanishing at infinity will be denoted  $Cl_\Gamma(X)$. A similar {$C^*$}-subalgebra of $Cl_\mathfrak{g}(X)$ generated by scalar functions $C_0(X)\subset Cl_\mathfrak{g}(X)$ together with the subspace $\{f'^*(w)\}$ for all covector fields $w\in \tau^*$ vanishing at infinity is isomorphic to $Cl_\Gamma(X)$ because the spaces of continuous sections $\{f'(v),\; v\in C_0(\mathfrak{g}_X)\}$ and $\{f'^*(w),\; w\in C_0(\tau^*)\}$ are isometrically isomorphic (see subsection 5.1). (We will not distinguish between these two {$C^*$}-algebras).

This definition can be simplified. Let us use the Riemannian metric on each orbit induced by the quadratic form $q_x(\xi)$ (see subsection 5.1). We get a continuous field of {$C^*$}-algebras $\{Cl_\tau(O_x),\, x\in X\}$ paramerized by $X$. Of course, $Cl_\tau(O_x)=Cl_\tau(O_y)$ if $x$ and $y$ belong to the same orbit, so actually we get the field of {$C^*$}-algebras parametrized by $X/G$. The algebra of continuous sections of this field (vanishing at infinity in $X/G$) is isomorphic to $Cl_{\Gamma}(X)$.

\medskip

Let us start with a simple example, very similar to example 4.19. 

\begin{ex} {\rm We consider again the case of the group $G=S^1$ acting on the disk $X=D^2$ by rotations, but we will use the algebra $Cl_{\Gamma}(X)$ instead of the Hilbert module $E=C(X)$.

We will give two versions of the Dirac operator. Both are based on the operator ${\cal D}$ of example 4.19: ${\cal D}=-i\,\partial/\partial\theta$, where $\theta$ is the angle. We put $F={\cal D}({\cal D}^2+1)^{-1/2}$. This defines a multiplier of $C^*(S^1,Cl_{\Gamma}(D^2))$, an operator of degree $0$. But we need an operator of degree $1$. 

Note that $Cl_{\Gamma}(S^1)\simeq C(S^1)\otimes Cl_1$. The generator of the Clifford algebra $Cl_1$ (which we will call here $\epsilon$) satisfies: $\epsilon^2=1$. We put ${\cal F}=c(\epsilon)F$, where $c(\epsilon)$ means the Clifford multiplication by $\epsilon$. This gives the multiplier of degree $1$ of $C^*(S^1,Cl_{\Gamma}(D^2))$. It has the Lie symbol $c(\eta)(1+\eta^2)^{-1/2}$. The space symbol of this operator is $rc(\xi)(1+r^2\xi^2)^{-1/2}$, where $r$ is the radius of the circle. The operator $1-{\cal F}^2$ is the convolution operator and an element of $C^*(S^1,Cl_{\Gamma}(D^2))$ (see 4.19).

The second version of the Dirac operator is an element of ${\cal L}({L^2_{lf}}(Cl_{\Gamma}(D^2)))$. We put $\tilde{\cal D}=c(\epsilon){\cal D}$ for $r>0$ and $\tilde{\cal D}=0$ for $r=0$. This operator has grading degree $1$. In the basis $\{e^{in\theta}\}$, we have: $\tilde{\cal D}(e^{in\theta})=c(\epsilon)ne^{in\theta}$. 

Since ${\cal D}$ is essentially self-adjoint, we can consider the bounded operator $F=({\cal D}^2)^{-1/2}{\cal D}$. (The operator $({\cal D}^2)^{-1/2}$ is well defined on the range of ${\cal D}$.) The operator $F$ has the Lie symbol $c(\eta)/|\eta|$ and the space symbol $c(\xi)/|\xi|$. In the basis $\{e^{in\theta}\}$, we have: $F(e^{in\theta})=c(\epsilon){\rm sgn}(n)e^{in\theta}$. This gives a bounded, selfadjoint, $G$-invariant element of the algebra ${\cal L}({L^2_{lf}}(Cl_{\Gamma}(D^2)))$. The operator $1-F^2$ is the projection onto the subspace of constant functions in $L^2(S^1_r)$ for all $r$, therefore it belongs to ${\cal K}({L^2_{lf}}(Cl_{\Gamma}(D^2)))$.}
\end{ex}

Now let us consider the general case. We will construct the Dirac operator on $C^*(G,Cl_{\Gamma}(X))$. During the construction, we will use the implementation of $Cl_{\Gamma}(X)$ as a subalgebra of $Cl_\mathfrak{g}(X)$. We start with the Dirac differential operator as an unbounded multiplier of ${\cal K}({L^2_{lf}}(Cl_{\Gamma}(X)))$. 

\medskip

\noindent{\bf Construction.} 

For each orbit $O_x\subset X$, there is a canonical differential Dirac operator on smooth, compactly supported elements of $Cl_{\Gamma}(O_x)=Cl_\tau(O_x)$. We define the norm on $Cl_\tau(O_x)$ using the Riemannian metric on $O_x$ given by the form $q_x(\xi)$. We recall that all spaces $L^2(O_x)$ are considered isometrically isomorphic to the corresponding spaces $L^2(G/M_x)$, so they do not depend on any choice of a Riemannian metric. But the metric of $Cl_\tau(O_x)$ plays a role in the definition of the Dirac operator. 

For any $v\in \mathfrak{g}$, let $g_v(t)$ be the one-parameter subgroup of $G$ corresponding to the vector $v$. Denote by $\partial/\partial v$ the derivation operator on $Cl_{\mathfrak{g},c}^\infty(X)$: $\partial/\partial v(b)=d/dt(g_v(t)(b))|_{t=0}$ at any point $x\in X$. We have: $Cl_{\Gamma}(G/M_x))\simeq (C_0(G)\otimes Cl_{\mathfrak{m}^\perp_x})^{M_x}$. Let us choose any basis $\{\tilde v_k\}$ in the space $\mathfrak{m}^\perp_x$ and the dual basis $\{v_k\}$ in the space $\mathfrak{g}_x/\mathfrak{m}_x$. Define the differential Dirac operator by the formula: ${\cal D}_{O_x}=(-i)\sum_k c(\tilde v_k)\partial/\partial v_k$, where $c$ means Clifford multiplication in $Cl_{\mathfrak{m}^\perp_x}$. This definition does not depend on the choice of the basis and gives an unbounded, essentially self-adjoint $G$-invariant operator on $(C_c^\infty(G)\otimes Cl_{\mathfrak{m}^\perp_x})^{M_x}$. Of course, this operator depends on the orbit and not on the point $x$ of this orbit. We will denote it ${\cal D}_O$ (for the orbit $O$). Note that in the interpretation of our Hilbert space as $(L^2(G)\otimes Cl_{\mathfrak{m}^\perp_x})^{M_x}$, the derivatives of the operator ${\cal D}_O$ are \emph {right} translation derivatives. (This is because doing the left translation $g\in G$ on the orbit $O_x$ we automatically change the identification $O_x\simeq G/M_x$ into $O_{g(x)}\simeq g\cdot G/M_x \cdot g^{-1}$. Since we assume the $G$-action on $G/M_x$ to be the left-translation action, we have to compose this $Ad(g)$-action on $G/M_x$ which we actually get from orbit action with the right translation by $g$.)

\smallskip

Let us calculate the space symbol of the operator ${\cal D}_{O_x}$. At the point $(x,\xi)\in O_x$ the symbol is $$i(-i)\sum_k c(\tilde v_k)(f'_x(v_k),\xi)=\sum_k c(\tilde v_k)(v_k,f'^*_x(\xi))=c(f'^*_x(\xi).$$

Therefore, in the implementation of $Cl_{\Gamma}(X)$ as a subalgebra of $Cl_\mathfrak{g}(X)$, the symbol of ${\cal D}_{O_x}$ is $c(f'^*_x(\xi))$. But if we want it in the implementation of $Cl_{\Gamma}(X)$ as a subalgebra of $Cl_\tau(X)$, we have to use the isomorphism of the subsection 5.1 between the implementations of the field $\Gamma$ on $\mathfrak{G} X$ and on $TX$ provided by the isometry $(f'f'^*)^{-1/2}f':{\rm Im}\, f'^*\to {\rm Im}\, f'$. Applying this isometry, we get $\sigma_{{\cal D}_{O_x}}(x,\xi)=c((f'_xf'^*_x)^{-1/2}(f'_xf'^*_x)(\xi))=c(\varphi_x(\xi))$.

\medskip

Let us calculate now the square of ${\cal D}_{O_x}$. For brevity, we will denote $\partial/\partial v_k$ by $\partial_k$. The operator ${\cal D}_{O_x}^2$ can be rewritten in the form:
$$-(1/2)\sum_{k,l}(c(\tilde v_k \tilde v_l)+c(\tilde v_l \tilde v_k))\partial_k\partial_l
+(1/2)\sum_{k,l}c(\tilde v_l \tilde v_k)[\partial_k,\partial_l].$$
The sum $c(\tilde v_k \tilde v_l)+c(\tilde v_l \tilde v_k)$ is equal to the scalar product $2(\tilde v_k,\tilde v_l)$, and the first sum $-\sum_{k,l}(\tilde v_k,\tilde v_l)\partial_k\partial_l$ gives the orbital Laplace operator $\Delta_G$ (\cite{Ka16}, section 6) on the orbit $O_x$. (In the case of a compact orbit $O_x$ (and compact $G$), this operator coincides with the Casimir operator.) The second sum $(1/2)\sum_{k,l}$ is a ($G$-invariant) operator of order $1$. Let us denote it $\mathfrak{R}_{O_x}$. The principal symbols of ${\cal D}_{O_x}^2$ and $\Delta_G$ (restricted to the orbit $O_x$) are both equal $q_x(\xi)$.

\medskip

Finally, we have to normalize the differential operator ${\cal D}$ to order $0$. Consider the operator ${\cal D}_G$ on the group $G$ itself, i.e. the Dirac operator defined on $L^2(G)\otimes Cl_\mathfrak{g}$ (by the formula given above) with the symbol $c(\xi)$ at any point. Also consider on $L^2(G)$ a properly supported $G$-invariant pseudo-differential operator $P_G$ with the symbol $(1+||\xi||^2)^{-1/2}$. The product $F_G=(P_G\otimes 1){\cal D}_G$ on $L^2(G)\otimes Cl_\mathfrak{g}$ is a properly supported, bounded, $G$-invariant operator with the symbol $c(\xi)(1+||\xi||^2)^{-1/2}$ (according to the H\" ormander calculus). Applying proposition 5.5 \cite{Ka16} we see that the operator with such symbol defines an element of ${\cal M}(C^*(G)\otimes Cl_\mathfrak{g})$ (cf. also proposition 4.10 above and the discussion before proposition 5.5 \cite{Ka16}). The operator $1-F_G^2$ has the principal symbol $(1+||\xi||^2)^{-1}$. Both $P_G$ and $1-F_G^2$ give elements of $C^*(G)\otimes Cl_\mathfrak{g}$.

Now, for each orbit $O_x\simeq G/M_x$, we can use the map $f_x:G\to O_x$ to `push forward' the operator $F_G$ to $O_x$. We will do it for ${\cal D}_G$ and $P_G$ separately. First, we can apply the operator $f_x({\cal D}_G)$ to any element of $(L^2(G)\otimes Cl_{\mathfrak{m}^\perp_x})^{M_x}$ using the same definition for derivatives as in the initial definition of ${\cal D}_{O_x}$ above and noting that all derivatives for $v\in \mathfrak{m}_x$ vanish at the point $x$ (and the same for any $v\in \mathfrak{m}_y$ at any other point $y\in O_x$). Therefore ${\cal D}_G$ pushed to $O_x$ gives ${\cal D}_{O_x}$. 

Secondly, the algebra $C^*(G)$ naturally acts on $L^2(O_x)$ by left convolution. Therefore, $P_G$ corresponds to an operator $P_{O_x}$ on $L^2(O_x)$. The composition $(P_{O_x}\otimes 1){\cal D}_{O_x}$ gives the operator which we will denote $F_{O_x}$. The principal symbol of the operator $P_{O_x}$ (calculated using the change of variables formula in the H\" ormander calculus) is equal $(1+||f'^*_x(\xi)||^2)^{-1/2}=(1+q_x(\xi))^{-1/2}$. This implies that the principal symbol of $F_{O_x}$ is $\sigma_{\cal D}(x,\xi)=c(\varphi_x(\xi))(1+q_x(\xi))^{-1/2}$. The operator $1-F_{O_x}^2$ has the principal symbol $(1+q_x(\xi))^{-1}$ and acts on $L^2(Cl_{\Gamma}(O_x))$ as an element of $C^*(G)\otimes Cl_{\mathfrak{m}^\perp_x}$.

\medskip

To check that this family of operators defines a multiplier of $C^*(G,Cl_{\Gamma}(X))$, we need to verify the orbit-continuity condition 4.18 in the strict topology. Let $O_x\subset X$, and $U=G\times_{M_x}Z_x$ be its tubular neighborhood, where $Z_x\simeq D^n$. If $O_y\subset U$, we can assume that $y\in Z_x$, so that $M_y\subset M_x$. When $O_y$ tends to $O_x$, all generators of $\mathfrak{m}_y^\perp\cap\mathfrak{m}_x$ vanish in the limit, and the Dirac operator corresponding to the orbit $O_y$ transforms into the Dirac operator corresponding to $O_x$. (When we say that the generators of $\mathfrak{m}_y^\perp\cap\mathfrak{m}_x$ vanish in the limit, we mean the metric of $Cl_{\Gamma}(X)$.) 

\smallskip

It is also clear from the above construction that the family $1-F_{O_x}^2$ is orbit-continuous in norm. In fact, this family is obtained by `pushing forward' just one operator $1-F_G^2$ which can be considered as an element of $C^*(G)\otimes Cl_\mathfrak{g}$. 

Note that we have actually constructed the family $\{F_O\}$ which is an element of ${\cal M}({\cal K}({L^2_{lf}}(Cl_{\Gamma}(X)))^G)$, the multiplier algebra of $G$-invariant integral leaf-wise operators. But the family $1-F_O^2$ does not define an element of ${\cal K}({L^2_{lf}}(Cl_{\Gamma}(X)))^G$ because there is no orbit-continuity in norm.

\smallskip

\begin{defi} The family $\{F_O\}$ will be called the leaf-wise Dirac operator and denoted $\mathfrak{D}_\Gamma$.
\end{defi}

We have proved the following 

\begin{prop} There exists a leaf-wise Dirac operator $\mathfrak{D}_\Gamma$. It is represented by an element of ${\cal M}({\cal K}({L^2_{lf}}(Cl_{\Gamma}(X)))^G)$. This Dirac operator defines an element of ${\cal M}(G^*(G,Cl_{\Gamma}(X)))$. The Lie symbol of 
$\mathfrak{D}_\Gamma$ is $c(\eta)(1+||\eta||^2)^{-1/2}$ and the space symbol is $c(\varphi_x(\xi))(1+q_x(\xi))^{-1/2}$, where $c$ means Clifford multiplication. Moreover, $C_0(X)\cdot (1-\mathfrak{D}_\Gamma^2)\subset C^*(G,Cl_{\Gamma}(X))$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10
\end{prop}

\begin{rem} {\rm As a variation of this construction, one can use a `classical' symbol and define $F_O$ as $({\cal D}_O^2)^{-1/2}D_O$ (cf. example 5.10). This gives a bounded multiplier of ${\cal K}({L^2_{lf}}(Cl_{\Gamma}(X)))$. However, if we want the family $\{1-F_O^2\}$ to be orbit-continuous \emph{in norm} on ${\cal K}({L^2_{lf}}(Cl_{\Gamma}(X)))$, then this essentially requires that $\ker ({\cal D}_O)$ should coincide with $\ker (\Delta_G)$ on any orbit $O$. In particular, in the case of a compact group $G$, ${\cal D}^2$ must be strictly positive on any irreducible representation of $G$ except the trivial one. It may be technically tricky to prove this. Moreover, it is very unlikely that a similar strong  condition can be true in the case of general leaf-wise operators.}
\end{rem}

\begin{rem} {\rm $1^{\rm o}$. The same construction gives the Dirac element $\mathfrak{D}_\Gamma\in {\cal M}(C^*(G,{\cal K}(\Lambda^*(\tau))))$. On the $K$-theory level, it just comes by the embedding $Cl_{\Gamma}(X)\subset {\cal K}(\Lambda^*(\tau)$.

