
\section{Examples}\label{sec:example}

\subsection[]{}\label{subsec:general-linear-group}
$G=GL_n$ is a connected smooth reductive $O$-group scheme which
satisfies the conditions I), II) and III) of the subsection 
\ref{subsec:fundamental-setting}. 

\begin{prop}\label{prop:congruent-beta-for-gl-n}
If 
$\beta\equiv\lambda\cdot 1_n+\beta_0\npmod{\frak{p}^{l^{\prime}}}$ 
for $\beta, \beta_0\in\frak{gl}_n(O)$ with $\lambda\in O$, then there
  exists a group homomorphism 
$\mu:O_r^{\times}\to\Bbb C^{\times}$ such that 
$$
 \pi\mapsto(\mu\circ\det)\otimes\pi
$$
gives a bijection of $\text{\rm Irr}(G(O_r)\mid\psi_{\beta_0})$ onto 
$\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$.
\end{prop}
\begin{proof}
Let $\mu:O_r^{\times}\to\Bbb C^{\times}$ be a group homomorphism such
that $\mu((1+\varpi^l)_rx)=\tau(\varpi^{-l^{\prime}}\lambda x)$ for all 
$x\in O$. Then we have 
$$
 \psi_{\beta}(h)=\mu\circ\det(h)\cdot\psi_{\beta_0}(h)
$$
for all 
$h=1_n+\varpi^lX\npmod{\frak{p}^r}\in K_l(O_r)$ ($X\in
M_n(O)$), because
$$
 \det(1_n+\varpi^lX)\equiv 1+\varpi^l\text{\rm tr}(X)
 \npmod{\frak{p}^r}.
$$
\end{proof}

A similar argument shows 

\begin{prop}\label{prop:central-beta-for-gl-n}
If $\beta\in\frak{gl}_n(O)$ is central modulo $\frak{p}$, then there
exists a group homomorphism $\mu:O_r^{\times}\to\Bbb C^{\times}$ such
that, for any 
$\pi\in\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$, there exists a 
$\sigma\in\text{\rm Irr}(G(O_{r-1}))$ such that 
$\pi=(\mu\circ\det)\otimes(\sigma\circ\text{\rm proj})$ where 
$\text{\rm proj}:G(O_r)\to G(O_{r-1})$ is the canonical surjection.
\end{prop}
\begin{proof}
Put $\beta\equiv\lambda\cdot 1_n\npmod{\frak p}$ with 
$\lambda\in O$. 
Let $\mu:O_r^{\times}\to\Bbb C^{\times}$ be a group homomorphism such
that $\mu((1+\varpi^l)_rx)=\tau(\varpi^{-l^{\prime}}\lambda x)$ for all 
$x\in O$. 
Then, for any $\pi\in\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$, the
representation $(\mu\circ\det)^{-1}\otimes\pi$ of $G(O_r)$ is trivial
on $K_{r-1}(O_r)$, in other word, it factors through $G(O_{r-1})$.
\end{proof}

If $\beta\in\frak{gl}_n(O)$ is
smoothly regular with respect to $GL_n$ over $\Bbb F$, then 
$G_{\beta}$ is a smooth commutative $O$-group scheme (see Remark 
\ref{remark:smoothly-regular-in-gl-n}). If further 
$\overline\beta\in\frak{gl}_n(\Bbb F)$ is semisimple, 
then $c_{\beta,\rho}(g,h)=1$ for all $g,h\in G_{\beta}(\Bbb F)$ and 
$\rho\in\frak{gl}_{n,\beta}(\Bbb F)\sphat$ 
(see Remark
\ref{remark:v-g-is-trivial-if-trace-form-is-non-deg-on-g-beta}). 
In this case Theorem \ref{th:main-result} gives

\begin{prop}\label{prop:regular-semisimple-case-for-gl-n}
There exists a bijection 
$\theta\mapsto
 \text{\rm Ind}_{G(O_r,\beta)}^{G(O_r)}\sigma_{\beta,\theta}$ of the
 set 
$$
 \left\{\theta\in G_{\beta}(O_r)\sphat\;\;\;
         \text{\rm s.t. $\theta=\psi_{\beta}$ on 
               $G_{\beta}(O_r)\cap K_l(O_r)$}\right\}
$$
onto $\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$.
\end{prop}

There are $\frac 1n\sum_{d|n}\mu\left(\frak nd\right)\cdot q^d$ 
irreducible polynomials in $\Bbb F[t]$ of degree $n$ ($\mu$ is the
M\"obius function). Take a polynomial 
$$
 p(t)=t^n+a_1t^{n-1}+\cdots+a_{n-1}t+a_n\in O[t]
$$ 
such that $p(t)\npmod{\frak p}\in\Bbb F[t]$ is
irreducible. Then 
$$
 [p(t)]=\left\{\beta\in\frak{gl}_n(O)\mid 
           \chi_{\beta}(t)\equiv p(t)\npmod{\frak p}\right\}
$$
is stable under the adjoint action of $GL_n(O)$. 
Take a $\beta\in [p(t)]$. Then  
$K=F[\beta]$ is an unramified field extension of $F$ of degree $n$,
which is the splitting field of $p(t)$ over $F$, and
$O_K=O_F[\beta]$ is the integer ring of $K$ with the maximal ideal 
$\frak{p}_K=\varpi O_K$. If we identify 
$G_{\beta}(O_r)$ with $\left(O_K/\frak{p}_K^r\right)^{\times}$, then we have
$$
 G_{\beta}(O_r)\cap K_{l^{\prime}}(O_r)
 =(1+\frak{p}_K^{l^{\prime}})/(1+\frak{p}_K^r)
 \hookrightarrow
 \left(O_K/\frak{p}_K^r\right)^{\times}
$$
and 
$\psi_{\beta}\left((1+\varpi^lx)_r\right)
 =T_{K/F}(\varpi^{-l^{\prime}}x\beta)$ ($x\in O_K$). 
So Proposition \ref{prop:regular-semisimple-case-for-gl-n} gives

\begin{prop}\label{prop:shintani-gerardin-parametrization}
There exists a bijection 
$\theta\mapsto
 \text{\rm Ind}_{G(O_r,\beta)}^{G(O_r)}\sigma_{\beta,\theta}$ of the
 set 
$$
 \left\{\begin{array}{l}
        \theta:O_K^{\times}\to\left(O_K/\frak{p}_K^r\right)^{\times}
               \to\Bbb C^{\times}:\text{\rm group homomorphism}\\
        \text{\rm such that 
  $\theta(1+\varpi^lx)=T_{K/F}(\varpi^{-l^{\prime}}\beta x)
   \;\;
   \forall x\in O_K$}
         \end{array}\right\}
$$
onto $\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$.
\end{prop}

This kind of parametrization of 
$\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$ is given first by Shintani 
\cite{Shintani1968} and then G\'erardin \cite{Gerardin1972}.