$2^{\rm o}$. As soon as the operator $\mathfrak{D}_\Gamma$ is defined on the dense subspace of $X$ consisting of all \emph{principal} orbits, it is automatically defined everywhere on $X$ by orbit-continuity in the strict topology.}
\end{rem}

\begin{lem} Let $F$ be a $G$-invariant, bounded PDO of order $0$ on a (possibly ${\bf Z}_2$-graded) vector bundle $E$ over $X$. Then the two operators $F\hat\otimes 1$ and $1\hat\otimes \mathfrak{D}_\Gamma$ on $L^2(E\hat\otimes \Lambda^*(\tau))$ commute (in the graded sense).
\end{lem}

\noindent {\em Proof.}  Recall that  by construction $\mathfrak{D}_\Gamma=P\cdot {\cal D}$, where the differential Dirac operator is ${\cal D}=(-i)\sum_k c(\tilde v_k)\partial/\partial v_k$. The partial derivatives $\partial/\partial v_k$ over one-parameter subgroups $g_k$ are limits of $(g_k(t)-1)/t$ when the parameter $t$ of the subgroup $g_k$ goes to $0$. Since all $g_k$ commute with $F\hat\otimes 1$, we see that $1\hat\otimes {\cal D}$ commutes $F\hat\otimes 1$. The operator $P$ is the orbit-wise left convolution with an element of $C^*(G)$, therefore it also commutes with $F\hat\otimes 1$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10


\subsection{General construction of leaf-wise operators}

Given a space leaf-wise symbol $\sigma$, we choose the corresponding Lie leaf-wise symbol $\mathfrak{b}(x,\eta)\in {\cal L}(E\otimes_{C_0(X)} C_0(\mathfrak{G} X))$. The construction of the corresponding operator in $C^*(G,{\cal K}(E))$ will be done orbit-wise. We will consider first the case of $G$-invariant symbols and operators (see assumptions 5.4, $1^{\rm o}$).

For any $x\in X$, there is the map $f_x:G/M_x\simeq O_x$. The Hilbert module $E|_{O_x}$ corresponds to the vector bundle $\tilde E_{G/M_x}$ on $G/M_x$. We restrict the symbol $\mathfrak{b}$ to the subbundle of covectors $\{\mathfrak{m}_y^\perp,\, y\in G/M_x\}$ and get the data $(\tilde E_{O_x}, \tilde \mathfrak{b})$ on the manifold $G/M_x$ sufficient for performing the construction of the operator on $G/M_x$ using the method of operator integration (see section 3).

More precisely, we apply the operator integration to the Fourier transform (in $\eta$) of the symbol $\tilde \mathfrak{b}$ and get an operator $\tilde F_x\in L^2(\tilde E_{G/M_x})$ on the manifold $G/M_x$. Because the original symbol was assumed $G$-invariant, we can average $\tilde F_x$ over $G$. According to proposition 4.10 and \cite{Ka16}, 5.5, we get an element of ${\cal M}(C^*(G)\otimes {\cal K}(E_x))$ satisfying certain right $M_x$-invariance conditions. 

Note that one can also perform this construction using the same procedure on $G$ (for every $x\in X$) instead of $G/M_x$. If the symbol is lifted to $G$ from $G/M_x$, then we will get the same element of ${\cal M}(C^*(G)\otimes {\cal K}(E_x))$ satisfying the same right $M_x$-invariance conditions.

Now we can return to the orbit $O_x$ using the same map $f_x:G/M_x\simeq O_x$: The algebra $C^*(G)\otimes {\cal K}(E_x)$ acts on $L^2(E|_{O_x})$ by left convolution. (Here the right $M_x$-invariance conditions are used.) We get an operator $F_x\in {\cal M}({\cal K}({L^2_{lf}}(E|_{O_x}))^G)$. Its symbol corresponds to the space symbol $\sigma$ on the orbit $O_x$ because $\mathfrak{b}$ and $\sigma$ are exactly related by the map $f'^*_x$. (See theorem 3.12 about the correspondence between the symbol and the operator.)

Actually, we have constructed too many operators: we need only one operator per orbit. It is clear however that the operators $F_x$ and $F_y$ corresponding to the points $x$ and $y$ of the same orbit have the same principal symbol and differ at most by an element of ${\cal K}({L^2_{lf}}(E|_{O_x}))^G$. 

Before we select one operator per orbit, we need to prove orbit-continuity of the constructed family in the strict topology in the sense of definition 4.17. Let $U$ be a small tubular neighborhood of the orbit $O_x$, and $y\in Z_x$. Let us take the Fourier transforms of the symbol $\mathfrak{b}$ in the direction of $\eta$ tangent to the orbit $O_x$ for the points $x$ and $y$. Definition 5.5, $2^{\rm o}$ says that these Fourier transforms are $\epsilon$-close to each other. On the other hand, when we take $\eta$ in the direction tangent to the slice $Z_x$, we know that $L^2(E|_{O_y\cap Z_x})$ vanishes when $y\to x$. So we need only to mention that $\mathfrak{b}(y,\eta)$ remains bounded as $y\to x$, and this is true on compact subsets in $x\in X$ because $\mathfrak{b}$ is globally bounded by definition. Therefore the full Fourier transform of $\mathfrak{b}(y,\eta)$ converges to that of $\mathfrak{b}(x,\eta)$ in the strict topology.

Let us cover $X$ with tubular neighborhoods of orbits and choose a subcovering $\{U\}$ which is locally finite as a covering of $X/G$. Take the corresponding partition of unity in $X/G$: $\sum_U \alpha_U=1$. For each tubular neighborhood $U$ of our covering choose the slice $Z_U$. This will associate to $U$ the operator $F_U$ corresponding to the family of operators $\{F_y\}$ for $y\in  Z_U$. Define the leaf-wise operator ${\cal F}$ as the sum $\sum_U \alpha_U F_U$. 

It is clear that ${\cal F}\in {\cal M}({\cal K}({L^2_{lf}}(E))^G)$, therefore, by proposition 4.11, it is also an element of ${\cal M}(C^*(G,{\cal K}(E)))$. This is the required leaf-wise operator.  

\begin{rem} {\rm The construction of the Dirac operator (of subsection 5.3) can also be performed by the method described here.} 
\end{rem}

In the stabilizer-invariant case $2^{\rm o}$ of 5.4, the construction is the following: We start again with the Lie symbol $\mathfrak{b}$. Fix an orbit $O$ in $X$. For each point $x\in X$, consider an auxiliary symbol $\tilde\mathfrak{b}_x(y,\eta)$ on the orbit $O$ which is $G$-invariant and coincides with $\mathfrak{b}$ at the point $x$, i.e. $\tilde\mathfrak{b}_x(x,\eta)=\mathfrak{b}(x,\eta)$. Such symbol $\tilde\mathfrak{b}_x$ is defined uniquely. Using the above $G$-equivariant construction we get for each symbol $\tilde\mathfrak{b}_x$ the multiplier of the convolution algebra. This operator acts on $L^2(E|{O_x})$, which gives an element $F_x\in{\cal M}({\cal K}(L^2(E|_O))^G)$, and consequently, an element of ${\cal M}(C^*(G,{\cal K}(E|_O)))$ by proposition 4.11. Note that the principal symbol of the operator $F_x$ is still $\sigma$ at the point $x$. 

Now we apply the operator integration to the family $\{F_x\}$ on each orbit $O$ and get the operator ${\cal F}_O$. (Compact operators here are defined as $C^*(G,{\cal K}(E|_O))$.) Again this family is orbit-continuous in the strict topology (by the reasoning given above), so we get an operator ${\cal F}\in{\cal M}(C^*(G,{\cal K}(E)))$ as a result. This concludes the construction in the stabilizer-invariant case.

\medskip

Finally, let us define `compact operators' for ${\cal M}(C^*(G,{\cal K}(E)))$ as elements of $C^*(G,{\cal K}(E))$. \emph{We reiterate definition 3.10 of `negative order' operators as those which become compact after being multiplied by a function from $C_0(X)$.} 

\begin{lem} If the space symbol is of `negative order', then the corresponding operator is of `negative order'.
\end{lem}

\noindent {\em Proof.}  The Lie leaf-wise symbol algebra $C_0(\mathfrak{G} X)$ has the function $(1+||\eta||^2)^{-1}$ as its pointwise (for each $x\in X$) strictly positive element. (The corresponding space leaf-wise symbol is $(1+q_x(\xi))^{-1}$.) In subsection 5.3 (proposition 5.12), we have already proved our assertion for the operator with this symbol. It remains to note that the operator integration method preserves the relation of order, modulo compact operators (see theorem 3.11). Therefore, if our assertion is true for the operator with the Lie symbol $(1+||\eta||^2)^{-1}$, it is true for the operator with any `negative order' symbol. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10


\section{Poincare duality}
 
 We will need a certain generalization of the Poincare duality \cite{Ka16}, theorem 4.6. The proof of this result is exactly the same as the proof of \cite{Ka16}, 4.6. A sketch of the proof will be given below.
 
 \smallskip
 
Let us start with the local Bott element. Let $U$ be a small tubular neighborhood of the diagonal $\Delta\subset X\times X$ (homeomorphic to $TX$) such that for any $(x,y)\in U$ there is a minimal geodesic connecting $x$ and $y$. The local Bott element $[{\cal B}_X]\in {\cal R} KK^G(X;C_0(X), C_0(U)\otimes_{C_0(X)} C_0(TX))$ (defined in \cite{Ka16}, 2.9) is a family of point-wise Bott elements $\{\beta_x\}, x\in X$. Each $\beta_x$ is the Bott element in $K_0(C_0(U_x)\otimes C_0(T_x(X)))$, where $U_x=U\cap (x\times X)$. Embedding $U$ into $X\times X$ we get $[{\cal B}_X]\in {\cal R} KK^G(X;C_0(X),C_0(X)\otimes C_0(TX))$.

If $A$ is a $C_0(X)$-algebra, we can tensor the local Bott element with $1_A$ over $C_0(X)$. We get an element $[{\cal B}_A]\in {\cal R} KK(X;A, A\otimes C_0(TX))$.  

\begin{theo} Let $X$ be a complete Riemannian manifold with an isometric proper action of a locally compact group $G$. For any separable $G-C_0(X)$-algebra $A$ and a separable $G$-algebra $B$,
$$KK^G_*(A,B)\simeq {\cal R}KK^G_*(X;A,C_0(TX)\otimes B)$$
$$\simeq {\cal R} KK^G_*(X;C_0(TX)\otimes_{C_0(X)}A,C_0(X)\otimes B).$$ 
\end{theo}

\noindent\emph{Sketch of proof.}  To deal with the first isomorphism, we define the homomorphisms:
$$\mu=[{\cal B}_A]\otimes_{A}:KK_*^G(A,B)\to {\cal R} KK_*^G(X;A,C_0(TX)\otimes B)$$
and $$\nu=\otimes_{C_0(TX)} [{\cal D}_X]: {\cal R} KK_*^G(X;A,C_0(TX)\otimes B)\to KK_*^G(A,B),$$
where $[{\cal D}_X]\in K^0_G(C_0(TX))$ is the Dolbeault element (\cite{Ka16}, 2.8). The associativity of the product and theorem 2.10 \cite {Ka16} easily imply that $\nu\cdot \mu=id$. In the opposite direction, the proof is exactly the same as in \cite{Ka16}, 4.6.

The second isomorphism comes from the Bott periodicity. The groups ${\cal R} KK^G_*(X;C_0(TX)\otimes_{C_0(X)}A,C_0(X)\otimes B)$ and ${\cal R} KK^G_*(X;A,C_0(TX)\otimes B)$ are isomorphic: take a ${\cal R} KK$-product (over $C_0(X)$) with $1_{C_0(TX)}$ and use the ${\cal R} KK$-equivalence (Bott periodicity) $C_0(TX)\otimes_{C_0(X)}C_0(TX)\simeq_{KK} C_0(X)$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\begin{rem} {\rm It is useful to have Poincare duality also for manifolds with boundary. We will indicate the necessary changes. Let $\bar X$ be a manifold with boundary $\partial X$ and interior $X$. To construct $[{\cal B}_{\bar X}]\in {\cal R} KK^G(\bar X;C_0(\bar X), C_0(\bar X)\otimes C_0(TX))$, we can extend $X$ to a complete Riemannian manifold $Y=\bar X\cup_{\partial X} (\partial X\times [0,\infty))$ and then restrict the element $[{\cal B}_Y]$ to $\bar X$ using the fact that $X$ and $Y$ are diffeomorphic. 

Now $A$ will be a $C_0(\bar X)$-algebra. We tensor $[{\cal B}_{\bar X}]$ with $1_A$ over $C_0(\bar X)$ and get an element $[{\cal B}_A]\in {\cal R} KK^G(\bar X;A, A\otimes C_0(TX))$. The Dolbeault element does not change.
The resulting Poincare duality isomorphism will be the following: $KK^G_*(A,B)\simeq {\cal R}KK^G_*(\bar X;A,C_0(TX)\otimes B)$.}
\end{rem}
 

\section{K-theory of symbol algebras}

Now we come to the symbol algebras. The symbol algebra $\mathfrak{S}_{lf}(X)$ for leaf-wise `negative order' operators was defined in 5.1 as the image of $C_0(\mathfrak{G} X)$ in $C_b(TX)$ under the map $f'^*:TX\to \mathfrak{G} X$. For any $x\in X$, the strictly positive element of $C_0(\mathfrak{G} X)$ in the fiber over $x$ is $(1+||\eta||^2)^{-1}$. Therefore the algebra $\mathfrak{S}_{lf}(X)$ is fiberwise generated (inside $C_b(TX)$) by the field of elements $(1+||f'^*_x(\xi)||^2)^{-1}=(1+q_x(\xi))^{-1}$, with an additional condition that elements of $\mathfrak{S}_{lf}(X)$ vanish at infininity in $x\in X$ uniformly in $\xi$. 

Recall now a similar algebra for transversal PDOs. It was defined in \cite{Ka16} (definitions 6.2, 6.3) and denoted $\mathfrak{S}_G(X)$ and $\mathfrak{S}_\Gamma(X)$. Here we change slightly this definition according to our more flexible approach to PDOs. We also prefer to change slightly the notation for this algebra. 

\begin{defi} The symbol algebra $\mathfrak{S}_{tr}(X)$ for `negative order' transversal PDOs is a subalgebra of the algebra of bounded functions $b\in C_b(TX)$ satisfying the conditions:

$1^{\rm o}$ $b(x,\xi)$ is continuous in $x$ uniformly in $\xi$ and vanishes (uniformly in $\xi$) when $x\to\infty$. 

$2^{\rm o}$ $b(x,\xi)$ is differentiable in $\xi$, and for the exterior derivative in $\xi$, there is an estimate: $||d_\xi(b)||\le C\cdot (1+||\xi||)^{-1}$ for any compact subset $K\subset X$, with the constant $C$ which depends only on $b$ and $K$ and not on $(x,\xi)$. 