\subsection[]{}\label{subsec:group-of-symplectic-similitude}
Let $G=GSp_{2n}$ be the $O$-group scheme such that
$$
 GSp_{2n}(L)=\{g\in GL_{2n}(L)\mid gJ_n\,^t\!g=\nu(g)\cdot J_n
                \;\text{\rm with $\nu(g)\in L^{\times}$}\}
$$
 ($J_n=\begin{bmatrix}
        0&1_n\\
       -1_n&0
       \end{bmatrix}$) 
for all commutative $O$-algebra $L$. Then $G$ is a connected smooth
reductive $O$-group scheme. The Lie algebra $\frak{gsp}_{2n}$ of $G$ 
is an affine $O$-subscheme of $\frak{gl}_{2n}$ such that
$$
 \frak{gsp}_{2n}(L)=\{X\in\frak{gl}_{2n}(L)\mid
                        XJ_n+J_n\,^t\!X=\nu(X)\cdot J_n\;
                      \text{\rm with $\nu(X)\in L$}\}
$$
for all commutative $O$-algebra $L$. 
Assume that the characteristic of $\Bbb F$ does not divide $n$. Then
$G$ satisfies the conditions I), II) and III) of the subsection 
\ref{subsec:fundamental-setting}. The same arguments as Propositions 
\ref{prop:congruent-beta-for-gl-n} and \ref{prop:central-beta-for-gl-n} 
show

\begin{prop}\label{prop:congruent-beta-for-gsp-2n}
If 
$\beta\equiv\lambda\cdot 1_n+\beta_0\npmod{\frak{p}^{l^{\prime}}}$ 
for $\beta, \beta_0\in\frak{gsp}_{2n}(O)$ with $\lambda\in O$, then there
  exists a group homomorphism 
$\mu:O_r^{\times}\to\Bbb C^{\times}$ such that 
$$
 \pi\mapsto(\mu\circ\det)\otimes\pi
$$
gives a bijection of $\text{\rm Irr}(G(O_r)\mid\psi_{\beta_0})$ onto 
$\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$.
\end{prop}

and

\begin{prop}\label{prop:central-beta-for-gsp-2n}
If $\beta\in\frak{gsp}_{2n}(O)$ is central modulo $\frak{p}$, then there
exists a group homomorphism $\mu:O_r^{\times}\to\Bbb C^{\times}$ such
that, for any 
$\pi\in\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$, there exists a 
$\sigma\in\text{\rm Irr}(G(O_{r-1}))$ such that 
$\pi=(\mu\circ\det)\otimes(\sigma\circ\text{\rm proj})$ where 
$\text{\rm proj}:G(O_r)\to G(O_{r-1})$ is the canonical surjection.
\end{prop}

If $\beta\in\frak{gsp}_{2n}(O)$ is
smoothly regular with respect to $G$ over $F$ and over $\Bbb F$, then 
$G_{\beta}$ is a smooth commutative $O$-group scheme. If further 
$\overline\beta\in\frak{gsp}_{2n}(\Bbb F)$ is semisimple, 
then $c_{\beta,\rho}(g,h)=1$ for all 
$g,h\in G_{\beta}(\Bbb F)^{(c)}=G_{\beta}(\Bbb F)$ and 
$\rho\in\frak{gsp}_{2n,\beta}(\Bbb F)\sphat$
(see Remark
\ref{remark:v-g-is-trivial-if-trace-form-is-non-deg-on-g-beta}). 
In this case Theorem \ref{th:main-result} gives

\begin{prop}\label{prop:regular-semisimple-case-for-gsp-2n}
There exists a bijection 
$\theta\mapsto
 \text{\rm Ind}_{G(O_r,\beta)}^{G(O_r)}\sigma_{\beta,\theta}$ of the
 set 
$$
 \left\{\theta\in G_{\beta}(O_r)\sphat\;\;\;
         \text{\rm s.t. $\theta=\psi_{\beta}$ on 
               $G_{\beta}(O_r)\cap K_l(O_r)$}\right\}
$$
onto $\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$.
\end{prop}

Let us consider a $\beta\in\frak{gsp}_{2n}(O)\subset\frak{gl}_{2n}(O)$
such that the characteristic polynomial $\chi_{\beta}(t)\in O[t]$ is
irreducible modulo $\frak{p}$. In this case $F[\beta]$ is unramified
field extension of $F$ of degree $2n$ such that $O[\beta]$ is the
integer ring of $F[\beta]$ so that $\beta\in O[\beta]^{\times}$. 

Now fix an unramified field extension $K$ of $F$ of degree $2n$. Let 
$O_K$ be the integer ring of $K$ with the maximal ideal 
$\frak{p}_K=\varpi O_K$. The finite field $\Bbb F=O/\frak{p}$ is
identified with a subfield of $\Bbb K=O_K/\frak{p}_K$. Then $K/F$ is a
Galois extension whose Galois group $\text{\rm Gal}(K/F)$ is
isomorphic to $\text{\rm Gal}(\Bbb K/\Bbb F)$ by the mapping which
sends $\sigma\in\text{\rm Gal}(K/F)$ to 
$\overline\sigma\in\text{\rm Gal}(\Bbb K/\Bbb F)$ where 
$(x\npmod{\frak{p}_K})^{\overline\sigma}=x^{\sigma}\npmod{\frak{p}_K}$. 

Let $\tau\in\text{\rm Gal}(K/F)$ be the unique element of order $2$. 
Fix an $\varepsilon\in O_K^{\times}$ such that 
$\varepsilon^{\tau}+\varepsilon=0$. Then the $F$-vector space $K$ is a
symplectic $F$-space with respect to the non-degenerate alternating
form 
$$
 D_{\varepsilon}(x,y)=T_{K/F}(\varepsilon xy^{\tau})
 \qquad
 (x,y\in K)
$$
on $K$ with a polarization $K=K_-\oplus K_+$ where 
$$
 K_{\pm}=\{x\in K\mid x^{\tau}=\pm x\},
$$
and $F\subset K_+\subset K$ is a subfield such that $(K_+:F)=n$. 
Let $\{v_1,\cdots,v_n\}$ be a $O$-basis of $O_{K_+}=O_K\cap K_+$ the
integer ring of $K_+$. Since $K/F$ is unramified, we have a $O$-basis
of $O_{K_+}$ $\{v_1^{\ast},\cdots,v_n^{\ast}\}$ such that 
$T_{K_+/F}(v_iv_j^{\ast})=\delta_{ij}$. Put 
$u_i=\varepsilon^{-1}v_i^{\ast}\subset K_-$. Then 
$\{u_1,\cdots,u_n,v_1,\cdots,v_n\}$ 
is a symplectic $F$-basis of $K=K_-\oplus K_+$ and an $O$-basis of 
$O_K=\varepsilon^{-1}O_{K_+}\oplus O_{K_+}$. 

Now our $O$-group scheme $G=GSp_{2n}$ and its Lie algebra 
$\frak{gsp}_{2n}$ are defined by
$$
 GSp_{2n}(L)=\left\{g\in GL_L(O_K{\otimes}L)\biggm|
                    \begin{array}{l}
                    D_{\varepsilon}(gx,gy)=\nu(g)D_{\varepsilon}(x,y)\\
                    \text{\rm for $\forall x,y\in O_K{\otimes}_OL$ with 
                              $\nu(g)\in L^{\times}$}
                    \end{array}\right\}
$$
and by
$$
 \frak{gsp}_{2n}(L)
  =\left\{X\in\text{\rm End}_L(O_K{\otimes}L)\biggm|
           \begin{array}{l}
            D_{\varepsilon}(Xx,y)+D_{\varepsilon}(x,Xy)
                       =\nu(X)D_{\varepsilon}(x,y)\\
            \text{\rm for $\forall x,y\in O_K{\otimes}_OL$ 
                               with $\nu(X)\in L$}
           \end{array}\right\}
$$
for all commutative $O$-algebra $L$. 