$3^{\rm o}$ For any $\varepsilon>0$, there exists $c>0$ such that for any $x\in X,\;\xi\in T^*_x(X)$,
$|b(x,\xi)|\le c\cdot (1+q_x(\xi))(1+||\xi||^2)^{-1}+\varepsilon.$

Similarly defined, symbols of operators on a finite-dimensional vector bundle $E$ over $X$ form a Hilbert module $\mathfrak{S}_{tr}(E)$ over $\mathfrak{S}_{tr}(X)$.
\end{defi}

\begin{rem} {\rm It is clear from the condition $3^{\rm o}$ that the algebra $\mathfrak{S}_{tr}(X)$ is fiberwise (over $X$) generated (inside $C_b(TX)={\cal M}(C_0(TX))$) by the field of elements $(1+q_x(\xi))(1+||\xi||^2)^{-1}$. Because the products $(1+q_x(\xi))^{-1}\cdot (1+q_x(\xi))(1+||\xi||^2)^{-1}=(1+||\xi||^2)^{-1}$ form a field of strictly positive elements in the fibers of the algebra $C_0(TX)$ over $X$ we have: $\mathfrak{S}_{lf}(X)\cdot \mathfrak{S}_{tr}(X)= C_0(TX)$. In particular, there is a natural map of $C_0(X)$-algebras: $\mathfrak{S}_{lf}(X)\otimes_{C_0(X)} \mathfrak{S}_{tr}(X)\to C_0(TX)$.}
\end{rem}

\begin{defi-lem} Denote the Hilbert $C_0(X)$-module corresponding to the field of Hilbert spaces $\{L^2(\Lambda^*(\tau_x))\}$ over $X$ by ${\cal T}$. There is the Bott-Dirac operator on each space $L^2(\Lambda^*(\tau_x))$, namely, ${\cal C}_x=(1+D_x^2)^{-1/2}$, where $D_x=(d_\xi+d^*_\xi+{\rm ext\,}(\xi)+{\rm \,int\,}(\xi))$. Denote the corresponding operator on ${\cal T}$ by ${\cal C}$. The homomorphism $\psi:Cl_{\Gamma}(X)\otimes_{C_0(X)}\mathfrak{S}_{lf}(X)\to {\cal L}({\cal T})$ is just multiplication on the $\mathfrak{S}_{lf}(X)$ part, and on the Clifford part of $Cl_{\Gamma}(X)$ it is given on (real) covectors by $\xi\mapsto {\rm ext\,}(\xi)+{\rm \,int\,}(\xi))$. The triple $({\cal T},\psi, {\cal C})$ defines a Dirac type element of the group ${\cal R} KK^G(X;Cl_{\Gamma}(X)\otimes_{C_0(X)}\mathfrak{S}_{lf}(X),C_0(X))$ which will be denoted $[d_{lf}]$.
\end{defi-lem}

\noindent {\em Proof.}  We need only to prove the compactness of commutators between ${\cal C}$ and the image of $\psi$. Any element $a\in \psi(Cl_{\Gamma}(X)\otimes_{C_0(X)}\mathfrak{S}_{lf}(X))$ can be considered as a function of $x\in X$ and $\xi\in \tau_x^*$. We can assume that $a$ is a smooth covector field. For any $x\in X$, the direction of the covector $a_x(\xi)$ belongs to the subspace $\Gamma_x$ tangent to the orbit passing through $x$. Moreover, because $a$ belongs to the image of $\psi$, as a function of $\xi$, $a_x(\xi)$ is constant in $\xi$ in the directions orthogonal to $\Gamma_x$, and we can assume that in the direction of $\Gamma_x$ it has compact support in $\xi$ (since we can assume that an element of $\mathfrak{S}_{lf}(X)$ is an image of a compactly supported element of $C_0(\mathfrak{G} X)$).

It is clear from the above that the graded commutator $[a_x(\xi),{\rm ext\,}(\xi)+{\rm \,int\,}(\xi)]$, which is equal to the scalar product $2(a_x(\xi),\xi)$, vanishes for any $\xi$ orthogonal to $\Gamma_x$. On the other hand, for $\xi$ in the direction of $\Gamma_x$, the above commutator is bounded because we assumed that $a_x(\xi)$ has compact support in that direction. Therefore, the commutator $[a_x(\xi),{\rm ext\,}(\xi)+{\rm \,int\,}(\xi)]$ is always bounded uniformly in $x$. 

Since $a_x(\xi)$ has compact support in the direction of $\Gamma_x$ and is constant in the direction orthogonal to $\Gamma_x$, all first $\xi$-derivatives of $a_x$ vanish at infinity in $\xi$. Therefore the (graded) commutator $[a_x,d_\xi+d_\xi^*]$ is bounded on $L^2(\Lambda^*(\tau_x))$. 

Now we will use the method of lemma 4.2 \cite{Ka16}. Write ${\cal C}_x=(2/\pi)\int_0^\infty D_x(1+\lambda^2+D_x^2)^{-1}d\lambda$. For any $a_x$ as above, the (graded) commutator $[a_x,D_x]$ is bounded on $L^2(\Lambda^*(\tau_x))$ and depends norm-continuously on $x\in X$. The operators $(1+\lambda^2+D_x^2)^{-1}$ and $D_x(1+\lambda^2+D_x^2)^{-1}$ are compact on $L^2(\Lambda^*(\tau_x))$ and depend norm-continuously on $x\in X$. The commutator $[a_x, {\cal C}_x]$ is presented by the integral 
$$(2/\pi)\int_0^\infty (1+\lambda^2+D_x^2)^{-1}((1+\lambda^2)[a,D_x]+D_x[a,D_x]D_x)(1+\lambda^2+D_x^2)^{-1}d\lambda$$
which converges uniformly in $\lambda$, so the result is compact and depends norm-continuously on $x\in X$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10 

\begin{defi} Two separable $G-C_0(X)$-algebras $D_1$ and $D_2$ will be called ${\cal R} KK$-dual if there are elements 
$$\alpha\in {\cal R} KK^G(X;D_1\hat\otimes_{C_0(X)} D_2,C_0(X)), \;\; \beta\in {\cal R} KK^G(X;C_0(X),D_1\hat\otimes_{C_0(X)} D_2)$$ 
such that the product with these two elements gives isomorphisms:
$${\cal R} KK^G(X; A\hat\otimes_{C_0(X)} D_1,B) \simeq {\cal R} KK^G(X; A,B\hat\otimes_{C_0(X)} D_2)$$
and
$${\cal R} KK^G(X; A\hat\otimes_{C_0(X)} D_2,B) \simeq {\cal R} KK^G(X; A,B\hat\otimes_{C_0(X)} D_1)$$
for any two separable $G-C_0(X)$-algebras $A$ and $B$. 
\end{defi}

\begin{rem} {\rm Assume that the above elements $\beta$ and $\alpha$ have their product over ${D_1\hat\otimes_{C_0(X)}D_2}$ equal to $1_{C_0(X)}\in {\cal R} KK^G(X; C_0(X),C_0(X))$. Then the ${\cal R} KK$-duality of $D_1$ and $D_2$ can be proved using the same kind of rotation homotopy that is used in the Atiyah proof of Bott periodicity. Both algebras $D_1$ and $D_2$ must allow this rotation homotopy. 

Indeed, for example, taking the product of $a\in {\cal R} KK^G(X; A,B\hat\otimes_{C_0(X)} D_2)$ first with $1_{D_1}$ and then with $1_{D_2}$, we arrive at an element of the group
$${\cal R} KK^G(X; A\hat\otimes_{C_0(X)} D_1\hat\otimes_{C_0(X)} D_2,B\hat\otimes_{C_0(X)} D_2\hat\otimes_{C_0(X)} D_1\hat\otimes_{C_0(X)} D_2).$$ 
Then we need to take the product with $\beta$ and $\alpha$. We would like the result to be equal to the initial element $a$. This would have been the case if we took the product with $\alpha$ over the last couple of $D_1\hat\otimes_{C_0(X)} D_2$. But in fact it is the previous couple $D_2\hat\otimes_{C_0(X)} D_1$ that is required to be used for the product. So we need a rotation homotopy which interchanges the two copies of $D_2$ in the last argument of the ${\cal R} KK$-group.}
\end{rem}

\begin{theo} The algebras $\mathfrak{S}_{lf}(X)$ and $Cl_{\Gamma}(X)$ are ${\cal R} KK$-dual. The algebra $Cl_{\Gamma}(X)\otimes_{C_0(X)}\mathfrak{S}_{lf}(X)$ is ${\cal R} KK$-equivalent to $C_0(X)$.
\end{theo}

\noindent {\em Proof.}  The element $[d_{lf}]$ of definition 7.3 will serve as $\alpha$. The element of  ${\cal R} KK^G(X;C_0(X),Cl_{\Gamma}(X)\otimes_{C_0(X)}\mathfrak{S}_{lf}(X))$ which will be denoted $[\mathfrak{f}_\Gamma]$, and will serve as $\beta$, is defined as follows: The Hilbert module is $Cl_{\Gamma}(X)\otimes_{C_0(X)}\mathfrak{S}_{lf}(X)$. The operator $\mathfrak{f}_\Gamma$ is defined by the covector field $\{\mathfrak{f}_{\Gamma_x}(\xi)=\varphi_x(\xi)(1+q_x(\xi))^{-1/2},\;x\in X\}$.

Let $D=Cl_{\Gamma}(X)\otimes_{C_0(X)}\mathfrak{S}_{lf}(X)$. To show that the product $\beta\otimes_D \alpha$ is equal to $1_{C_0(X)}$, we need to produce a homotopy of the product operator $N_1\mathfrak{f}_\Gamma+N_2{\cal C}$ to ${\cal C}$. (The operators $N_1,N_2$ are those used in the $KK$-product construction.)

The graded commutator $[N_1\mathfrak{f}_\Gamma+N_2{\cal C},{\cal C}]$ is non-negative modulo compact operators. Indeed, $[\varphi_x(\xi),d_\xi+d^*_\xi]$ is constant in $\xi$ and $[\varphi_x(\xi),{\rm ext\,}(\xi)+{\rm \,int\,}(\xi)]=2(\varphi_x(\xi),\xi)\ge 0$. Using the same integral presentation as in lemma 7.3, both for ${\cal C}$ and $\mathfrak{f}_\Gamma$, as well as the very similar calculation of a commutator in the proof of theorem 8.9 \cite{Ka16}, we find that $[\mathfrak{f}_\Gamma,{\cal C}]$ is non-negative modulo compact operators. Now the existence of the homotopy joining $N_1\mathfrak{f}_\Gamma+N_2{\cal C}$ to ${\cal C}$ follows from lemma 11 \cite{Sk84}.

In the remaining part of the proof we will use a rotation homotopy. For each $x\in X$, the fiber of $\mathfrak{S}_{lf}(X)$ over $x$ is the image of $C_0(\mathfrak{g}_x)$ in $C_b(\tau_x)$.  So we will use the rotation of $\mathfrak{g}_x\times \mathfrak{g}_x$. Similarly for $Cl_{\Gamma}(X)$, where the fiber over $x$ is just the corresponding Clifford algebra.

Let us first deal with $\alpha\otimes_{C_0(X)}\beta=1_D$. This is very similar to the proof of the Bott periodicity in \cite{Ka80}, theorem 5.7. This product can be written as $(\phi:D\to {\cal L}({\cal T}\otimes_{C_0(X)} D), {\cal C}\times\hspace{-15pt}\times \mathfrak{f}_\Gamma)$. The map $\phi$ factors through ${\cal M}(D\hat\otimes D)$. There is an obvious rotation homotopy of $D\hat\otimes D$, as already explained. The operator $\mathfrak{f}_\Gamma$ rotates together with $\phi$. The operator ${\cal C}$ does not rotate. At the end of this rotation homotopy we arrive at the product $(\beta\otimes_D \alpha)\otimes_{C_0(X)} 1_D=1_D\in {\cal R} KK^G(D,D)$. The details on choosing the homotopy of operators $N_1,N_2$ involved in the $KK$-product construction see \cite{Ka80}, the proof of theorem 5.7.

To complete the proof, it remains to show the existence of a rotation homotopy indicated in the remark 7.5. Let $D_1=Cl_{\Gamma}(X), D_2=\mathfrak{S}_{lf}(X)$, and consider the rotation for the ${\cal R} KK$-group in the displayed formula in the remark 7.5. If ${\cal E}$ is the Hilbert module for $a\in {\cal R} KK^G(X; A,B\hat\otimes_{C_0(X)} D_2)$, we need to compose the homomorphism 
$$A\hat\otimes_{C_0(X)} D_1\hat\otimes_{C_0(X)} D_2 \to {\cal L}({\cal E}\hat\otimes_{B\hat\otimes_{C_0(X)}D_2} B\hat\otimes_{C_0(X)}D_2\hat\otimes_{C_0(X)}D_1\hat\otimes_{C_0(X)} D_2)$$
with the rotation interchanging the last two copies of $D_2$. Clearly, there is a map of $D_2\hat\otimes_{C_0(X)} D_2$ into the right hand side, and we can compose it with the rotation of the map $D_2\to {\cal M}(D_2\hat\otimes_{C_0(X)} D_2)$, as already explained. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\medskip

Before we move further, we need to correct an error in \cite{Ka16} at the end of definition 7.1, where it was claimed that the algebra $Cl_{\Gamma}(X)\hat\otimes_{C_0(X)}Cl_{\Gamma}(X)$ is isomorphic to the algebra of compact operators on the Hilbert module $C_0(\Lambda^*_\Gamma(X))$ over $C_0(X)$. Unfortunately, this is a wrong statement, which was used a number of times in sections 7 and 8 of \cite{Ka16}. The algebras $Cl_{\Gamma}(X)\hat\otimes_{C_0(X)}Cl_{\Gamma}(X)$ and $C_0(X)$ are not Morita equivalent in general. They are Morita equivalent only in the special case when $\Gamma$ is defined by a vector bundle. But for a general $\Gamma$ we will use the ${\cal R} KK$-duality between $Cl_{\Gamma}(X)$ and $C^*_\Gamma(X)$  provided by the previous theorem:

\begin{cor} If $A$ and $B$ are any separable $G-C_0(X)$-algebras, then 
$${\cal R} KK^G(X; A\hat\otimes_{C_0(X)}Cl_{\Gamma}(X),B)\simeq {\cal R} KK^G(X; A,B\hat\otimes_{C_0(X)}\mathfrak{S}_{lf}(X)),$$
$${\cal R} KK^G(X; A\hat\otimes_{C_0(X)}\mathfrak{S}_{lf}(X),B)\simeq{\cal R} KK^G(X; A,B\hat\otimes_{C_0(X)}Cl_{\Gamma}(X)).$$
 \end{cor}

Combined with the Poincare duality 6.1, this gives a corrected version of theorem 7.8 \cite{Ka16} -- see theorem 7.11 below. Here is another corollary of Poincare duality 6.1 and ${\cal R} KK$-duality:

\begin{prop} Let $G$ be a compact group and $X$ a compact $G$-manifold. If the $G-C(X)$-algebras $A$ and $A^{\rm dual}$ are ${\cal R} KK$-dual then 
$$K^*_G(A)\simeq K_*^G(A^{\rm dual}\hat\otimes_{C_0(X)} C_0(TX)).$$
\end{prop}

\noindent {\em Proof.}  The left side is isomorphic to ${\cal R} KK_*^G(X;A, C_0(TX))$ by theorem 6.1, which in its turn is isomorphic to ${\cal R} KK_*^G(X;C_0(X),A^{\rm dual}\hat\otimes_{C_0(X)} C_0(TX))$ by ${\cal R} KK$-duality, and coincides with the right side in the statement because $X$ is compact. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10
 
 \begin{notat} {\rm We simplify the notation $Cl_\Gamma(X)\otimes_{C_0(X)} C_0(TX)$ to $Cl_{\Gamma}(TX)$. (Note that the latter \emph{is not} related with the orbits of $G$ on the manifold $TX$.) We will also denote the algebra $Cl_\tau(X)\hat\otimes_{C_0(X)}Cl_{\Gamma}(X)$ by $Cl_{\tau\oplus\Gamma}(X)$.}
\end{notat}

Now we are going to restate and reprove theorem 7.4 of \cite{Ka16} which relates the symbol algebra $\mathfrak{S}_{tr}(X)$ with the algebra $Cl_{\Gamma}(TX)$. We remind that the symbol algebra $\mathfrak{S}_{tr}(X)$ is not separable, so to state the $KK$-equivalence one needs a separable version of the $KK$-theory introduced by G. Skandalis in \cite{Sk88}, pp. 571-572 (cf. also the beginning of section 7 in \cite{Ka16}). 

\medskip

Here are some definitions from \cite{Ka16} that we will need: 

\medskip

First, a family of Dirac operators on tangent spaces of $X$ (\cite{Ka16}, 2.5). For any $x\in X$, let $H_x=L^2(T_x^*(X))\otimes Cl_{\tau_x}$ be the Hilbert module  over $Cl_{\tau_x}$. The Dirac operator on this Hilbert module is defined by $\mathfrak{d}_x=\sum_{k=1}^{\dim X} (-i)c(e_k)\partial/\partial \xi_k$ in any orthonormal basis $\{e_k\}$ of $T^*_x(X)$, where $c(e_k)$ are the left Clifford multiplication operators and $\xi_k$ are the coordinates of $T^*_x(X)$ in the basis $\{e_k\}$. The family of operators $\{\mathfrak{d}_x/(1+\mathfrak{d}_x^2)^{-1/2}\}$ on the family of Hilbert modules $\{H_x,\,x\in X\}$ defines an element ${\cal R} KK^G(X; C_0(TX),Cl_\tau(X))$ denoted $[d_\xi]$. 