Take a $\beta\in K_-\cap O_K$ such that $O_K=O[\beta]$. Identify 
$\beta\in K$ with the element $x\mapsto x\beta$ of 
$\frak{gsp}_{2n}(O)\subset\frak{gl}_{2n}(O)$. 
Then the characteristic polynomial $\chi_{\beta}(t)\in O[t]$ of 
$\beta\in\frak{gl}_{2n}(O)$ is irreducible modulo $\frak{p}$. We have 
$$
 G_{\beta}(O_r)=G(O_r)\cap\left(O_K/\frak{p}_K^r\right)^{\times}
               =\{\gamma\npmod{\frak{p}_K^r}\mid
                    \gamma\in U_{K/F}\}
$$
where
$$
 U_{K/F}=\{\gamma\in O_K^{\times}\mid 
            \gamma\cdot\gamma^{\tau}\in O^{\times}\}
$$
is a subgroup of $O_K^{\times}$. In this case we have 
$\psi_{\beta}(h)=\tau(\varpi^{-l^{\prime}}T_{K/F}(\beta x))$ for all 
$$
 h=1+\varpi^lx\npmod{\frak{p}_K^r}\in K_l(O_r)\cap G_{\beta}(O_r)
 \subset\left(O_K/\frak{p}_K^r\right)^{\times}.
$$
Then Proposition \ref{prop:regular-semisimple-case-for-gsp-2n} gives 

\begin{prop}\label{prop:shintani-gerardin-type-parametrization-for-gsp-2n}
There exists a bijection 
$\theta\mapsto
 \text{\rm Ind}_{G(O_r,\beta)}^{G(O_r)}\sigma_{\beta,\theta}$ of the
 set 
$$
 \left\{\begin{array}{l}
        \theta:U_{K/F}\to\left(O_K/\frak{p}_K^r\right)^{\times}
               \to\Bbb C^{\times}:\text{\rm group homomorphism}\\
        \text{\rm such that 
  $\theta(\gamma)
   =\tau(\varpi^{-l^{\prime}}T_{K/F}(\varpi^{-l^{\prime}}\beta x))
   \;\;
   \forall\gamma=1+\varpi^lx\in U_{K/F}$}
         \end{array}\right\}
$$
onto $\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$.
\end{prop}



\subsection[]{}\label{subsec:general-orthogonal-group}
Take a $S\in M_n(O)$ such that $^tS=S$ and $\det S\in O^{\times}$. 
Let $G=GO(S)$ be the $O$-group scheme such tha
$$
 G(L)=\{g\in GL_n(L)\mid gS^t\!g=\nu(g)\cdot S\;
          \text{\rm with $\nu(g)\in L^{\times}$}\}
$$
for all commutative $O$-algebra $L$. Then $G$ is a connected smooth
reductive $O$-group scheme. The Lie algebra $\frak{g}=\frak{go}(S)$ of
$G$ is an affine $O$-subscheme of $\frak{gl}_n$ such that
$$
 \frak{g}(L)=\{X\in\frak{gl}_n(L)\mid 
                 XS+S^t\!X=\nu(X)\cdot S\;
                 \text{\rm with $\nu(X)\in L$}\}
$$
for all commutative $O$-algebra $L$. 
Assume that the characteristic of $\Bbb F$ does not divide $n$. Then
$G$ satisfies the conditions I), II) and III) of the subsection 
\ref{subsec:fundamental-setting}. 
The same arguments as Propositions 
\ref{prop:congruent-beta-for-gl-n} and \ref{prop:central-beta-for-gl-n} 
show

\begin{prop}\label{prop:congruent-beta-for-go(s)}
If 
$\beta\equiv\lambda\cdot 1_n+\beta_0\npmod{\frak{p}^{l^{\prime}}}$ 
for $\beta, \beta_0\in\frak{g}(O)$ with $\lambda\in O$, then there
  exists a group homomorphism 
$\mu:O_r^{\times}\to\Bbb C^{\times}$ such that 
$$
 \pi\mapsto(\mu\circ\det)\otimes\pi
$$
gives a bijection of $\text{\rm Irr}(G(O_r)\mid\psi_{\beta_0})$ onto 
$\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$.
\end{prop}

and

\begin{prop}\label{prop:central-beta-for-go(s)}
If $\beta\in\frak{g}(O)$ is central modulo $\frak{p}$, then there
exists a group homomorphism $\mu:O_r^{\times}\to\Bbb C^{\times}$ such
that, for any 
$\pi\in\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$, there exists a 
$\sigma\in\text{\rm Irr}(G(O_{r-1}))$ such that 
$\pi=(\mu\circ\det)\otimes(\sigma\circ\text{\rm proj})$ where 
$\text{\rm proj}:G(O_r)\to G(O_{r-1})$ is the canonical surjection.
\end{prop}

If $\beta_s\in\frak{g}(L)$ ($L=F$ or $L=\Bbb F$) is semisimple, then the
centralizer $Z_{G{\otimes}_OL}(\beta_s)$ is connected and its center is
also connected. Hence if $\beta\in\frak{g}(O)$ is smoothly regular
over $F$ and over $\Bbb F$, then $G_{\beta}$ is a smooth commutative
$O$-group scheme. If further $\overline\beta\in\frak{g}(\Bbb F)$ is
semisimple, then $c_{\beta,\rho}(g,h)=1$ for all 
$g,h\in G_{\beta}(\Bbb F)^{(c)}=G_{\beta}(\Bbb F)$ and all 
$\rho\in\frak{g}_{\beta}(\Bbb F)\sphat$ (see Remark 
\ref{remark:v-g-is-trivial-if-trace-form-is-non-deg-on-g-beta}). 
In this case Theorem \ref{th:main-result} gives

\begin{prop}\label{prop:regular-semisimple-case-for-go(s)}
There exists a bijection 
$\theta\mapsto
 \text{\rm Ind}_{G(O_r,\beta)}^{G(O_r)}\sigma_{\beta,\theta}$ of the
 set 
$$
 \left\{\theta\in G_{\beta}(O_r)\sphat\;\;\;
         \text{\rm s.t. $\theta=\psi_{\beta}$ on 
               $G_{\beta}(O_r)\cap K_l(O_r)$}\right\}
$$
onto $\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$.
\end{prop}