Next, the fiber-wise Bott element  
$[{\cal B}_{\xi,\Gamma}]\in {\cal R} KK^G(X;Cl_{\tau\oplus\Gamma}(X),\mathfrak{S}_{tr}(X))$ (\cite{Ka16}, 7.2), which is given by the pair $(\mathfrak{S}_{tr}(E),\beta)$, where $E=\Lambda^*(X)$ is considered as a Hilbert module over $C_0(X)$, $\mathfrak{S}_{tr}(E)=E\otimes_{C_0(X)}\mathfrak{S}_{tr}(X)$, the left action of $Cl_{\tau\oplus\Gamma}(X)$ on $\mathfrak{S}_{tr}(E)$ is defined on (real) covectors by 
$$\xi_1\oplus\xi_2\mapsto i({\rm ext\,}(\xi_1)-{\rm \,int\,}(\xi_1))+({\rm ext\,}(\xi_2)+{\rm \,int\,}(\xi_2)).$$
Here $\xi_1$ is a section of $\tau=T(X)$ and $\xi_2$ a section of $\Gamma \subset \tau$. The operator $\beta$ is the Bott operator defined by $\beta(\xi)=({\rm ext\,} (\xi)+{\rm \,int\,} (\xi))/(1+||\xi||^2)^{1/2}$.

If we forget about $\Gamma$ in this definition of $[{\cal B}_{\xi,\Gamma}]$ and replace $\mathfrak{S}_{tr}(X)$ with $C_0(TX)$ we get the tangent Bott element $[{\cal B}_\xi]\in {\cal R} KK^G(X;Cl_\tau(X),C_0(TX))$ (\cite{Ka16}, 2.6).

We will consider here the operator $\mathfrak{f}_\Gamma$ defined in the proof of theorem 7.6 as an element of ${\cal M}(Cl_\Gamma(TX))$. The element $[\mathfrak{f}_\Gamma]\in {\cal R} KK^G(X;\mathfrak{S}_{tr}(X),Cl_\Gamma(TX))$ is given by the natural homomorphism $\mathfrak{S}_{tr}(X)\to {\cal M}(Cl_\Gamma(TX))$, together with the operator $\mathfrak{f}_\Gamma=\{\varphi_x(\xi)(1+q_x(\xi))^{-1/2},\;x\in X\}$.

\begin{theo} The algebras $Cl_{\Gamma}(TX)$ and $Cl_{\tau\oplus\Gamma}(X)$ are ${\cal R} KK^G$-equivalent, and $\mathfrak{S}_{tr}(X)$ is ${\cal R} KK^G_{sep}$-equivalent to each of them.
\end{theo}

\noindent {\em Proof.}  The ${\cal R} KK^G$-equivalence between the algebras $Cl_{\Gamma}(TX)$ and $Cl_{\tau\oplus\Gamma}(X)$ follows from the ${\cal R} KK^G$-equivalence between $C_0(TX)$ and $Cl_\tau(X)$ (theorem 2.7 \cite{Ka16}) by tensoring it over $C_0(X)$ (i.e. $\otimes_{C_0(X)}$) with $Cl_{\Gamma}(X)$. This ${\cal R} KK^G$-equivalence in fact is given by the product with the elements $[d_\xi]$ and $[{\cal B}_\xi]$ defined above which are inverses of each other (\cite{Ka16}, theorem 2.7). 

The product $[{\cal B}_{\xi,\Gamma}]\otimes_{\mathfrak{S}_{tr}(X)}[\mathfrak{f}_\Gamma]$ is given by the pair 
$$(\psi:Cl_{\tau\oplus\Gamma}(X)\to {\cal L}(E\hat\otimes_{C_0(X)}Cl_{\Gamma}(TX)), N_1(\beta\hat\otimes 1)+N_2(1\hat\otimes \mathfrak{f}_\Gamma)),$$
where $E=\Lambda^*(X)$. Here $N_1,N_2$ are the operators entering the $KK$-product construction: $N_1\cdot \mathfrak{S}_{tr}(X)\subset C_0(TX)$ and $N_2\cdot \mathfrak{S}_{lf}(X)\subset C_0(TX)$. 

Let us make a rotation homotopy of this element. Consider the homomorphism $Cl_{\Gamma}(X)\hat\otimes_{C_0(X)}Cl_{\Gamma}(X)\to {\cal L}(E\otimes_{C_0(X)}Cl_{\Gamma}(TX))$, where the first copy of $Cl_{\Gamma}(X)$ maps according to the map $\psi$ and the second copy maps into ${\cal M}(Cl_{\Gamma}(TX))$. We will rotate this homomorphism by using the rotation of $\Gamma_x\times\Gamma_x$ ($x\in X$). The result of this homotopy will be $\mathfrak{f}_\Gamma$ acting on $E$ and $Cl_{\Gamma}(X)$ acting on $Cl_{\Gamma}(TX)$. The operator will turn into $N_1(\beta\hat\otimes 1)+N_2(\mathfrak{f}_\Gamma\hat\otimes 1)$. 

Finally, observe that this operator is homotopic to $\beta\hat\otimes 1$ by lemma 11 \cite{Sk84} because the graded commutator $[N_1(\beta\hat\otimes 1)+N_2(\mathfrak{f}_\Gamma\hat\otimes 1),\beta\hat\otimes 1]\ge 0$ modulo compact operators. (In fact $[\mathfrak{f}_\Gamma(\xi),\beta(\xi)]$ differs by a positive multiple from the scalar product $(\varphi_x(\xi),\xi)$.) 

As a result, we get the equality: $[{\cal B}_{\xi,\Gamma}]\otimes_{\mathfrak{S}_{tr}(X)}[\mathfrak{f}_\Gamma]=[{\cal B}_\xi]\otimes_{C_0(X)} 1_{Cl_{\Gamma}(X)}$. Because $[{\cal B}_\xi]$ is an invertible element, we have proved that both $[{\cal B}_{\xi,\Gamma}]$ and $[\mathfrak{f}_\Gamma]$ are one-sided invertible. To show that these two elements are actually invertible, we will prove that the product $[\mathfrak{f}_\Gamma]\otimes_{C_0(X)}([d_\xi]\otimes_{C_0(X)}1_{Cl_{\Gamma}(X)})\otimes_{C_0(X)}[{\cal B}_{\xi,\Gamma}]$ is equal $1_{\mathfrak{S}_{tr}(X)}$.

Let us first look at the product $([d_\xi]\otimes_{C_0(X)}1_{Cl_{\Gamma}(X)})\otimes_{C_0(X)}[{\cal B}_{\xi,\Gamma}]$. We will denote it $[d_{tr}]$. It is given by the pair $(\pi:Cl_{\Gamma}(TX)\to {\cal L}({\cal T}\hat\otimes_{C_0(X)} \mathfrak{S}_{tr}(X)), F)$, where the Hilbert $C_0(X)$-module ${\cal T}$ is defined in lemma 7.3. The map $\pi$ of $Cl_{\Gamma}(TX)$ is given by multiplication by functions of $C_0(\tau_x)$ on $L^2(\tau_x)$ (for each tangent space $\tau_x,\,x\in X$), and on (real) covectors of $Cl_{\Gamma}(TX)$ the map $\pi$ is $\xi\mapsto {\rm ext\,}(\xi)+{\rm \,int\,}(\xi)$. The operator $F=N_1(A\hat\otimes 1)+N_2(1\hat\otimes \beta)$, where $A=(d_\xi+d_\xi^*)(1+(d_\xi+d_\xi^*)^2)^{-1/2}$, both $d_\xi+d_\xi^*$ and $\beta$ use the same exterior algebra variables, but $d_\xi+d_\xi^*$ acts on the $L^2(\tau_x)$ spaces, whereas $\beta(\xi)$ is the multiplication operator on $\mathfrak{S}_{tr}(X)$.

The product $[\mathfrak{f}_\Gamma]\otimes_{Cl_{\Gamma}(TX)} [d_{tr}]$ is given by the pair $(\tilde\pi:\mathfrak{S}_{tr}(X)\to {\cal L}({\cal T}\otimes_{C_0(X)}\mathfrak{S}_{tr}(X)), N_1(A\hat\otimes 1)+N_2(1\hat\otimes (\beta+\mathfrak{f}_\Gamma))$. Here $\mathfrak{S}_{tr}(X)$ acts on ${\cal T}$ and $\mathfrak{f}_\Gamma$ is the multiplication operator on $\mathfrak{S}_{tr}(X)$. 

We first deform the latter product using the operator homotopy $N_1(A\hat\otimes 1)+N_2(1\hat\otimes (\beta+t\mathfrak{f}_\Gamma)),\; 1\ge t\ge 0$. After that we make a rotation homotopy using the fact that $\tilde\pi$ factors through the homomorphism ${\cal M}(\mathfrak{S}_{tr}(X)\otimes_{C_0(X)} \mathfrak{S}_{tr}(X)) \to {\cal L}({\cal T}\otimes_{C_0(X)}\mathfrak{S}_{tr}(X))$. As a result of this rotation homotopy, the map $\tilde\pi$ transforms into the identity map $\mathfrak{S}_{tr}(X)\to\mathfrak{S}_{tr}(X)$, and $\beta$ transforms into the multiplication operator which acts on the $L^2(\tau_x)$  spaces. Therefore the operator will be homotopic to the operator ${\cal C}$ of lemma 7.3. This means that our product turns into $1_{\mathfrak{S}_{tr}(X)}$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10
 
 \medskip

We collect the most important results of this section in the next theorem:

\begin{theo}
 $${\cal R} KK^G_*(X;C_0(X),\mathfrak{S}_{lf}(X))\simeq {\cal R} KK^G_*(X;Cl_{\Gamma}(X),C_0(X))$$
 $$\simeq {\cal R} KK^G_*(X;Cl_{\tau\oplus\Gamma}(X),Cl_\tau(X))\simeq K^*_G(Cl_{\tau\oplus\Gamma}(X)).$$
 
$${\cal R} KK^G_*(X;C_0(X),\mathfrak{S}_{tr}(X))\simeq {\cal R} KK^G_*(X;C_0(X),Cl_{\tau\oplus\Gamma}(X))$$
$$\simeq {\cal R} KK^G_*(X;\mathfrak{S}_{lf}(X),Cl_\tau(X))
\simeq K^*_G(\mathfrak{S}_{lf}(X)).$$
\end{theo}

\noindent {\em Proof.}  The first isomorphism comes from corollary 7.7. The next isomorphism is the Clifford periodicity provided by the fact that $\tau$ is a vector bundle. The final isomorphism in the second line comes from the Poincare duality theorem 6.1. In the third line, the first isomorphism is given by theorem 7.10. The next isomorphism comes from corollary 7.7. The last isomorphism of the fourth line is provided by the Poincare duality 6.1.
{\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10 


\section{Elliptic symbols and index groups}

We start with a definition of certain special $KK$-groups related with crossed products. Then we give an overview of symbol and index groups for various classes of operators. In the last part of this section, there are some important results on interrelations between different index and symbol groups.

\subsection{Special $KK$-groups for crossed products}

Let $B$ be a $G$-algebra. Consider the canonical unitary representation $u:G\to{\cal M}(C^*(G,B)): g\mapsto u_g$. Then the $G$-action on $C^*(G,B)$ is given by $g(b)=u_gbu_g^{-1}$, where $b\in C^*(G,B)$. For any Hilbert module ${\cal E}$ over $C^*(G,B)$, one can define a structure of a $G-C^*(G,B)$-module as follows.  For $e\in {\cal E}$, put $g(e)=e\cdot u_g^{-1}$. An easy check shows that this definition of the $G$-action on ${\cal E}$ satisfies the axioms of a $G-C^*(G,B)$-module (\cite{KaJOT}, definition 2.1, $5^{\rm o}$ - $6^{\rm o}$).

When $B$ is a $G-C_0(X)$-algebra, where $X$ is a proper $G$-space, there exists also a canonical right action of $C_0(X)$ on $C^*(G,B)$ given by the right multiplication. Moreover, a similar right action of $C_0(X)$ exists also for any Hilbert module ${\cal E}$ over $C^*(G,B)$. It is given by $e\cdot f=\lim_{\alpha\to 0}e(e,e)((e,e)+\alpha)^{-1}f$, where $e\in {\cal E},\,f\in C_0(X)$. In the next definition we will use this $C_0(X)$-action. 

\begin{defi} Let $A$ and $B$ be $G-C_0(X)$-algebras. We define the group $KK^{G,X}(A,C^*(G,B))$ in the same way as the usual $KK^G(A,C^*(G,B))$, but with an additional assumption that for any triple $({\cal E},\psi:A\to{\cal L}({\cal E}), F)$ and any $a\in A,\, f\in C_0(X)$, one has: $\psi(a)\cdot f=\psi(fa)$ in ${\cal L}({\cal E})$, where the right multiplication by $f$ on ${\cal E}$ is defined above. \end{defi}
 
 \begin{prop}
 The group $KK^{G,X}(A,C^*(G,B))$ has the following additional properties. 

$1^{\rm o}$. There is a natural map: 
$$KK^{G,X}(A,C^*(G,B))\to KK^{G,X}(A\hat\otimes_{C_0(X)} D,C^*(G,B\hat\otimes_{C_0(X)} D))$$
given by the product with $1_D$.

\smallskip

$2^{\rm o}$. If $\alpha\in KK^{G,X}(A,C^*(G,B))$ and $\beta\in 
{\cal R} KK^G(X;B,D)$, then the product $\alpha\otimes_{C^*(G,B)}j^G(\beta)$ belongs to $KK^{G,X}(A,C^*(G,D))$. 

\smallskip

$3^{\rm o}$. There is a natural product: 
$$KK^{G,X}(A,C^*(G,B))\otimes {\cal R} KK^G(X;D,E)$$
$$\to KK^{G,X}(A\hat\otimes_{C_0(X)} D,C^*(G,B\hat\otimes_{C_0(X)} E)).$$
This product is commutative.

$4^{\rm o}$. There is an isomorphism:
$$KK^{G,X}(\mathfrak{S}_{lf}(X),C^*(G,C_0(X)))\simeq KK^{G,X}(C_0(X),C^*(G,Cl_{\Gamma}(X))).$$
\end{prop}

\noindent {\em Proof.}  The first two statements are clear. The third one is a composition of the first two. For the commutativity of the product in $3^{\rm o}$, one has to identify $A\hat\otimes_{C_0(X)} D$ with $D\hat\otimes_{C_0(X)} A$ and $B\hat\otimes_{C_0(X)} E$ with $E\hat\otimes_{C_0(X)} B$ via $a_1\hat\otimes a_2\mapsto (-1)^{{\rm deg}a_1\cdot {\rm deg}a_2}a_2\hat\otimes a_1$. This is similar to \cite{Ka88}, theorem 2.14, $8^{\rm o}$. The last statement follows from theorem 7.6 and its proof. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\subsection{Overview of definitions}

\emph{We will use the equivariant $KK^G$ in all notation for the symbol and index groups. In the non-equivariant case one just has to drop the `$G$'.}

\medskip

In the definition of ellipticity we will follow \cite{Ka16}, section 3, for elliptic operators and \cite{Ka16}, section 6, for t-elliptic operators. As in \cite{Ka16} we consider two options: 

\medskip

$1^{\rm o}$ Self-adjoint operators of degree $1$ on ${\bf Z}_2$-graded spaces have index in the $K^0$-groups.

\medskip

$2^{\rm o}$ Self-adjoint operators on ungraded spaces have index in the $K^1$-groups.

\begin{defi} {\bf (Elliptic case.)} A self-adjoint PDO acting on sections of a vector bundle $E$ is elliptic if its symbol $\sigma\in {\cal L}(C_0(p^*(E)))$ satisfies the condition: $f\cdot (\sigma^2-1)\in {\cal K}(C_0(p^*(E)))$ for any $f\in C_0(X)$ (both in the graded and non-graded case).
\end{defi}

When the symbol $\sigma$ is elliptic, the coarse PDO construction of section 3 gives the index element in the group $K^*(C_0(X))$. If a locally compact group $G$ acts on $X$ properly and isometrically and the symbol $\sigma$ is elliptic and $G$-invariant then $[{\cal F}]=(L^2(E),{\cal F})\in K^*_G(C_0(X))$.

Recall from \cite{Ka16} that the symbol group for elliptic operators is defined as ${\cal R} KK^G_*(X;C_0(X),C_0(TX))$. If one wants to consider elliptic operators with coefficients in a unital $G$-algebra $B$ (see \cite{MiFo}), the symbol group will be ${\cal R} KK^G_*(X;C_0(X),B\otimes C_0(TX))$ and the index group will be $KK^G_*(C_0(X),B)$. 