Suppose $n=2m$ is even. Let $K$ be an unramified field extension of
$F$ of degree $n$ with the integer ring $O_K$. Then 
$\frak{p}_K=\varpi O_K$ is the maximal ideal of $O_K$, and 
$\Bbb F=O/\frak{p}$ is canonically identified with a subfield of 
$\Bbb K=O_K/\frak{p}_K$. Let $\tau\in\text{\rm Gal}(K/F)$ be the
unique element of order $2$. Fix an $\varepsilon\in O_K^{\times}$ such
that $\varepsilon^{\tau}=\varepsilon$ and put
$$
 S_{\varepsilon}(x,y)=T_{K/F}(\varepsilon xy^{\tau})
 \qquad
 (x,y\in K).
$$
Then $S$ is a regular $F$-quadratic form on $K$. 
Fix a $O$-basis $\{u_1,\cdots,u_n\}$  of $O_K$ and put 
$B=\left(u_i^{\sigma_j}\right)_{1\leq i,j\leq n}\in M_n(O_K)$ where 
$\text{\rm Gal}(K/F)=\{\sigma_1,\cdots,\sigma_n\}$. Then we have 
$$
 \left(S_{\varepsilon}(u_i,u_j)\right)_{1\leq i,j\leq n}
 =B\begin{bmatrix}
     \varepsilon^{\sigma_1}&      &                      \\
                           &\ddots&                      \\
                           &      &\varepsilon^{\sigma_n}
   \end{bmatrix}\,^tB^{\tau}
$$ 
so that the discriminant of the quadratic form $S_{\varepsilon}$ is
$$
 \det(S_{\varepsilon}(u_i,u_j))_{1\leq i,j\leq n}=
 (-1)^m(\det B)^2N_{K/F}(\varepsilon). 
$$
Note that $(\det B)^{\sigma}=-\det B$ for a generator $\sigma$ of 
$\text{\rm Gal}(K/F)$. Since $K/F$ is unramified extension,
its discriminant $D(K/F)=(\det B^2)$ is trivial, in other word 
$\det(S_{\varepsilon}(u_i,u_j))_{1\leq i,j\leq n}\in O^{\times}$. So
the $O$-group 
scheme $G=GO(S_{\varepsilon})$ and its Lie algebra
$\frak{g}=\frak{go}(S_{\varepsilon})$ is 
defined by 
$$
 G(L)=\left\{g\in GL_L(O_K{\otimes}L)\biggm|
                    \begin{array}{l}
                    S_{\varepsilon}(xg,yg)=\nu(g)S_{\varepsilon}(x,y)\\
                    \text{\rm for $\forall x,y\in O_K{\otimes}_OL$ with 
                              $\nu(g)\in L^{\times}$}
                    \end{array}\right\}
$$
and by 
$$
 \frak{g}(L)
  =\left\{X\in\text{\rm End}_L(O_K{\otimes}L)\biggm|
           \begin{array}{l}
            S_{\varepsilon}(xX,y)+S_{\varepsilon}(x,yX)
                       =\nu(X)S_{\varepsilon}(x,y)\\
            \text{\rm for $\forall x,y\in O_K{\otimes}_OL$ 
                               with $\nu(X)\in L$}
           \end{array}\right\}
$$
for all commutative $O$-algebra $L$. Note that $\text{\rm End}_F(K)$
acts on $K$ from the right side.

Take a $\beta\in O_K$ such that $O_K=O[\beta]$ and $\beta^{\tau}=-\beta$. 
Identify $\beta\in K$ with the element $x\mapsto x\beta$ of 
$\frak{g}(O)\subset\text{\rm End}_O(O_K)=\frak{gl}_{2n}(O)$. 
Then the characteristic polynomial $\chi_{\beta}(t)\in O[t]$ of 
$\beta\in\frak{gl}_{2n}(O)$ is irreducible modulo $\frak{p}$. We have 
$$
 G_{\beta}(O_r)=G(O_r)\cap\left(O_K/\frak{p}_K^r\right)^{\times}
               =\{\gamma\nnpmod{\frak{p}_K^r}\mid
                    \gamma\in U_{K/F}\}
$$
where
$$
 U_{K/F}=\{\gamma\in O_K^{\times}\mid 
            \gamma\cdot\gamma^{\tau}\in O^{\times}\}
$$
is a subgroup of $O_K^{\times}$. In this case we have 
$\psi_{\beta}(h)=\tau(\varpi^{-l^{\prime}}T_{K/F}(\beta x))$ for all 
$$
 h=1+\varpi^lx\npmod{\frak{p}_K^r}\in K_l(O_r)\cap G_{\beta}(O_r)
 \subset\left(O_K/\frak{p}_K^r\right)^{\times}.
$$
Then Proposition \ref{prop:regular-semisimple-case-for-go(s)} gives 

\begin{prop}
\label{prop:shintani-gerardin-type-parametrization-for-go(s)-of-even-n}
There exists a bijection 
$\theta\mapsto
 \text{\rm Ind}_{G(O_r,\beta)}^{G(O_r)}\sigma_{\beta,\theta}$ of the
 set 
$$
 \left\{\begin{array}{l}
        \theta:U_{K/F}\to\left(O_K/\frak{p}_K^r\right)^{\times}
               \to\Bbb C^{\times}:\text{\rm group homomorphism}\\
        \text{\rm such that 
  $\theta(\gamma)
   =\tau(\varpi^{-l^{\prime}}T_{K/F}(\varpi^{-l^{\prime}}\beta x))
   \;\;
   \forall\gamma=1+\varpi^lx\in U_{K/F}$}
         \end{array}\right\}
$$
onto $\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$.
\end{prop}

Let us consider the case of $n=2m+1$ being odd. In this case 
$\det\beta=0$ for all $\beta\in\frak{g}(O)$ so that the characteristic
polynomial $\chi_{\beta}(t)\in O[t]$ of $\beta$ is of the form 
$\chi_{\beta}(t)=t\cdot p(t)$ with $p(t)\in O[t]$. We will give
examples in which $p(t)$ is irreducible modulo $\frak{p}$. 