Note that in the case of a compact manifold $X$, the symbol group `shortens' to $K_*^G(C_0(TX))$. But it is important to emphasize that for a non-compact $X$, one can still consider symbols $\sigma$ belonging to $K_*^G(C_0(TX))$ if the group $G$ is compact. Such symbol has an additional property that the corresponding operator is Fredholm, because $\sigma^2-1$ is of strongly negative order (see 3.7 and 3.11, $3^{\rm o}$).

\begin{defi} {\bf (T-elliptic case.)} A self-adjoint PDO is t-elliptic if its symbol $\sigma\in {\cal L}(\mathfrak{S}_{tr}(E))$ satisfies the condition: $f\cdot (\sigma^2-1)\in {\cal K}(\mathfrak{S}_{tr}(E))$ for any $f\in C_0(X)$ (both in the graded and non-graded case).
\end{defi}

For t-elliptic operators (\cite{Ka16}, section 6), the symbol group is defined as ${\cal R} KK^G_*(X;C_0(X),\mathfrak{S}_{tr}(X))$, and in the case of operators with coefficients in a unital $G$-algebra $B$, the symbol group is ${\cal R} KK_*^G(X;C_0(X),B\otimes \mathfrak{S}_{tr}(X))$. The corresponding index groups are $K^*(C^*(G,C_0(X)))$ and $KK_*^G(C^*(G,C_0(X)),B)$ respectively. The index of an operator $F$ is denoted ${\rm ind}(F)$. Another index $[F]\in K^*_G(\mathfrak{S}_{lf}(X))$, called analytical index, is defined in 10.5. 

Also for t-elliptic operators there are Clifford and tangent-Clifford symbol groups: ${\cal R} KK_*^G(X;C_0(X),Cl_{\tau\oplus\Gamma}(X))$ and ${\cal R} KK_*^G(X;C_0(X),Cl_{\Gamma}(TX))$ respectively. They are isomorphic to ${\cal R} KK_*^G(X;C_0(X),\mathfrak{S}_{tr}(X))$ by theorem 7.10. 

When the manifold $X$ and the group $G$ are compact, the symbol groups simplify to $K_*^G(\mathfrak{S}_{tr}(X))\simeq K_*^G(Cl_{\tau\oplus\Gamma}(X))\simeq K_*^G(Cl_{\Gamma}(TX))$. However, when $G$ is compact but $X$ is not, we can still have symbols in the latter groups. The corresponding operators have a distributional index in $K^0(C^*(G))$ (see remark 10.4).

\begin{defi} {\bf (Leaf-wise elliptic case.)} A leaf-wise self-adjoint PDO is elliptic if its symbol $\sigma\in {\cal L}(\mathfrak{S}_{lf}(E))$ satisfies the condition: $f\cdot (\sigma^2-1)\in {\cal K}(\mathfrak{S}_{lf}(E))$ for any $f\in C_0(X)$ (both in the graded and non-graded case).
\end{defi}

For leaf-wise operators, the symbol group is ${\cal R} KK_*^G(X;C_0(X),\mathfrak{S}_{lf}(X))$, which is the same as
${\cal R} KK_*^G(X;C_0(X),C^*_\Gamma(X))$. In the case of operators with values in an admissible $G-C_0(X)$-algebra $B$ (definition 4.14), the symbol group is ${\cal R} KK_*^G(X;C_0(X),B\otimes_{C_0(X)} \mathfrak{S}_{lf}(X))$. The corresponding index groups are $KK_*^{G,X}(C_0(X),C^*(G,C_0(X)))$ and $KK_*^{G,X}(C_0(X),C^*(G,B))$ respectively.

\medskip

\noindent{\bf \large Adding a new variable.}

\smallskip

However, it is very useful to generalize these definitions. Instead of just a `lonely' operator ${\cal F}$ acting on a Hilbert space or a Hilbert module (which commutes modulo compact operators with the left action of $C_0(X)$) we can consider the case when a $C_0(X)$-unital $G-C_0(X)$-algebra $A$ (definition 4.14) acts on the same Hilbert module. 

\smallskip

For elliptic operators, the symbol group will be ${\cal R} KK^G_*(X;A,B\otimes C_0(TX))$ and the index group  $KK^G_*(A,B)$. 

\smallskip

For t-elliptic operators, we will get the symbol group ${\cal R} KK^G_*(X;A,B\otimes \mathfrak{S}_{tr}(X))$ and the index group $KK^G_*(C^*(G,A),B)$. 

\smallskip

For leaf-wise operators, the symbol group is ${\cal R} KK^G_*(X;A,B\otimes_{C_0(X)} \mathfrak{S}_{lf}(X))$ and the index group is $KK^{G,X}_*(A,C^*(G,B))$.

\begin{ex} {\rm Let $B={\bf C}$ and $A=Cl_{\tau\oplus\Gamma}(X)$. The Bott element $[{\cal B}_{\xi,\Gamma}]\in {\cal R} KK^G(X;Cl_{\tau\oplus\Gamma}(X),\mathfrak{S}_{tr}(X))$ defined before theorem 7.10 (see also definition 7.2 \cite{Ka16}) can be considered as an element of a t-elliptic symbol group. The corresponding element $[d_{X,\Gamma}]$ of the index group $K^0(C^*(G,Cl_{\tau\oplus\Gamma}(X)))$ is defined in \cite{Ka16}, 8.8. Here is this definition.

The Dirac element $[d_{X,\Gamma}]$ is given by the pair $(H,F_X)$, with $H=L^2(\Lambda^*(X))$, the action of $Cl_{\tau\oplus\Gamma}(X)$ on $H$ defined on (real) covectors by $\xi_1\oplus\xi_2\mapsto {\rm ext\,}(\xi_1)+{\rm \,int\,}(\xi_1)+i({\rm ext\,}(\xi_2)-{\rm \,int\,}(\xi_2))$.
Here $\xi_1$ is a section of $\tau=T(X)$ and $\xi_2$ is a section of $\Gamma$. The representation of $C^*(G,Cl_{\tau\oplus\Gamma}(X))$ on $H$ is induced by the covariant representation defined by the action of $G$ and the above action of $Cl_{\tau\oplus\Gamma}(X)$, the operator $d_X$ is the exterior derivation operator, and $F_X=(d_X+d^*_X)(1+(d_X+d^*_X)^2)^{-1/2}$.}
\end{ex}

\begin{rem} {\rm In the t-elliptic case, the group ${\cal R} KK_*^G(X; \mathfrak{S}_{lf}(X), C_0(TX))$ can also be considered as the symbol group and $K^*_G(\mathfrak{S}_{lf}(X))$ as the index group. See theorems 7.11 and 10.6.}
\end{rem}

\begin{ex} {\rm Another important element is the leaf-wise Dirac element $[D_\Gamma]\in KK^{G,X}(C_0(X),C^*(G,Cl_{\Gamma}(X)))$. It is given by the pair $(C^*(G,Cl_{\Gamma}(X)),\break \mathfrak{D}_\Gamma)$, where the Dirac operator $\mathfrak{D}_\Gamma$ is constructed in subsection 5.3,  proposition 5.12. The symbol corresponding to the element $[D_\Gamma]$ is equal to the element $[\mathfrak{f}_\Gamma]\in {\cal R} KK(X;C_0(X),Cl_{\Gamma}(X)\otimes_{C_0(X)}\mathfrak{S}_{lf}(X))$ (constructed in the proof of theorem 7.6).

The product of the element $[D_\Gamma]$ with $1_{Cl_\tau(X)}$ in the sense of proposition 8.2, $1^{\rm o}$ will be denoted $[D_{\Gamma,\tau}]\in KK^{G,X}(Cl_\tau(X),C^*(G,Cl_{\tau\oplus\Gamma}(X)))$.

A similarly product of the element $[D_\Gamma]$ with $1_{\mathfrak{S}_{lf}(X)}$  gives an element of the group $KK^{G,X}(\mathfrak{S}_{lf}(X),C^*(G,\mathfrak{S}_{lf}(X)\hat\otimes_{C_0(X)}Cl_{\Gamma}(X)))$. Using                        theorem 7.6, we get the element $[D_{\Gamma,C_0(X)}]\in KK^{G,X}(\mathfrak{S}_{lf}(X),C^*(G,C_0(X)))$ which is very important for the index theorems of the following sections. 

Here is a direct definition of the element $[D_{\Gamma,C_0(X)}]$ which will be used later: Recall from the definition-lemma 7.3 the Hilbert $C_0(X)$-module ${\cal T}$ and the operator ${\cal C}$ on it. The Hilbert module for our element will be ${\cal Q}={\cal T}\hat\otimes_{C_0(X)} C^*(G,C_0(X))$. $\mathfrak{S}_{lf}(X)$ acts on it is by multiplication over ${\cal T}$ as in 7.3. Let us present ${\cal Q}$ as the product $\{L^2(\tau_x)\}\hat\otimes_{C_0(X)} \Lambda^*(\tau)\hat\otimes_{C_0(X)}C^*(G,C_0(X))$. By remark 5.14, $1^{\rm o}$, we can consider the operator $\mathfrak{D}_\Gamma$ as an operator on  $\Lambda^*(\tau)\hat\otimes_{C_0(X)}C^*(G,C_0(X))$. The full operator for this element can be written as the $KK$-product $N_1({\cal C}\hat\otimes 1)+N_2(1\hat\otimes \tilde \mathfrak{D}_\Gamma)$, where $\tilde \mathfrak{D}_\Gamma$ is a $K$-theoretic connection for $\mathfrak{D}_\Gamma$. Here $C_0(X)\cdot N_1\cdot {\cal K}({\cal T})\subset {\cal K}({\cal Q})$ and $C_0(X)\cdot N_2\cdot (1-\tilde \mathfrak{D}_\Gamma^2)\subset {\cal K}({\cal Q})$. (The operator $1-\mathfrak{D}_\Gamma^2$ has `negative order'.) Concerning the compactness of commutators between the operator ${\cal C}$ and the Clifford variables of the operator $\mathfrak{D}_\Gamma$, see the proof of 7.3.}
\end{ex}

\subsection{Theorems}

Now we come to a couple of important results. The first one is theorem 8.9 \cite{Ka16}:

\begin{theo} $[D_{\Gamma,\tau}]\otimes_{C^*(G,Cl_{\tau\oplus\Gamma}(X))}[d_{X,\Gamma}]=[d_X]$.
\end{theo}

\noindent {\em Proof.}  The proof of theorem 8.9 \cite{Ka16} works as it written there, except that it needs a small adjustment: one has to replace $\varphi_x(\xi)=f'_xf'^*_x(\xi)$ used in \cite{Ka16} with $\varphi_x(\xi)=(f'_xf'^*_x)^{1/2}(\xi)$ as it is defined in subsection 5.1 of the present paper. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\medskip

The second result is part of theorem 8.15 \cite{Ka16}. Because of the confusion in \cite{Ka16} related with the ${\cal R} KK$-duality of $Cl_{\Gamma}(X)$ with itself (instead of with $\mathfrak{S}_{lf}(X)$ as would be correct), we need to restate and reprove this important result.

\begin{theo} The following composition:
$${\cal R} KK^G_*(X; C_0(X),Cl_{\tau\oplus\Gamma}(X))\to KK^G_*(C^*(G,C_0(X)),C^*(G,Cl_{\tau\oplus\Gamma}(X)))$$
$$\to K_G^*(C^*(G,C_0(X)))\to K_G^*(\mathfrak{S}_{lf}(X))$$
coincides with the isomorphism of theorem 7.11.
Here the first map is $j^G$, the second one is the product with the element $[d_{X,\Gamma}]$, and the last one is the product with the element $[D_{\Gamma,C_0(X)}]$.
 \end{theo}

The proof is based on a lemma which copies lemma 8.16 \cite{Ka16}:

\begin{lem} Let $a\in {\cal R} KK_*^G(X;C_0(X),Cl_{\tau\oplus\Gamma}(X))$. Then the triple product 
$$[D_{\Gamma,C_0(X)}]\otimes_{C^*(G,C_0(X))} j^G(a)\otimes_{C^*(G,Cl_{\tau\oplus\Gamma}(X))}[d_{X,\Gamma}]\in K^*_G(\mathfrak{S}_{lf}(X))$$ 
is equal to the product $\tilde a\otimes_{Cl_\tau(X)}[d_X]$, where $\tilde a\in {\cal R} KK^G_*(X;\mathfrak{S}_{lf}(X),Cl_\tau(X))$ is ${\cal R} KK$-dual to a.
\end{lem}

\noindent \emph{Proof of the lemma.} We will use the formula for $[d_X]$ from theorem 8.9. Then we need only to show that 
$$[D_{\Gamma,C_0(X)}]\otimes_{C^*(G,C_0(X))} j^G(a)=\tilde a \otimes_{Cl_\tau(X)} [D_{\Gamma,\tau}].$$
The element $[D_{\Gamma,C_0(X)}]$ is the product of $1_{\mathfrak{S}_{lf}(X)}\otimes_{C_0(X)}[D_\Gamma]$ and $j^G([d_{lf}])$, where $[d_{lf}]\in{\cal R} KK(X;Cl_{\Gamma}(X)\otimes_{C_0(X)}\mathfrak{S}_{lf}(X),C_0(X))$ is defined in 7.3 and used for ${\cal R} KK$-duality in theorem 7.6. We have: $\tilde a=a\otimes_{Cl_{\Gamma}(X)} [d_{lf}]$ and $[d_{lf}]\otimes_{C_0(X)} a=1_{Cl_{\Gamma}(X)}\otimes_{C_0(X)} \tilde a$. Using this, it is easy to see that both sides of the  previous displayed formula are equal to the product of $[D_\Gamma]$ and $\tilde a$ in the sense of proposition 8.2, $3^{\rm o}$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\medskip

\noindent{\it Proof of theorem 8.10.} The composition displayed in the statement of the theorem corresponds to the triple product from lemma 8.11. The isomorphism of theorem 7.11 corresponds to the product $\tilde a\otimes_{Cl_\tau(X)}[d_X]$ from lemma 8.11. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10


\section{Index theorem for elliptic operators}

The index theorem for elliptic operators is covered in great detail in section 4 of \cite{Ka16}, theorems 4.1-4.2. However, we suggest here another proof which will serve as a model for the proofs of index theorems for t-elliptic and leaf-wise elliptic operators in the next two sections.

Here is a reminder concerning the definition of the Dolbeault element (cf. \cite{Ka16}, 2.8). The manifold $TX$ is an almost complex manifold: the cotangent space $T^*_{(x,\xi)}(TX)$ at any point $(x,\xi)$ is the direct sum of the two orthogonal subspaces isomorphic to $T^*_x(X)$: the horizontal space $\tau_x^h=p^*(T^*_x(X))$ and the space dual to the vertical tangent space  $\tau_x^v=T_{\xi}(T_x(X))$. This allows to define the complex structure on the tangent space of the manifold $TX$ for any $(x,\xi)\in TX$. We will identify the complex cotangent bundle $T^*(TX)$ with the complexification of the lifted cotangent bundle of $X:\; p^*(T^*(X))\otimes {\bf C}$. The exterior algebra of this bundle is denoted by $\Lambda^*_{\bf C}(TX)$. The coordinates in the cotangent space $T^*_{(x,\xi)}(TX)$ will be denoted by $(\chi, \zeta)$. 

We define the symbol of the Dolbeault operator ${\cal D}$ as
$$\sigma_{\cal D}(x,\xi,\chi,\zeta)={\rm ext\,} (-\chi+i\zeta)+{\rm \,int\,} (-\chi+i\zeta)$$
on the exterior algebra bundle $\Lambda^*_{\bf C}(TX)$. (Note that ${\rm \,int\,}={\rm ext\,}^*$ in the Hermitian metric.) The differential operator ${\cal D}$ is essentially self-adjoint. The algebra $C_0(TX)$ acts on $L^2(\Lambda^*_{\bf C}(TX))$ by multiplication. The $K$-homology class $[{\cal D}_X]\in K^0_G(C_0(TX))$, called the Dolbeault element, is defined by the pair $(L^2(\Lambda^*_{\bf C}(TX)), {\cal D}(1+{\cal D}^2)^{-1/2})$.

\begin{theo} Let $X$ be a complete Riemannian manifold and $G$ a second countable locally compact group which acts on $X$ properly and isometrically. Let $F$ be a bounded, properly supported $G$-invariant elliptic operator on $X$ of order $0$ with the symbol $[\sigma_F]\in {\cal R} KK^G_*(X;C_0(X),C_0(TX))$. Then the formula for the index $[F]$ of this operator is 
$$[F]=[\sigma_F]\otimes_{C_0(T^*(X))} [{\cal D}_X]\in K^*_G(C_0(X)),$$
where $[{\cal D}_X]\in K^*_G(C_0(TX))$ is the Dolbeault element. 