Let us use the notations used in the case of even $n$ and fix an 
$\eta\in O^{\times}$. Then $F$-quadratic form on $V=K\times F$ is
defined by 
$$
 S_{\varepsilon,\eta}((x,s),(y,t))
 =S_{\varepsilon}(x,y)+\eta\cdot st.
$$
Then the $O$-group scheme $G=GO(S_{\varepsilon,\eta})$ and its Lie
algebra $\frak{g}=\frak{go}(S_{\varepsilon,\eta})$ is defined by
$$
 G(L)=\left\{g\in GL_L((O_K\times O){\otimes}L)\biggm|
                    \begin{array}{l}
                    S_{\varepsilon,\eta}(ug,vg)
                         =\nu(g)S_{\varepsilon,\eta}(u,v)
                    \;\;\text{\rm for}\\
                    \text{\rm $\forall u,v\in(O_K\times O){\otimes}_OL$ with 
                              $\nu(g)\in L^{\times}$}
                    \end{array}\right\}
$$
and
\begin{align*}
 &\frak{g}(L)\\
 &=\left\{X\in\text{\rm End}_L((O_K\times O){\otimes}L)\biggm|
           \begin{array}{l}
            S_{\varepsilon,\eta}(uX,v)+S_{\varepsilon,\eta}(u,vX)
                       =\nu(X)S_{\varepsilon,\eta}(u,v)\\
            \text{\rm for $\forall u,v\in(O_K\times O){\otimes}_OL$ 
                               with $\nu(X)\in L$}
           \end{array}\right\}
\end{align*}
for all commutative $O$-algebra $L$. An element 
$X\in\text{\rm End}_L((O_K\times O){\otimes}_OL)$ is denoted by
$$
 X=\begin{bmatrix}
    A&B\\
    C&D
   \end{bmatrix}\;\text{\rm with}\;
 \begin{cases}
  A\in\text{\rm End}_L(O_K{\otimes}_OL),
    &B\in\text{\rm Hom}_L(O_K{\otimes}_OL,L),\\
  C\in\text{\rm Hom}_L(L,O_K{\otimes}_OL),
    &D\in\text{\rm End}_L(L)=L.
 \end{cases}
$$
Take a $\beta_0\in O_K^{\times}$ such that $O_K=O[\beta_0]$ and 
$\beta_0^{\tau}=-\beta_0$. Then the element 
$(x,s)\mapsto(x\beta_0,0)$ of $\frak{g}(O)$ is denoted by
$$
 \beta=\begin{bmatrix}
        \beta_0&0\\
           0   &0
       \end{bmatrix}\in\frak{g}(O)
      \subset\text{\rm End}_O(O_K\times O),
$$
where $\beta_0\in O_K^{\times}$ is identified with 
$x\mapsto x\beta_0$ an element of $\text{\rm End}_O(O_K)$.  Now the
characteristic polynomial of 
$\beta\in\frak{g}(O)\subset\text{\rm End}_O(O_K\times O)$ is 
$\chi_{\beta}(t)=t\cdot\chi_{\beta_0}(t)$ where 
$\chi_{\beta_0}(t)\in O[t]$ is the characteristic polynomial of 
$\beta_0\in\text{\rm End}_O(O_K)$ which is irreducible modulo
$\frak{p}$. In particular $\beta\npmod{\frak{p}}\in\frak{g}(\Bbb F)$
is semisimple. We have
\begin{align*}
 G_{\beta}(O_r)
 &=G(O_r)\cap O_r[\beta\npmod{\frak{p}^r}]^{\times}\\
 &=\left\{\begin{bmatrix}
           \gamma\npmod{\frak{p}_K^r}&0\\
           0&a\npmod{\frak{p}^r}
          \end{bmatrix}\biggm|
           \begin{array}{l}
            \gamma\in U_{K/F},\;a\in O^{\times}\\
            \gamma\gamma^{\tau}\equiv a^2\npmod{\frak{p}^r}
           \end{array}
          \right\}.
\end{align*}
In this case 
$\psi_{\beta}(h)=\tau\left(\varpi^{-l^{\prime}}T_{K/F}(\beta_0x)\right)$
for all
$$
 h=\begin{bmatrix}
    1+\varpi^lx\npmod{\frak{p}_K^r}&0\\
    0&1+\varpi^ly\npmod{\frak{p}^r}
   \end{bmatrix}
 \in K_l(O_r)\cap G_{\beta}(O_r).
$$
The Proposition \ref{prop:regular-semisimple-case-for-go(s)} gives

\begin{prop}
\label{prop:shintani-gerardin-type-parametrization-for-go(s)-of-odd-n}
There exists a bijection 
$\theta\mapsto
 \text{\rm Ind}_{G(O_r,\beta)}^{G(O_r)}\sigma_{\beta,\theta}$ of the
 set 
$$
 \left\{\begin{array}{l}
        \theta:U_{K/F}{\times}_{\frak{p}^r}O^{\times}
          \to\left(O_K/\frak{p}_K^r\right)^{\times}O_r^{\times}
               \to\Bbb C^{\times}\\
    \hphantom{\theta}:\text{\rm group homomorphism such that}\\
    \hphantom{theta:}\theta(\gamma,a)
   =\tau(\varpi^{-l^{\prime}}T_{K/F}(\varpi^{-l^{\prime}}\beta x))\\
    \hphantom{theta:}
      \text{\rm for $\forall(\gamma,a)=(1+\varpi^lx,1+\varpi^ly)
           \in U_{K/F}{\times}_{\frak{p}^r}O^{\times}$}
         \end{array}\right\}
$$
onto $\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$. Here
$$
 U_{K/F}{\times}_{\frak{p}^r}O^{\times}
 =\{(\gamma,a)\in U_{K/F}\times O^{\times}\mid
     \gamma\gamma^{\tau}\equiv a^2\npmod{\frak{p}^r}\}
$$
is a subgroup of $U_{K/F}\times O^{\times}$.
\end{prop}


\subsection[]{}\label{sebsec:unitary-group}
Let $K/F$ be the unramified quadratic field extension and put 
$\text{\rm Gal}(K/F)=\langle\tau\rangle$. Let us denote by $O_K$ the
integer ring of $K$ and $\frak{p}_K=\varpi O_K$ the maximal ideal of
$O_K$. Fix a $S\in M_n(O_K)$ such that $^tS^{\tau}=S$ and 
$\det S\in O_K^{\times}$. Then the unitary group $G=U(S)$ associated
with the Hermitian form $S$ is the $O$-group scheme defined by
$$
 G(A)=\{g\in GL_n(O_K{\otimes}_O A)\mid gS\,^t\!g^{\tau\otimes 1}=S\}
$$
for all commutative $O$-algebra $A$. Its Lie algebra
$\frak{g}=\frak{u}(S)$ is an affine $O$-scheme defined by 
$$
 \frak{g}(A)=\{X\in\frak{gl}_n(O_K{\otimes}_O A)\mid 
                XS+S\,^tX^{\tau\otimes 1}=0\}.
$$
We can take an $\varepsilon\in O^{\times}$ such that
$O_K=O[\sqrt\varepsilon]$. If there exists an $\eta\in A$ such that
$\varepsilon 1_A=\eta^2$, then we can identify $O_K{\otimes}_O A$ with 
$A\oplus A$ such a way that $(x,y)^{\tau\otimes 1}=(y,x)$. Then 
we have identifications $G(A)=GL_n(A)$ and
$\frak{g}(A)=\frak{gl}_n(A)$. If we put 
$$
 K=\left\{\begin{bmatrix}
            x&\varepsilon y\\
            y&x
           \end{bmatrix}\biggm| x,y\in F\right\},
 \quad
 O_K=\left\{\begin{bmatrix}
            x&\varepsilon y\\
            y&x
           \end{bmatrix}\biggm| x,y\in O\right\}
$$
then
$$
 M_n(O_K)=\{X\in M_{2n}(O_F)\mid I_n(\varepsilon)X=XI_n(\varepsilon)\}
$$
where
$$
 I_n(\varepsilon)=\begin{bmatrix}
                   I(\varepsilon)&      &              \\
                                 &\ddots&              \\
                                 &      &I(\varepsilon)
                  \end{bmatrix}\in M_{2n}(O_F)
 \;\text{\rm with $I(\varepsilon)=\begin{bmatrix}
                                   0&\varepsilon\\
                                   1&0
                                  \end{bmatrix}$}.
$$
Then we have 
$$
 G(A)=\{g\in GL_{2n}(A)\mid I_n(\varepsilon)g=gI_n(\varepsilon), 
                            gS^t\!g^{\tau}=S\}
$$
and
$$
 \frak{g}(A)=\{X\in\frak{gl}_{2n}(A)\mid 
                I_n(\varepsilon)X=XI_n(\varepsilon),\;
                XS+S^t\!X^{\tau}=0\}
$$
for all commutative $O$-algebra $A$. In this way $G$ is a closed
$O$-group subscheme of $GL_{2n}$. Then $G$ satisfies the conditions
I), II) and III) of the subsection \ref{subsec:fundamental-setting}. 
The center of $\frak{g}(O)$ is
$$
 Z(\frak{g}(O))=\{\lambda\cdot 1_n\in GL_n(O_K)\mid
                   \lambda\in O_K\;
                   \text{\rm s.t. $\lambda^{\tau}+\lambda=0$}\}.
$$
The similar arguments as Propositions 
\ref{prop:congruent-beta-for-gl-n} and \ref{prop:central-beta-for-gl-n} 
show