If the group $G$ is compact and the symbol $[\sigma_F]$ belongs to $K^G_*(C_0(TX))$, then $F$ is Fredholm and defines an element of $R^*(G)=K_*^G(C_0({\rm point}))$ which is calculated by the same $KK$-product formula as above. 
\end{theo}

\noindent {\em Proof.}  Both assertions of the theorem will be proved together. The fact that under the assumptions of the second assertion the operator $F$ is Fredholm follows from the coarse PDO construction (sections 2 and 3) because $||1-\sigma^2_F(x,\xi)||\to 0$ uniformly in $\xi$ when $x\to \infty$ (cf. corollary 2.3 and theorem 3.11).

On the manifold $TX$, let ${\cal C}=\{{\cal C}_x, x\in X\}$ be the family of Bott-Dirac operators on ${\cal T}=\{L^2(\Lambda^*(\tau_x))\}$ (see lemma 7.3), namely, ${\cal C}_x=D_x(1+D_x^2)^{-1/2}$ where $D_x=(d_\xi+d^*_\xi+{\rm ext\,}(\xi)+{\rm \,int\,}(\xi))$. Each of these operators has symbol 
$({\rm ext\,}(\xi+i\zeta)+{\rm \,int\,}(\xi+i\zeta))(1+||\xi||^2+||\zeta||^2)^{-1/2}$ 
and index $1$. Here $\zeta$ is the derivation variable, $\xi$ the multiplication variable. The family $\{{\cal C}_x\}$ represents the element $1_{C_0(X)}\in KK^G(C_0(X),C_0(X))$. 
Taking the $KK$-product with $[F]\in K_G^*(C_0(X))$ 
we will get the same element $[F]\in K^*_G(C_0(X))$.

We want to prove that the $KK$-products 
$[{\cal C}]\times\hspace{-15pt}\times F$ and $\sigma_F\times\hspace{-15pt}\times [{\cal D}_X]$ give the same element of $K_G^*(C_0(X))$. Both these products are represented on the Hilbert space $H={\cal T}\otimes_{C_0(X)} L^2(E)$. Here we assume that $F$ acts on $L^2$-sections of the vector bundle $E$ over $X$. We can present the first product in the form: $(H, N_1({\cal C}\hat\otimes 1)+N_2(1\hat\otimes\tilde F))$, where $\tilde F$ is the $KK$-theoretic connection for $F$, and $N_1, N_2$ the operators entering the $KK$-product construction. According to lemma 3.14, we can take for $\tilde F$ a PDO on $TX$ with the symbol equal to the symbol of $F$ lifted to $TX$. 

The second product can be presented as $(H, N'_1\tilde{\cal D}_X+N'_2(1\hat\otimes \sigma_F))$, where $\tilde{\cal D}_X$ is the Dolbeault operator on $\{L^2(\Lambda^*(\tau_x))\}\otimes_{C_0(X)} L^2(X)$ lifted to $H$ (i.e. with the same symbol as the symbol of ${\cal D}_X$).

The homotopy that will join the two products will be the rotation homotopy which will go in the tangent spaces to $TX$ pointwise (see notation before the statement of the theorem). The rotation interchanging the derivation variable $\chi$ of $\tau_x^h$ and the multiplication variable $\xi$ of $\tau_x^v$ will be performed in $\tau_x^h\oplus \tau_x^v$.

We will present both products as pseudo-differential operators on $TX$ and produce a homotopy between their symbols. The symbol of ${\cal C} \times\hspace{-15pt}\times F$ on $TX$ can be written as 
$$N_1(({\rm ext\,}(\xi+i\zeta)+{\rm \,int\,}(\xi+i\zeta))\hat\otimes 1)/(||\xi||^2+||\zeta||^2+1)^{1/2})+N_2(1\hat\otimes\sigma_F(x,\chi))$$
where 
$$N_1^2=(||\xi||^2+||\zeta||^2+1)/(||\xi||^2+||\chi||^2+||\zeta||^2+1),\;\;N_2^2=1-N_1^2.$$
The symbol of $\sigma_F\times\hspace{-15pt}\times {\cal D}$ on $T^*(X)$ can be written as
$$N'_1(({\rm ext\,}(-\chi+i\zeta)+{\rm \,int\,}(-\chi+i\zeta))\hat\otimes 1)/(||\chi||^2+||\zeta||^2+1)^{1/2}+N'_2(1\hat\otimes\sigma_F(x,\xi))$$
where 
$$N'^2_1=(||\chi||^2+||\zeta||^2+1)/(||\xi||^2+||\chi||^2+||\zeta||^2+1),\;\;N'^2_2=1-N'^2_1.$$ 

Here $\chi$ and $\zeta$ are the derivation variables, $x$ and $\xi$ the multiplication variables. 

The conditions required by definition 3.7 are easy to check. It is also easy to show that both symbols are elliptic in the sense that for each of these two symbols $\sigma$, on compact subsets of $X$ one has: $1-\sigma^2(x,\xi,\chi,\zeta)\to 0$ when $||\xi||^2+||\chi||^2+||\zeta||^2\to 0$. The $K$-theoretic connection property and the positivity property for the products (\cite{Ka88}, 2.10, (b) and (c)) are straightforward (cf. theorem 3.11).

The homotopy which joins the two operators with the symbols written above is given by the rotation in the $(\xi,\chi)$-variables: 
$$\xi\mapsto \cos t\cdot  \xi -\sin t\cdot \chi,\;\;\chi\mapsto \sin t\cdot \xi+ \cos t\cdot \chi,$$
with $0\le t\le \pi/2$. It is again easy to see that in the course of this homotopy the symbols remain elliptic. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10


\section{Index theorems for t-elliptic operators}

Recall the definitions of the Clifford and the tangent-Clifford symbols for a t-elliptic operator $F$ with the symbol $[\sigma_F]\in {\cal R} K_*^G(X;C_0(X),\mathfrak{S}_{tr}(X))$(cf. \cite{Ka16}, definitions 8.11 and 8.13): 

\begin{defi} We define the tangent-Clifford symbol as
$$[\sigma_F^{tcl}]=[\sigma_F]\otimes_{\mathfrak{S}_{tr}(X)}[\mathfrak{f}_\Gamma]\in {\cal R} KK^G_*(X;C_0(X), Cl_\Gamma(TX)),$$
where the element $\mathfrak{f}_\Gamma\in {\cal R} KK^G(X;\mathfrak{S}_{tr}(X),Cl_{\Gamma}(TX))$ is defined before theorem 7.10. 

The Clifford symbol of a t-elliptic operator $F$ is defined as
$$[\sigma_F^{cl}]=[\sigma_F^{tcl}]\otimes_{Cl_{\Gamma}(TX)}[d_{\xi}]\in {\cal R} K^G_*(X;C_0(X), Cl_{\tau\oplus\Gamma}(X)),$$
where $[d_\xi]\in {\cal R} KK^G(X;Cl_{\Gamma}(TX),Cl_{\tau\oplus\Gamma}(X))$ is also defined before theorem 7.10.
\end{defi}

\begin{rem}{\rm In the present paper we will not use the explicit formulas for the tangent-Clifford symbol given in \cite{Ka16}, 8.13. Instead, we will use the following formula for the tangent-Clifford symbol: 
$$\sigma_F^{tcl}(x,\xi)=\sigma_F(x,\xi)\hat\otimes 1+(1-\sigma_F^2)^{1/2}\hat\otimes \mathfrak{f}_\Gamma(x,\xi)\in {\cal L}(C_0(p^*(E))\hat\otimes_{C_0(X)}Cl_{\Gamma}(X))$$
where $\mathfrak{f}_\Gamma(x,\xi)=\varphi_x(\xi)(1+q_x(\xi))^{-1/2}$. Here $p:TX\to X$ is the projection and $E$ the vector bundle where the operator $F$ acts. We assume that $||\sigma_F||\le 1$ by cutting it above that level if necessary.   

Note that $d_\xi (\sigma_F^{tcl}(x,\xi))$ vanishes at infinity in $\xi$ uniformly in $x\in X$ on compact subsets of $X$. Indeed, this is true for the $\sigma_F\hat\otimes 1$ summand because of the condition $2^{\rm o}$ of definition 7.1. Consequently, the same is true for $(1-\sigma_F^2)^{1/2}$ as well. Also it is easy to show that $||d_\xi(\mathfrak{f}_\Gamma(x,\xi))||\le {\rm const}\cdot  (1+q_x(\xi))^{-1/2}$. In view of remark 7.2, this proves our claim. 

Note also that $1-(\sigma_F^{tcl})^2=(1-\sigma_F^2)(1+q_x(\xi))^{-1}$, and using remark 7.2, we see that $C_0(X)\cdot (1-(\sigma_F^{tcl})^2)$ is contained in ${\cal K}(C_0(p^*(E)))\hat\otimes_{C_0(X)}Cl_{\Gamma}(X)$. In fact, both $(1-\sigma_F^2)(1+q_x(\xi))^{-1}$ and $(1-\sigma_F^2)\varphi_x(\xi)(1+q_x(\xi))^{-1}$ vanish at infinity in $\xi$ uniformly on compact subsets in $x$ by 7.2. 

The only essential property of $\sigma_F^{tcl}(x,\xi)$ that is missing is the continuity in $x$ uniformly in $\xi$ (condition 3.7, $1^{\rm o}$).}
\end{rem}

The basic Dirac and Dolbeault elements for the next theorem are defined in \cite{Ka16}, 8.8 and 8.17. We have already recalled the definition of the Dirac element $[d_{X,\Gamma}]$ in example 8.6 above. Here is the definition of the Dolbeault elements:

The Clifford Dolbeault element $[{\cal D}^{cl}_{X,\Gamma}]\in K^0(C^*(G,Cl_{\Gamma}(TX)))$ (\cite{Ka16}, 8.17) comes from $[d_{X,\Gamma}]$ via the ${\cal R} KK$-equivalence between $Cl_{\Gamma}(TX)$ and $Cl_{\tau\oplus\Gamma}(X)$ (see theorem 7.10). The explicit description of the element $[{\cal D}^{cl}_{X,\Gamma}]$ is as follows: Let $[{\cal D}_X]\in K^0_G(C_0(TX))$ be the Dolbeault element (see definition before theorem 9.1). We keep the Hilbert space and the operator ${\cal D}_X$ as in that definition, but extend the action of $C_0(TX)$ on $L^2(\Lambda^*_{\bf C}(TX))$ to the action of $Cl_\Gamma(TX)$. On (real) covectors of the Clifford part, this action is given by: $\chi\mapsto -({\rm ext\,} (\chi)+{\rm \,int\,} (\chi))$ (where $\chi$ is a section of $\Gamma \subset \tau=T(X)$). The resulting pair gives the element $[{\cal D}^{cl}_{X,\Gamma}]$. 

The Dolbeault element $[{\cal D}_{X,\Gamma}]\in K^0_G(C^*(G,\mathfrak{S}_\Gamma(X)))$ comes from $[{\cal D}^{cl}_{X,\Gamma}]$ via the ${\cal R} KK$-equivalence between $Cl_{\Gamma}(TX)$ and $\mathfrak{S}_\Gamma(X)$ of theorem 7.10.

\smallskip

Recall from \cite{Ka16}, 8.1, that we consider $K^*(C^*(G,C_0(X)))$ as a subgroup of $K^*_G(C^*(G,C_0(X)))$ (see the beginning of subsection 8.1 for the definition of the $G$-action on $C^*(G,C_0(X))$). Namely, a triple $(\psi:C^*(G,C_0(X))\to{\cal L}(H),T)$ representing an element of $K^*(C^*(G,C_0(X)))$ (assuming that $\psi$ is non-degenerate) defines an element of $K_G^*(C^*(G,C_0(X)))$, where the $G$-action on $H$ is induced by $\psi$. 

\begin{theo} {\rm (Index theorem 8.18 \cite{Ka16}).} Let $X$ be a complete Riemannian manifold and $G$ a Lie group which acts on $X$ properly and isometrically. Let $F$ be a properly supported, $G$-invariant, $L^2$-bounded transversally elliptic operator on $X$ of order $0$. Then 
$${\rm ind}(F)=j^G([\sigma_F^{cl}])\otimes_{C^*(G,Cl_{\tau\oplus\Gamma}(X))} [d_{X,\Gamma}]\in K^*(C^*(G,C_0(X))),$$ 
and
$${\rm ind}(F)=j^G([\sigma_F])\otimes_{C^*(G,\mathfrak{S}_{tr}(X))} [{\cal D}_{X,\Gamma}]\in K^*(C^*(G,C_0(X))),$$
and
$${\rm ind}(F)=j^G([\sigma^{tcl}_F])\otimes_{C^*(G,Cl_\Gamma(TX))} [{\cal D}^{cl}_{X,\Gamma}]\in K^*(C^*(G,C_0(X))).$$
\end{theo}

\begin{rem}{\rm When the manifold $X$ and the group $G$ are compact, as noted already in subsection 8.2, the symbol group simplifies to $K_*^G(\mathfrak{S}_{tr}(X))$, or to $K_*^G(Cl_{\tau\oplus\Gamma}(X))$, or to $K_*^G(Cl_{\Gamma}(TX))$. In this case, there exists the distributional index which is an element of $K^*(C^*(G))$. We will denote it ${\rm ind}^{\rm dist}(F)$. 

To obtain this index, one has to use the isomorphism   $K_*^G(\mathfrak{S}_{tr}(X))\simeq KK_*^G({\bf C},\mathfrak{S}_{tr}(X))$, apply the map $j^G$ and take the product with the element ${\cal D}_{X,\Gamma}$ as in the second formula of the theorem. Similarly can be modified the two other formulas of the theorem. This is a direct corollary of the theorem because the restriction map $KK_*^G(C(X),\mathfrak{S}_{tr}(X)) \to KK_*^G({\bf C},\mathfrak{S}_{tr}(X))$ for the inclusion ${\bf C}\subset C(X)$ corresponds to the restriction map $K^*(C^*(G,C_0(X)))\to K^*(C^*(G))$ for the inclusion $C^*(G)\subset C^*(G,C_0(X))$. 

In fact, it is enough to assume only that $G$ is compact, but not necessarily that $X$ is compact. Like in the case of elliptic operators (see theorem 9.1) if we assume that $[\sigma_F]$ is an element of $K_*^G(\mathfrak{S}_{tr}(X))$, then $\sigma_F^2-1\in \mathfrak{S}_{tr}(E)$. Following the coarse PDO construction of sections 2 and 3 (see in particular corollary 2.3 and theorem 3.11) and the proof of proposition 6.4 \cite{Ka16}, one can show that ${\rm ind}(F)\in K^*(C^*(G))$. This is the distributional index of $F$. The formulas of the theorem give the calculation of this distributional index as indicated above. The proof of this statement requires only minor modifications in the proof that follows.

We remark that in the case when $G$ is compact and $[\sigma_F]\in K_*^G(\mathfrak{S}_{tr}(X))$, one can use the  homomorphism $K_*^G(\mathfrak{S}_{tr}(X))\to KK^G_*(C(X^+),\mathfrak{S}_{tr}(X))$, where $X^+$ is a one-point compactification of $X$. The formulas of the theorem applied to the symbol in the latter group give the index in $K^*(C^*(G,C(X^+)))$. This index, which we will denote ${\rm Ind}(F)$ allows to obtain both ${\rm ind}(F)$ and ${\rm ind}^{\rm dist}(F)$ by restricting $C(X^+)$ to $C_0(X)$ or, respectively, to the constant functions in $C(X^+)$.}
\end{rem}

\noindent {\em Proof.}  All three formulas are equivalent. We will prove the third one. 

Let $F$ be a transversally elliptic operator of order $0$ acting on sections of the vector bundle $E$ over $X$. We will use the expression for the tangent-Clifford symbol given in remark 10.2. The product $j^G([\sigma_F^{tcl}])\otimes [{\cal D}^{cl}_{X,\Gamma}]$ is presented by the pair 
$$(C^*(G,C_0(X))\to {\cal L}({\cal E}\hat\otimes_{Cl_{\Gamma}(X)} L^2(\Lambda^*(X))),N_1(\sigma_F^{tcl}\hat\otimes 1)+ N_2 (1\hat\otimes \widetilde{{\cal D}^{cl}_{X,\Gamma}})),$$
where ${\cal E}$ is the Hilbert module of sections of the family of Hilbert spaces $\{E_x\hat\otimes Cl_{\Gamma_x}\otimes L^2(\tau_x)\}$ parametrized by  $x\in X$; $N_1, N_2$ are the operators involved in the $KK$-product construction, and $\widetilde{{\cal D}^{cl}_{X,\Gamma}}$ is the $K$-theoretic connection for ${\cal D}^{cl}_{X,\Gamma}$. All Clifford variables act over $\Lambda^*(X)$.

We will follow closely the proof of theorem 9.1. In particular, we can choose the operators $N_1,N_2$ as in the proof of theorem 9.1. These will be the PDOs with the symbols:
$$\sigma_{N_1}=||\xi||(||\xi||^2+||\chi||^2+||\zeta||^2+1)^{-1/2}, \;\sigma_{N_2}=(1-\sigma_{N_1}^2)^{1/2}.$$
Here $\xi$ is the multiplication variable, $\chi$ and $\zeta$ are the derivation variables. The variable $\xi$ is used in $\sigma_F$ and $\mathfrak{f}_\Gamma$. The variables $\chi$  and $\zeta$ are used in the symbol of ${\cal D}^{cl}_{X,\Gamma}$. 

One of the differences between the elliptic case (theorem 9.1) and the current t-elliptic one is that the symbol $\sigma_F^{tcl}$ is not a usual symbol of a PDO treated in section 3, although it has many similarities with such, as explained in remark 10.2. However, $\sigma_F^{tcl}$ is not continuous in $x$ uniformly in $\xi$, and the construction of the operator out of $\sigma_F^{tcl}$ requires the techniques of both sections 3 and 5. 

In order to deal with $\sigma_F^{tcl}$ along the rotation homotopy that will follow, we will treat the two summands of it differently. For example, $\sigma_{N_1}(\sigma_F\hat\otimes 1\hat\otimes 1)$ satisfies the conditions of definition 3.7, so this is a symbol of a PDO on the manifold $TX$. On the other hand, if the operator $(1-\sigma_F^2)^{1/2}\hat\otimes \mathfrak{f}_\Gamma\hat\otimes 1$ commutes with some other PDO on a symbol level (like $N_1$ or $N_2$), then on the operator level the commutator becomes compact after multiplication by $C_0(X)$. This follows from proposition 3.3 (where we take $h=\varphi_x, k=(1-\sigma_F^2)^{1/2}$). The conditions of proposition 3.3 are satisfied - see remark 10.2.

The very specific feature of the t-elliptic case compared to the elliptic case of theorem 9.1 is that $\mathfrak{f}_\Gamma(x,\xi)$ does not commute well with the operator $\widetilde{{\cal D}^{cl}_{X,\Gamma}}$ because these two operators use the same Clifford variables. This deficiency of commutation is repaired by the presence of the algebra $C^*(G,C_0(X))$ which essentially provides an additional multiple $(1+\Delta_G)^{-1}$ (where $\Delta_G$ is the orbital Laplacian, the operator with the symbol $1+q_x(\chi)$) killing the non-trivial commutators. 

We will use the rotation homotopy as in the proof theorem 9.1. The homotopy of all components of our operator, except $\mathfrak{f}_\Gamma$ goes as in the proof of theorem 9.1. For the homotopy of $\mathfrak{f}_\Gamma$, we use the rotation homotopy of the Lie symbol of the orbital Dirac operator $\mathfrak{D}_\Gamma$. The variable corresponding to $\xi$ for the Lie symbol will be $\eta$, as in definition 5.6. The analogous variable corresponding to $\chi$ will be called $\theta$. The rotation of the Lie symbol variables will go between $\eta$ and $\theta$.

\smallskip

Here is an important remark: 

Along this rotation homotopy, the commutator $[\mathfrak{f}_\Gamma,\widetilde{{\cal D}^{cl}_{X,\Gamma}}]$ will not be killed by the multiplication with $C^*(G,C_0(X))$. The part of the commutator that is not killed is $\cos t \sin t [\varphi_x(\xi),\xi]$ multiplied by certain other positive operators. But since $[\varphi_x(\xi),\xi]\ge 0$, the commutator $[N_1 (\sigma_F^{tcl}\hat\otimes 1),N_2 (1\hat\otimes \widetilde{{\cal D}^{cl}_{X,\Gamma}})]$ is also positive modulo compact operators after multiplication with $C^*(G,C_0(X))$. Therefore, in the process of the homotopy we get $(N_1 (A\hat\otimes 1)+ N_2 (1\hat\otimes \widetilde{{\cal D}^{cl}_{X,\Gamma}}))^2\ge 1$ modulo compact operators after multiplication by $C^*(G,C_0(X))$. So we can correct our homotopy by dividing $N_1 (A\hat\otimes 1)+ N_2 (1\hat\otimes \widetilde{{\cal D}^{cl}_{X,\Gamma}})$ by some positive invertible operator. At the end of the homotopy (at $t=\pi/2$) the non-trivial commutator again vanishes because $\sin t \cos t=0$.

\smallskip

Let us look now at the operator corresponding to the symbol $\sigma_F^{tcl}$. We write it as $T=F\hat\otimes 1+(1- F^2)^{1/2}(1\hat\otimes \mathfrak{D}_\Gamma)$. It acts on $L^2(E\hat\otimes \Lambda^*(\tau))$. We assume that $||F||\le 1$ by cutting $F$ using functional calculus. Note that the operator $F$ is by definition $G$-invariant.

According to lemma 5.15, the graded commutator $[F\hat\otimes 1, 1\hat\otimes \mathfrak{D}_\Gamma]$ is $0$. Therefore $1-T^2=(1-F^2)(1-\mathfrak{D}_\Gamma^2)$. The symbol of $1-F^2$ belongs to the subalgebra of ${\cal L}(C_0(p^*(E))\hat\otimes \Lambda^*(\tau))$ with the strictly positive element $(1+q_x(\xi))(1+||\xi||^2)^{-1}$. On the other hand, the symbol of $1-\mathfrak{D}_\Gamma^2$ belongs to the subalgebra with the strictly positive element $(1+q_x(\xi))^{-1}$. The reasoning of the proof of proposition 6.4 \cite{Ka16} shows that the operator $1-T^2=(1-F^2)(1-\mathfrak{D}_\Gamma^2)$ becomes compact after being multiplied by any function from $C_0(X)$. 

After the rotation, $\sigma_F^{tcl}(x,\chi)=\sigma_F(x,\chi)\hat\otimes 1+(1-\sigma_F^2(x,\chi))^{1/2}\hat\otimes \mathfrak{f}_\Gamma(x,\chi)$, which corresponds to the operator $T$. The Dolbeault part is transformed into the family of Bott-Dirac operators (of index $1$) ${\cal C}=\{{\cal C}_x, x\in X\}$ on $\{L^2(\Lambda^*(\tau_x))\}$, namely, ${\cal C}_x=D_x(1+D_x^2)^{-1/2}$ where $D_x=(d_\xi+d^*_\xi+{\rm ext\,}(\xi)+{\rm \,int\,}(\xi))$. We can remove the operator $\mathfrak{D}_\Gamma$ from the operator $T$ because $\mathfrak{D}_\Gamma$ is killed by multiplication with $C_0(X)\cdot (1+\Delta_G)^{-1}\in C^*(G,C_0(X))$. After that we can remove the family ${\cal C}$ as well because it corresponds to the element $1\in KK(C^*(G,C_0(X)),C^*(G,C_0(X))$. Therefore we arrive at the element ${\rm ind}(F)$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\medskip

The following definition replaces definition 8.10 \cite{Ka16} of the Clifford index of a t-elliptic operator.

\begin{defi} Let $F$ be a transversally elliptic operator with the index ${\rm ind}(F)\in K^*(C^*(G,C_0(X))$, as defined in \cite{Ka16}, proposition 6.4. We will call $[F]=[D_{\Gamma,C_0(X)}]\otimes_{C^*(G,C_0(X))} {\rm ind}(F)
\in K^*_G(\mathfrak{S}_{lf}(X))$ the analytical index of $F$. Here $[D_{\Gamma,C_0(X)}]\in KK^{G,X}(\mathfrak{S}_{lf}(X),C^*(G,C_0(X)))$ is the leaf-wise Dirac element (example 8.8).
\end{defi}

Recall that ${\cal R} KK^G_*(X;C_0(X),\mathfrak{S}_{tr}(X))\simeq {\cal R} KK^G_*(X;\mathfrak{S}_{lf}(X),C_0(TX))$, and both these groups are isomorphic to $K_G^*(\mathfrak{S}_{lf}(X))$ (theorem 7.11). In particular, we can consider the symbol $[\sigma_F]$ of a t-elliptic operator $F$ acting on sections of $E$ as an element $(C_0(p^*(E)),\sigma_F)\in {\cal R} KK^G_*(X;\mathfrak{S}_{lf}(X),C_0(TX))$. Here $E$ is a vector bundle over $X$ and $p$ the projection $TX\to X$. 

\medskip 

The following theorem replaces the Clifford index theorems 8.12 and 8.14 of \cite{Ka16}.

\begin{theo} For a t-elliptic operator $F$ on a complete Riemannian manifold $X$ with a proper isometric action of the group $G$, the symbol $[\sigma_F]\in {\cal R} KK^G_*(X;\mathfrak{S}_{lf}(X),C_0(TX))$ and the analytical index $[F]\in K^*_G(\mathfrak{S}_{lf}(X))$ are related by the isomorphism of theorem 7.11. More precisely, $[\sigma_F]\otimes_{C_0(TX)} [{\cal D}_{X}]=[F]$, where $[{\cal D}_{X}]\in K^0_G(C_0(TX))$ is the Dolbeault element.
\end{theo}

\noindent {\em Proof.}  Combine theorems 8.10 and 10.3. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\medskip

\noindent\emph{Sketch of the direct proof.} 

We refer to the proof of theorem 10.3 for the details concerning the operator $T=F\hat\otimes 1+(1- F^2)^{1/2}(1\hat\otimes \mathfrak{D}_\Gamma)$ on $L^2(E\hat\otimes \Lambda^*(\tau))$. It was shown in the proof of theorem 10.3 that $C_0(X)\cdot (1-T^2)$ is contained in compact operators. We will also use the notation of definition-lemma 7.3 and example 8.8: the Hilbert $C_0(X)$-module ${\cal T}$ and the operator ${\cal C}$.

The element $[F]$ is given by the pair $(\mathfrak{S}_{lf}(X)\to {\cal L}({\cal T}\otimes_{C_0(X)} L^2(E), A)$, where $A=N_1 ({\cal C}\hat\otimes 1)+N_2(1\hat\otimes \tilde T)$. Here $N_1, N_2$ are the operators used in the $KK$-product construction, and $\tilde T$ is the $KK$-theoretic connection for $T$.  

The situation here is very similar to the one in the proof of theorem 10.3: the role of the algebra $C^*(G,C_0(X))$ is played here by $\mathfrak{S}_{lf}(X)$. The operator $\mathfrak{D}_\Gamma$ has Clifford variables which do not commute well with the ${\rm ext\,}(\xi)+{\rm \,int\,}(\xi)$ part of the operator ${\cal C}$. However, this (graded) commutator is killed by multiplication with $\mathfrak{S}_{lf}(X)$ (see the proof of 7.3). 

We can perform a rotation homotopy very similar to the one that was used in the proofs of theorems 9.1 and 10.3. Namely, $\xi\mapsto \cos t\cdot  \xi +\sin t\cdot \chi,\;\chi\mapsto -\sin t\cdot \xi+ \cos t\cdot \chi,\, 0\le t\le \pi/2$. In the process of this homotopy there will be a non-trivial commutator between $\mathfrak{D}_\Gamma$ and ${\cal C}$, but its part related with $\xi$ will still be killed by multiplication with $\mathfrak{S}_{lf}(X)$. The part of the commutator related with $\chi$ (the derivation part) will be again `positive', and we will be able to `normalize' this homotopy. (See the proof of theorem 8.9 \cite{Ka16} for the positivity.)

At the end of the homotopy, the family of operators ${\cal C}$ will turn into the Dolbeault operator ${\cal D}_X$, and the operator $T$ will turn into the tangent-Clifford symbol $\sigma_F^{tcl}$. The $\varphi_x(\xi)(1+q_x(\xi))^{-1/2}$ part of $\sigma_F^{tcl}$ can be dropped because it is killed by multiplication with $\mathfrak{S}_{lf}(X)$. Therefore we arrive at the product $[\sigma_F]\otimes_{C_0(TX)}[{\cal D}_X]$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

 
\section{Index theorem for leaf-wise elliptic operators}

We will consider here the two cases: the one of $G$-invariant operators and the other of stabilizer-invariant operators (see assumptions 5.4) simultaneously. We will keep the superscript `$G$' (which is needed in the first case and not needed in the second one) in the notation of the $KK$-groups. 

\begin{theo} The symbol $[\sigma_F]\in {\cal R} KK^G_*(X;C_0(X),\mathfrak{S}_{lf}(X))$ and the index ${\rm ind}(F)\in KK^{G,X}_*(C_0(X),C^*(G,C_0(X)))$ of a leaf-wise elliptic operator $F$ are related by the formula: ${\rm ind}(F)=[\sigma_F]\otimes_{\mathfrak{S}_{lf}(X)} [D_{\Gamma,C_0(X)}]$, where $[D_{\Gamma,C_0(X)}]\in KK^{G,X}(\mathfrak{S}_{lf}(X),C^*(G,C_0(X)))$ is the leafwise Dirac element (see example 8.8 concerning $[D_{\Gamma,C_0(X)}]$ and proposition 8.2 concerning the $KK$-product).
\end{theo}

\noindent {\em Proof.}  Recall from 5.12 that $\mathfrak{D}_\Gamma$ is the leaf-wise operator with the space symbol $\mathfrak{f}_\Gamma(x,\xi)=\varphi_x(\xi)(1+q_x(\xi))^{-1/2}$. Also recall from example 8.8 the presentation $(\mathfrak{S}_{lf}(X)\to{\cal L}({\cal T}\hat\otimes_{C_0(X)} C^*(G,C_0(X))), {\cal C} \times\hspace{-15pt}\times \mathfrak{D}_\Gamma)$ of the element $[D_{\Gamma,C_0(X)}]$.

The product of $[\sigma_F]\in {\cal R} KK(X;C_0(X),\mathfrak{S}_{lf}(X))$ and $[D_{\Gamma,C_0(X)}]$ is given by the pair 
$$({\cal T}\hat\otimes_{C_0(X)}E\hat\otimes_{C_0(X)}C^*(G,C_0(X)), N_1({\cal C}\hat\otimes 1\hat\otimes 1)+N_2(1\hat\otimes\sigma_F\hat\otimes 1)+N_3(1\hat\otimes 1\hat\otimes \tilde \mathfrak{D}_\Gamma)).$$ 
Here $E$ is the Hilbert module where the operator $F$ acts, $\tilde \mathfrak{D}_\Gamma$ is the $KK$-theoretic connection for $\mathfrak{D}_\Gamma$, and $N_1,N_2,N_3$ are the operators used in the $KK$-product construction. 

Recall that both $F$ and $\mathfrak{D}_\Gamma$ admit a construction using the operator integration (subsection 5.4). We will apply the rotation homotopy to the operator representing the product. The space symbols $\sigma_F$ and $\sigma_{\mathfrak{D}_\Gamma}$ come from the Lie symbols 
$$\mathfrak{b}_F\in{\cal L}(E\hat\otimes_{C_0(X)}C_0(\mathfrak{G} X))\;\; {\rm and} \;\;\mathfrak{b}_{D_\Gamma}\in{\cal L}(\Lambda^*(\tau)\hat\otimes C_0(\mathfrak{G} X))$$ via the map ${\cal M}(C_0(\mathfrak{G} X))\to {\cal M}(\mathfrak{S}_{lf}(X))$. Considering $\mathfrak{b}_F\hat\otimes \mathfrak{b}_{D_\Gamma}$ as an element of the tensor product 
$${\cal L}(E\hat\otimes_{C_0(X)}\Lambda^*(\tau)\hat\otimes_{C_0(X)}C_0(\mathfrak{G} X)\hat\otimes_{C_0(X)} C_0(\mathfrak{G} X))$$
we will rotate (fiber-wise over $X$) the subspace of
$\mathfrak{G} X\times \mathfrak{G} X$ which is used for the operator integration. 