\begin{prop}\label{prop:congruent-beta-for-u(s)}
If 
$\beta\equiv\lambda\cdot 1_n+\beta_0\npmod{\frak{p}^{l^{\prime}}}$ 
for $\beta, \beta_0\in\frak{g}(O)$ with $\lambda\in O_K$, then there
  exists a group homomorphism 
$\mu:O_{K,r}^{\times}\to\Bbb C^{\times}$ such that 
$$
 \pi\mapsto(\mu\circ\det)\otimes\pi
$$
gives a bijection of $\text{\rm Irr}(G(O_r)\mid\psi_{\beta_0})$ onto 
$\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$.
\end{prop}

and

\begin{prop}\label{prop:central-beta-for-u(s)}
If $\beta\in\frak{g}(O)$ is central modulo $\frak{p}$, then there
exists a group homomorphism $\mu:O_{K,r}^{\times}\to\Bbb C^{\times}$ such
that, for any 
$\pi\in\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$, there exists a 
$\sigma\in\text{\rm Irr}(G(O_{r-1}))$ such that 
$\pi=(\mu\circ\det)\otimes(\sigma\circ\text{\rm proj})$ where 
$\text{\rm proj}:G(O_r)\to G(O_{r-1})$ is the canonical surjection.
\end{prop}

If $\beta_s\in\frak{g}(L)$ ($L=F$ or $L=\Bbb F$) is semisimple, then the
centralizer $Z_{G{\otimes}_OL}(\beta_s)$ is connected and its center is
also connected. Hence if $\beta\in\frak{g}(O)$ is smoothly regular
over $F$ and over $\Bbb F$, then $G_{\beta}$ is a smooth commutative
$O$-group scheme. If further $\overline\beta\in\frak{g}(\Bbb F)$ is
semisimple, then $c_{\beta,\rho}(g,h)=1$ for all 
$g,h\in G_{\beta}(\Bbb F)^{(c)}=G_{\beta}(\Bbb F)$ and all 
$\rho\in\frak{g}_{\beta}(\Bbb F)\sphat$ (see Remark 
\ref{remark:v-g-is-trivial-if-trace-form-is-non-deg-on-g-beta}). 
In this case Theorem \ref{th:main-result} gives

\begin{prop}\label{prop:regular-semisimple-case-for-u(s)}
There exists a bijection 
$\theta\mapsto
 \text{\rm Ind}_{G(O_r,\beta)}^{G(O_r)}\sigma_{\beta,\theta}$ of the
 set 
$$
 \left\{\theta\in G_{\beta}(O_r)\sphat\;\;\;
         \text{\rm s.t. $\theta=\psi_{\beta}$ on 
               $G_{\beta}(O_r)\cap K_l(O_r)$}\right\}
$$
onto $\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$.
\end{prop}

Assume that $n$ is odd. Let $L/F$ be the unramified field extension of
degree $2n$ so that $K$ is a subfield of $L$. There exists a 
$\tau\in\text{\rm Gal}(L/F)$ of order $2$ and 
$\text{\rm Gal}(K/F)=\langle\tau|_K\rangle$. Fix an 
$\varepsilon\in O_L^{\times}$ such that
$\varepsilon^{\tau}=\varepsilon$. Then 
$$
 S_{\varepsilon}(x,y)=T_{L/K}(\varepsilon xy^{\tau})
 \quad
 (x,y\in L)
$$
is a non-degenerate Hermitian form over $K$ on $L$. Let 
$\{u_1,\cdots,u_n\}$ be a $O_K$-basis of $O_L$. Then we have
$$
 (S_{\varepsilon}(u_i,u_j))_{1\leq i,j\leq n}
 =B\begin{bmatrix}
    \varepsilon^{\sigma_1}&      &                      \\
                          &\ddots&                      \\
                          &      &\varepsilon^{\sigma_n}
   \end{bmatrix}^t\!B^{\tau}
$$
where $B=(u_i^{\sigma_k})_{1\leq i,k\leq n}\in M_n(O_L)$ with 
$\text{\rm Gal}(L/K)=\{\sigma_1,\cdots,\sigma_n\}$. So the
discriminant of the Hermitian form $S_{\varepsilon}$ is 
$$
 \det(S_{\varepsilon}(u_i,u_j))_{1\leq i,j\leq n}
 =N_{L/K}(\varepsilon)\det B(\det B)^{\tau}\in O_K^{\times}
$$
because $L/K$ is unramified. Then the $O$-group scheme 
$G=U(S_{\varepsilon})$ and its Lie algebra
$\frak{g}=\frak{u}(S_{\varepsilon})$ are defined by
$$
 G(A)=\{g\in GL_A(O_L{\otimes}_OA)\mid 
          S_{\varepsilon}(gx,gy)=S_{\varepsilon}(x,y)\;
          \text{\rm for $\forall x,y\in L$}\}
$$
and
$$
 \frak{g}(A)=\{X\in\text{\rm End}_A(O_L{\otimes}_OA)\mid
                S_{\varepsilon}(Xx,y)+S_{\varepsilon}(x,Xy)=0\;
                \text{\rm for $\forall x,y\in L$}\}
$$
for all commutative $O$-algebra $A$. 

Take a $\beta\in O_L$ such that $\beta^{\tau}+\beta=0$ and 
$O_L=O[\beta]$. Identify $\beta\in L$ with the element 
$x\mapsto x\beta$ of 
$\frak{g}(O)\subset\text{\rm End}_O(O_L)=\frak{gl}_{2n}(O)$. Then the
characteristic polynomial $\chi_{\beta}(t)\in O[t]$ is irreducible
modulo $\frak{p}$. We have
$$
 G_{\beta}(O_r)=G(O_r)\cap\left(O_L/\frak{p}_L^r\right)^{\times}
  =\{\gamma\nnpmod{\frak{p}_L^r}\mid\gamma\in U_{L/F}\}
$$
where
$$
 U_{L/F}
 =\{\gamma\in O_L^{\times}\mid\gamma\gamma^{\tau}\in O^{\times}\}.
$$
In this case 
$\psi_{\beta}(h)=\tau(\varpi^{-l^{\prime}}T_{L/F}(\beta x))$ for all
$$
 h=1+\varpi^lx\npmod{\frak{p}_L^r}\in K_l(O_r)\cap G_{\beta}(O_r)
  \subset\left(O_L/\frak{p}_L^r\right)^{\times}.
$$
Then Proposition \ref{prop:regular-semisimple-case-for-u(s)} gives 

\begin{prop}\label{prop:shintani-gerardin-type-parametrization-for-u(s)}
There exists a bijection 
$\theta\mapsto
 \text{\rm Ind}_{G(O_r,\beta)}^{G(O_r)}\sigma_{\beta,\theta}$ of the
 set 
$$
 \left\{\begin{array}{l}
        \theta:U_{L/F}\to\left(O_L/\frak{p}_L^r\right)^{\times}
               \to\Bbb C^{\times}:\text{\rm group homomorphism}\\
        \text{\rm such that 
  $\theta(\gamma)
   =\tau(\varpi^{-l^{\prime}}T_{L/F}(\varpi^{-l^{\prime}}\beta x))
   \;\;
   \forall\gamma=1+\varpi^lx\in U_{K/F}$}
         \end{array}\right\}
$$
onto $\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$.
\end{prop}