For any $x\in X$ and any $t,\,0\le t\le \pi/2$, we will choose in $\mathfrak{g}_x\times \mathfrak{g}_x$ the subspace isomorphic to $\mathfrak{g}_x$ which corresponds to the second copy of $\mathfrak{g}_x\times \mathfrak{g}_x$ turned by the angle $t$, and we will apply the operator integration construction of subsection 5.4 using this turned second copy of $\mathfrak{g}_x$. Note that there is no rotation related with the operator ${\cal C}$. 

If we interpret the product operator written at the beginning as the one coming from the symbol $N'_1({\cal C}\hat\otimes 1\hat\otimes 1)+N'_2(1\hat\otimes\sigma_F\hat\otimes 1)+N'_3(1\hat\otimes 1\hat\otimes \mathfrak{f}_\Gamma))$ by operator integration over the second copy of $\mathfrak{G} X\times \mathfrak{G} X$, then at the end of the homotopy we will get the operator coming from the same symbol by operator integration over the first copy of $\mathfrak{G} X\times \mathfrak{G} X$.

At the end of this homotopy (at $t=\pi/2$), we will get the operator $\tilde N_1({\cal C}\hat\otimes 1\hat\otimes 1)+\tilde N_2(1\hat\otimes F\hat\otimes 1)+\tilde N_3(1\hat\otimes 1\hat\otimes\mathfrak{f}_\Gamma)$. As we have already seen in the proof of theorem 7.6, the product ${\cal C}\times\hspace{-15pt}\times\, \mathfrak{f}_\Gamma$ is equal (homotopic) to ${\cal C}$, so we can remove $[\mathfrak{f}_\Gamma]$. Now we are left with the product (in the sense of proposition 8.2) of the element $[{\cal C}]=1_{C_0(X)}$ and the element ${\rm ind}(F)$. So the result is ${\rm ind}(F)$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\begin{rem} {\rm If $X$ is a (complete Riemannian) $G$-manifold and $Y\subset X$ is a complete $G$-submanifold with boundary, we can restrict a leaf-wise operator $F$ from $X$ to $Y$ and the formula of theorem 12.1 will apply to $F|_Y$ (with $[\sigma_F]$ and ${\rm ind}(F)$ considered as elements of the corresponding groups on $Y$).}
\end{rem}


\section{Examples: t-elliptic operators on ${\bf R}^n$}

This section contains a discussion of some examples of t-elliptic operators. Throughout this section, we will consider the case when $X$ is a Euclidean space ${\bf R}^{2n}$ or ${\bf C}^n$. We will assume that a compact group $G$ acts  on $X$ orthogonally, via a spin-representation (i.e via $G\to Spin(2n)\to SO(2n)$), or, in the case of $X={\bf C}^n$, via a unitary representation. 

\medskip

For t-elliptic operators in this section we will be assuming that $[\sigma_F]\in K_0^G(\mathfrak{S}_{tr}(X))$. 

\medskip

We will use the isomorphism: $K^0(C^*(G))=\widehat{R(G)}={\rm Hom} (R(G),{\bf Z})$. We consider ${\rm Hom} (R(G),{\bf Z})$ as a module over $R(G)$, and the ring $R(G)$ as a submodule of $\widehat{R(G)}$. The $R(G)$-module structure on $K^*(C^*(G,B))$ for any $G$-algebra $B$ is given by the $KK$-product with the $j^G$-image of $R(G)$ in the group $KK(C^*(G,B),C^*(G,B))$. 


\subsection{Atiyah's operators}

In his foundational lecture series \cite{Ati:TrEll}, M. Atiyah gave a very elaborate study of a class of t-elliptic operators on ${\bf R}^{2n}$. The index that he was calculating was the distributional index. We will indicate here a way of calculating the (topological) distributional index in certain examples based on theorem 10.3 and remark 10.4. 

\begin{ex} {\rm The most famous example of a t-elliptic operator for $G=S^1, X={\bf C}$ is the differential operator $\tilde\partial_{\bar z}=\partial/\partial \bar z +z:L^2({\bf C}^1)\to L^2({\bf C}^1)\otimes\Lambda^1_{\bf C}({\bf C}^1)$. We consider it as a self-adjoint operator $\tilde\partial_{\bar z}+ \tilde\partial_{\bar z}^*$ with the symbol ${\rm ext\,}(z+i\xi)+{\rm \,int\,}(z+i\xi)$ on $\Lambda_{\bf C}^*({\bf C}^1)$ (where $\xi$ is a complex variable covector). We normalize it to order $0$ by dividing its symbol by $(1+||z||^2+||\xi||^2)^{-1/2}$. Let us denote the normalized operator by $F$. Its symbol $\sigma_F$ satisfies the strong decay condition at infinity of $X$: $||\sigma_F^2-1||_{z,\xi}\le {\rm const}\cdot (1+q_z(\xi))(1+||z||^2+||\xi||^2)^{-1}$, and defines an element $[\sigma_F]\in K_0^{S^1}(\mathfrak{S}_{tr}({\bf C}^1))$ (see \cite{Ka16}, 8.21).

Let us tensor $\Lambda_{\bf C}^*({\bf C}^1)$ with another copy of $\Lambda^*_{\bf C}({\bf C}^1)$ and consider the operator $(\tilde\partial_{\bar z}+ \tilde\partial_{\bar z}^*)\hat\otimes 1$ on $L^2({\bf C}^1)\otimes\Lambda^*_{\bf C}({\bf C}^2)$. There is a homotopy of this operator given by the homotopy of its symbol to an elliptic symbol: 
$$[\cos t ({\rm ext\,} z+{\rm \,int\,} z)\hat\otimes 1+\sin t (1\hat\otimes ({\rm ext\,} z+{\rm \,int\,} z))$$
$$+({\rm ext\,}(i\xi)+{\rm \,int\,}(i\xi))\hat\otimes 1](1+|z|^2+|\xi|^2)^{-1/2}$$ 
with $0\le t\le \pi/2$. At $t=\pi/2$, we get the symbol of the Bott-Dirac operator $[1\hat\otimes ({\rm ext\,} z+{\rm \,int\,} z)+({\rm ext\,}(i\xi)+{\rm \,int\,}(i\xi))\hat\otimes 1](1+|z|^2+|\xi|^2)^{-1/2}$, which has index $1\in R(G)$.   

If we denote the basic representation of $G=S^1$ on ${\bf C}={\bf R}^2$ by $t$, then the (${\bf Z}_2$-graded) representation of $S^1$ on $\Lambda^*_{\bf C}({\bf C}^1)$ corresponds to $[\Lambda^0]-[\Lambda^1]=1-t\in R(S^1)$. Note that the index of an elliptic operator can be calculated  by the formula of theorem 9.1, or theorem 10.3, and the results will be the same. Since we tensored our initial operator $F$ by $[\Lambda^*_{\bf C}({\bf C}^1)]=1-t$, then using theorem 10.3 together with remark 10.4, we obtain: $1={\rm ind}({\cal C})=(1-t)\cdot {\rm ind}^{\rm dist}(F)\in R(S^1)$. 

Therefore, ${\rm ind}^{\rm dist}(F)=(1-t)^{-1}=\sum_{k=0}^\infty t^k \in \widehat {R(S^1)}$. This gives a calculation of the \emph{topological} distributional index for the operator $F$. (For the analytical calculation of the same ${\rm ind}^{\rm dist}(F)$, see \cite{Ka16}, example 8.21.)} 
\end{ex}

\begin{ex} {\rm The previous example can be easily generalized to $X={\bf C}^n$ with the action of $U(n)$. The operator will be again ${\cal D}_X+{\rm potential}$, where ${\cal D}_X$ is the Dolbeault operator and the potential is the operator ${\rm ext\,}(z)+{\rm \,int\,}(z)$ with $z\in \Lambda^1({\bf C}^n)={\bf C}^n$. So the symbol of this operator is ${\rm ext\,}(z+i\xi)+{\rm \,int\,}(z+i\xi)$ on $\Lambda_{\bf C}^*({\bf C}^n)$ (where $\xi$ is a complex variable covector). We again normalize it to order $0$ by dividing its symbol by $(1+||z||^2+||\xi||^2)^{-1/2}$, and denote the normalized operator by $F$.

Tensoring $\Lambda_{\bf C}^*({\bf C}^n)$ with another copy of $\Lambda_{\bf C}^*({\bf C}^n)$ and doing the same homotopy of the operator as in example 12.1, we obtain the Bott-Dirac operator of index $1$. This time the element $[\Lambda_{\bf C}^*({\bf C}^n)]=[\Lambda^{ev}]-[\Lambda^{od}]\in R(U(n))$. Its restriction to the maximal torus $T^n\subset U(n)$ is $\prod_{k=1}^n (1-t_k)$, which is invertible as an endomorphism of $\widehat {R(T^n)}$. The final answer is ${\rm ind}^{\rm dist}(F)=[\Lambda_{\bf C}^*({\bf C}^n)]^{-1}$.}
\end{ex}


\subsection{Braverman's operators}

Let us consider now Braverman's operators (see \cite{Brav}).   The general setting is the following: $G$ is a compact Lie group, $X$ is a $G$-spin complete Riemannian manifold of dimension $2n$, ${\cal D}$ is a $G$-invariant Dirac operator acting on sections of the (${\bf Z}_2$-graded) vector bundle $E$ (over $X$) which is a left module over $Cl_\tau(X)$ (see, e.g., \cite{Ka16}, before definition 3.9). This is a differential operator with the symbol $\sigma_{\cal D}(x,\xi)=c(\xi)$, where $c$ means Clifford multiplication. When normalized to an operator $F$ of order $0$, it defines an element $[F]\in K^0_G(C_0(X))$. The symbol $[\sigma_F]$ of $F$ is an element of ${\cal R} KK^G_0(X;C_0(X),C_0(TX))$, and the Clifford symbol $[\sigma^{cl}_F]$ is an element of ${\cal R} KK^G_0(X;C_0(X),Cl_\tau(X))$ (cf. \cite{Ka16}, 3.9, 3.10).

The Lie algebra of $G$ will be denoted $\mathfrak{g}$. Braverman's `taming map' is an equivariant continuous map $\nu:X\to \mathfrak{g}$ such that $||\nu(x)||\le 1$ and $||\nu(x)|| =1$ outside of a compact subset of $X$. It can be considered as a $G$-invariant section of the vector bundle $\mathfrak{g}_X$ (see subsection 5.1). Using the $G$-action on $X$ (namely, the map $f':\mathfrak{g}_X\to T(X)$), one gets a vector field $v(x)$ on $X$ (which will be used as a grading degree $1$ multiplier of $Cl_\tau(X)$). This vector field has to be rescaled by multiplying it with a certain $G$-invariant positive scalar function $f(x)$ growing at infinity (see \cite{Brav}, definition 2.6). Then ${\cal D}_{fv}={\cal D}+c(fv)$ is a Braverman type operator. Its symbol $\sigma_{{\cal D}_{fv}}(x,\xi)=c(\xi+f(x)v(x))$ is t-elliptic. When normalized to order $0$, it defines an element $[\sigma_{F_{fv}}]$ of the group $K_0^G(\mathfrak{S}_{tr}(X))$. The corresponding index element will be denoted ${\rm ind}^{\rm dist}(F_{fv})$. Note that the vector field $v$ defines an element $[v]=(E,v)\in K_0^G(C_0(X))$. 

In the case of $X={\bf R}^{2n}$, the Clifford module bundle $E$ is the product $X\times E_0$, where $E_0$ (see \cite{BGV}, 3.19) is the canonical Clifford module (isomorphic $\Lambda^*({\bf C}^n)$ when the group $G$ acts via $U(n)$). The $G$-action on $E_0$ goes through the map $G\to Spin(2n)\to {\rm End}(E_0)$. Therefore, we have the corresponding element $[E_0^{ev}]-[E_0^{od}]\in R(G)$.

\begin{prop} Let $G=Spin(2n)$ and $D_{fv}$ a Braverman type operator on ${\bf R}^{2n}$. Then $([E_0^{ev}]-[E_0^{od}])\cdot [\sigma_{F_{fv}}]=[v]\otimes_{C_0(X)} [\sigma_F]\in K_0^G(\mathfrak{S}_{tr}(X))$. Furthermore, ${\rm ind}^{\rm dist}(F_{fv})=([E_0^{ev}]-[E_0^{od}])^{-1}\cdot ([v]\otimes_{C_0(X)} [F])\in \widehat{R(G)}$.
\end{prop}

\noindent{\bf Remark.} The product $[v]\otimes_{C_0(X)} [\sigma_F]$ is defined by the following composition: $K_0^G(C_0(X))\otimes_{C_0(X)}{\cal R} KK_0^G(X;C_0(X),C_0(TX))\to K_0^G(C_0(TX))\to K_0^G(\mathfrak{S}_{tr}(X))$. The product $[v]\otimes_{C_0(X)} [F]\in R(G)$ is the usual pairing. 

\medskip

\noindent {\em Proof.}  We tensor our operator with the space $E_0$ (as in examples 12.1 and 12.2). The resulting symbol $[\sigma_{F_{fv}}\hat\otimes 1]$ is homotopic to the product $[v]\,\times\hspace{-15pt}\times \,[\sigma_F]$ by the homotopy similar to the one in examples 12.1 and 12.2. This gives the first formula. The second formula follows directly from this by the application of theorem 10.3 and remark 10.4. However, we need to invert the element $[E_0^{ev}]-[E_0^{od}]\in R(G)$. The maximal torus $T^n$ of $G=Spin(2n)$ acts on $E_0\simeq \Lambda^*({\bf C}^n)$ in a coordinate-wise manner (as in example 12.2). Therefore, the restriction of $[E_0^{ev}]-[E_0^{od}]$ to the torus $T^n$ is $\prod_{k=1}^n (1-t_k)$, which is invertible as an endomorphism of $\widehat {R(T^n)}$.
{\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10

\medskip

Proposition 12.3 resembles theorem 2.5 \cite{LRS} on the symbol level. Here is the exact analog of that theorem on the level of symbols for an arbitrary complete Riemannian manifold $X$. We consider the general setting described at the beginning of this subsection. Recall from \cite{LRS} that the vector field $v$ defines, in fact, an element $[v]\in K_0^G(Cl_{\Gamma}(X))$. The Clifford symbol $[\sigma_F^{cl}]$ of the (normalized) Dirac operator $F$ is an element of ${\cal R} KK^G(X;C_0(X),Cl_\tau(X))$ (see \cite{Ka16}, 3.8, 3.9, 3.10). It can be mapped into ${\cal R} KK^G(X;Cl_{\Gamma}(X),Cl_{\tau\oplus\Gamma}(X))$ by tensoring it with $1_{Cl_{\Gamma}(X)}$. The Bott element $[{\cal B}_{\xi,\Gamma}]\in {\cal R} KK^G(X;Cl_{\tau\oplus\Gamma}(X),\mathfrak{S}_{tr}(X))$ is defined before theorem 7.10. 

\begin{prop} The symbol $[\sigma_{F_{fv}}]$ of the Braverman operator is equal to the triple product: $[v]\otimes_{C_0(X)}[\sigma_F^{cl}]\otimes_{C_0(X)}[{\cal B}_{\xi,\Gamma}]$.
\end{prop}

\noindent {\em Proof.}  Let $E$ be the Clifford module bundle where the operator ${\cal D}$ acts. Using the calculation of $[\sigma_F^{cl}]$ in \cite{Ka16}, 3.9, 3.10, we easily see that the Hilbert module for the triple product is the image of the projection $P$ of \cite{Ka16}, 3.9, on $E\hat\otimes_{C_0(X)}\Lambda^*(X)$. But $\Lambda^*(X)\simeq E\hat\otimes_{C_0(X)}E$, so the image of $P$ is just $E$. The Bott operator $\beta$ acts on this $E$ as $c(\xi)(1+||\xi||^2)^{-1/2}$, the element $[v]$ acts as Clifford multiplication $c(v(x))$, so $[v]\times\hspace{-15pt}\times [\beta]$ is obviously $[\sigma_{F_{fv}}]$. {\vbox{\hrule height 5pt width 5pt}\quad}\penalty -10