\section{Introduction}\label{sec:introduction}

Let $F$ be a non-dyadic non-archimedean local field. 
The integer ring of $F$ is denoted by $O$ with the maximal
ideal $\frak{p}$ generated by $\varpi$. The residue class field 
$\Bbb F=O/\frak{p}$ is a finite field of $q$ elements.
Fix a continuous unitary character $\tau$ of the additive group $F$ such
that $\{x\in F\mid\tau(xO)=1\}=O$, and define an additive
character $\widehat\tau$ of $\Bbb F$ by 
$\widehat\tau(\overline x)=\tau(\varpi^{-1}x)$. 
For an integer $l>0$ put $O_l=O/\frak{p}^l$ so that $\Bbb F=O_1$. 

If a connected reductive quasi-split linear group $\bold{G}$ over
$F$ is split over an
unramified extension of $F$, then there exists a smooth affine group
scheme $G$ over $O$ such that $G{\otimes}_{O}F=\bold{G}$ and 
$G{\otimes}_{O}\Bbb F$ is a connected reductive group over $\Bbb F$. 
In this case the locally compact group $G(F)=\bold{G}(F)$ of the
$F$-rational points has an open compact subgroup $G(O)$ wich is
called a hyperspecial compact subgroup of $G(F)$ 
\cite[3.8.1]{Tits1979}. 
An important problem in the harmonic analysis on $G(F)$
is to determine the irreducible unitary representations of the compact
group $G(O)$. Such a representation $\pi$ of $G(O)$ factors through  the 
canonical group homomorphism $G(O)\to G(O_r)$ for some
$r>0$ since the canonical group homomorphism is surjective due to 
the smoothness of the group scheme $G$ over $O$ and Hensel's
lemma, and $\pi$ is trivial on the kernel of the canonical group
homomorphism for some $r>0$. 
Hence the problem is reduced to determine the set 
$\text{\rm Irr}(G(O_r))$ of the equivalence classes of the 
irreducible complex representations of the finite group $G(O_r)$. 

This problem in the case $r=1$, that is the representation theory of
the finite reductive group $G(\Bbb F)$, has been studied extensively,
starting 
from Green \cite{Green1955} concerned with $GL_n(\Bbb F)$ to the
decisive paper of Deligne-Lusztig \cite{Deligne-Lusztig1976}. 

This paper treats the case $r>1$ where the study of the representation
theory of the finite group $G(O_r)$ is less complete. 
Shalika \cite{Shalika2004} treats the case $SL_2(O_r)$, Silberger 
\cite{Silberger1970} the case $PGL_2(O_r)$. Shintani \cite{Shintani1968} 
and G\'erardin \cite{Gerardin1972} treat cuspidal representations
of $GL_n(O_r)$ in order to construct supercuspidal representations of
$GL_n(F)$. The last two papers 
use Clifford theory and Weil representations over
finite fields. In the series of papers 
\cite{Hill1993,Hill1994,Hill1995-1,Hill1995-2}, Hill treats the case
$GL_n(O_r)$ systematically by means of Clifford theory, but 
different methods are used 
for representations associated with different type of adjoint orbits. 

In this paper, we will establish a parametrization of the irreducible
representations of $G(O_r)$ ($r>1$) associated with the regular (more 
precisely smoothly regular) adjoint orbits. Taking a representative 
$\beta$ of the adjoint orbit, the parametrization is given by means of
a subset of the character group of $G_{\beta}(O_r)$ where $G_{\beta}$
is the centralizer of $\beta$ in $G$ which is smooth commutative group
scheme over $O$. Our theory is based on Clifford theory
and Weil representations over finite fields, and 
it works well under an assumption
of the triviality of certain Schur multiplier of a finite commutative 
group $G_{\beta}(\Bbb F)$. We can verify the
assumption in the case of $GL_n$ with $n\leq 4$, and the discussions
in this paper show that the assumption is highly probable for the
reductive groups in general 

The main result of this paper is Theorem \ref{th:main-result}. The
situation is quite simple when $r$ is even, and almost all of this
paper is devoted to treat the case of $r=2l-1$ being odd. In this case
we need Weil representation over finite field to construct irreducible
representations of $G_{\beta}(O_r)\cdot K_{l-1}(O_r)$, 
where $K_{l-1}(O_r)$ is the kernel of the canonical group homomorphism
$G(O_r)\to G(O_{l-1})$, and at this
point appears the Schur multiplier as an obstruction to the
construction. Here we shall note that over a finite
field, Weil representation is a representation of a symplectic group, 
not of the $2$-fold covering group of it, and that the Schur
multiplier is coming not from Weil representation but from 
certain {\it twist} which occurs en route of connecting 
$K_{l-1}(O_r)$ with the Heisenberg group over finite field 
(see section \ref{sec:weil-representation-over-finite-field} for the
details). 

Several fundamental properties of the Schur multiplier will be
discussed in section \ref{sec:properties-of-c-beta-rho}. 
These properties, combined with the
results of \cite{Takase2016} in the case of $G=GL_n$, shows that it is
highly 
probable that the Schur multiplier is trivial for all reductive group
schemes over $O$ provided that $\beta$ is regular and that 
the residue characteristic is big enough. 

We will give some examples of classical groups where the
characteristic polynomial of $\beta$ is irreducible modulo
$\frak{p}$. In this case the
parametrization is given by a subset of the character group of unit
groups of unramified extensions of the base field $F$. See propositions 
\ref{prop:shintani-gerardin-parametrization} for a general linear
group, 
\ref{prop:shintani-gerardin-type-parametrization-for-gsp-2n} for a
group of symplectic similitudes, 
\ref{prop:shintani-gerardin-type-parametrization-for-go(s)-of-even-n} and 
\ref{prop:shintani-gerardin-type-parametrization-for-go(s)-of-odd-n} for a
general orthogonal group with respect to a quadratic form of even and
odd variables respectively, and 
\ref{prop:shintani-gerardin-type-parametrization-for-u(s)} for an
unitary group associated with Hermitian form of odd variables.

\section*{References}


\section{Main results}\label{sec:main-results}

\subsection[]{}\label{subsec:fundamental-setting}
Let $G\subset GL_n$ be a closed smooth $O$-group subscheme, and
$\frak{g}$ the Lie algebra of $G$ which is a closed affine
$O$-subscheme of $\frak{gl}_n$ the Lie algebra of $GL_n$. We may
  assume that the fibers $G{\otimes}_OK$ ($K=F$ or $K=\Bbb F$) are
  non-commutative algebraic $K$-group (that is smooth $K$-group
  scheme).  

For any
$O$-algebra $K$, the set of the $K$-valued points $\frak{gl}_n(K)$ is
  identified with the $K$-Lie algebra of square matrices $M_n(K)$ of size $n$
  with Lie bracket $[X,Y]=XY-YX$, and 
the group of $K$-valued points $GL_n(K)$ is
identified with the matrix group
$$
 GL_n(K)=\{g\in M_n(K)\mid \det g\in K^{\times}\}
$$  
where $K^{\times}$ is the multiplicative group of $K$. Hence
$\frak{g}(K)$ is identified with a matrix Lie subalgebra of $\frak{gl}_n(K)$
and $G(K)$ is identified with a matrix subgroup of $GL_n(K)$. Let 
$$
 B:\frak{gl}_n{\times}_O\frak{gl}_n\to
                \mathcal{O}=\text{\rm Spec}(O[t])
$$
be the trace form on $\frak{gl}_n$, that is $B(X,Y)=\text{\rm tr}(XY)$
for all $X,Y\in\frak{gl}_n(K)$ with any $O$-algebra $K$. The
smoothness of $G$ implies that we have a canonical isomorphism
$$
 \frak{g}(O)/\varpi^r\frak{g}(O)\,\tilde{\to}\,
 \frak{g}(O_r)=\frak{g}(O){\otimes}_{O}O_r
$$
(\cite[Chap.II, $\S 4$, Prop.4.8]{Demazure-Gabriel1970}) and that the
 canonical group homomorphism $G(O)\to G(O_r)$ is
 surjective due to Hensel's lemma. Then for any $0<l<r$ the canonical
 group homomorphism $G(O_r)\to G(O_l)$ is surjective whose kernel is
 denoted by $K_l(O_r)$. 

For any $g\in G(O)$ (resp. $X\in\frak{g}(O)$), 
the image under the canonical surjection onto $G(O_l)$ 
(resp. onto $\frak{g}(O_l)$) with $l>0$ is denoted by 
$$
 g_l=g\npmod{\frak{p}^l}\in G(O_l)
 \quad
(\text{\rm resp.} X_l=X\npmod{\frak{p}^l}\in\frak{g}(O_l)).
$$
Since the reduction modulo $\frak{p}$ plays a fundamental role in our
theory, let us use 
the notation $\overline g=g\npmod{\frak p}\in G(\Bbb F)$ 
(resp. $\overline X=X\npmod{\frak p}\in\frak{g}(\Bbb F)$) if $l=1$. 

We will pose the following three conditions;
\begin{itemize}
\item[I)] $B:\frak{g}(\Bbb F)\times\frak{g}(\Bbb F)\to\Bbb F$ is
  non-degenerate, 
\item[II)] for any integers $r=l+l^{\prime}$ with 
           $0<l^{\prime}\leq l<r$, we have a group isomorphism
$$
 \frak{g}(O_{l^{\prime}})\,\tilde{\to}\,K_l(O_r)
$$
           defined by 
$X\npmod{\frak{p}^{l^{\prime}}}\mapsto1+\varpi^lX\npmod{\frak{p}^r}$,
\item[III)] if $r=2l-1\geq 3$ is odd, then we have a mapping
$$
 \frak{g}(O)\to K_{l-1}(O_r)
$$
defined by 
$X\mapsto(1+\varpi^{l-1}X+2^{-1}\varpi^{2l-2}X^{2l-2})\npmod{\frak{p}^r}$.
\end{itemize}
The condition I) implies that 
$B:\frak{g}(O_l)\times\frak{g}(O_l)\to O_l$ 
is non-degenerate for all $l>0$, and so 
$B:\frak{g}(O)\times\frak{g}(O)\to O$ is also non-degenerate. 
The mappings of the conditions II) and III) from Lie algebras to
groups can be regarded as truncations of the exponential mapping.

See section \ref{sec:example} for the examples of classical groups
which satisfy these three fundamental conditions of our theory. 

The character group of an finite abelian group $\mathcal{G}$ 
is denoted by $\mathcal{G}\sphat$.

\subsection[]{}\label{subsec:clifford-theory-in-general}
From now on we will fix an integer $r\geq 2$ and put 
$r=l+l^{\prime}$ with the smallest integer $l$ such that 
$0<l^{\prime}\leq l$. In other word
$$
 l^{\prime}=\begin{cases}
             l&:r=2l,\\
             l-1&:r=2l-1.
            \end{cases}
$$
Take a $\beta\in\frak{g}(O)$ and define a
character $\psi_{\beta}$ of the finite abelian group 
$K_l(O_r)$ by 
$$
 \psi_{\beta}((1+\varpi^lX)\nnpmod{\frak{p}^r})
 =\tau(\varpi^{-l^{\prime}}B(X,\beta))
 \qquad
 (X\in\frak{g}(O)).
$$
Then $\beta\npmod{\frak{p}^{l^{\prime}}}\mapsto\psi_{\beta}$ 
gives an isomorphism of the
additive group $\frak{g}(O_{l^{\prime}})$ onto the character
group $K_l(O_r)\sphat$. For any 
$g_r=g\npmod{\frak{p}^r}\in G(O_r)$, we have
$$
 \psi_{\beta}(g_r^{-1}hg_r)
 =\psi_{\text{\rm Ad}(g)\beta}(h)
 \qquad
 (h\in K_l(O_r)).
$$
So the stabilizer of $\psi_{\beta}$ in $G(O_R)$ is 
$$
 G(O_r,\beta)
 =\left\{g_r\in G(O_r)\bigm|
         \text{\rm Ad}(g)\beta\equiv\beta
                    \pmod{\frak{p}^{l^{\prime}}}\right\}
$$
which is a subgroup of $G(O_r)$ containing $K_{l^{\prime}}(O_r)$.

Now let us denote by 
$\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$ 
(resp. $\text{\rm Irr}(G(O_r,\beta)\mid\psi_{\beta})$) the set of the
isomorphism classes of the irreducible complex representation $\pi$ of 
$G(O_r)$ (resp. $\sigma$ of $G(O_r,\beta)$) such
that 
$$
 \langle\psi_{\beta},\pi\rangle_{K_l(O_r)}
 =\dim_{\Bbb C}\text{\rm Hom}_{K_l(O_r)}(\psi_{\beta},\pi)
 >0
$$
(resp. 
$\langle\psi_{\beta},\sigma\rangle_{K_l(O_r)}>0$). Then Clifford's
 theory says that 
\begin{enumerate}
\item $\text{\rm Irr}(G(O_r))
       =\bigsqcup_{\beta\npmod{\frak{p}^{l^{\prime}}}}
           \text{\rm Irr}(G(O_r)\mid\psi_{\beta})$
  where $\bigsqcup_{\beta\npmod{\frak{p}^{l^{\prime}}}}$ 
  is the disjoint union over the
  representatives $\beta\npmod{\frak{p}^{l^{\prime}}}$ 
  of the $\text{\rm Ad}(G(O_{l^{\prime}}))$-orbits in 
  $\frak{g}(O_{l^{\prime}})$,
\item a bijection of 
      $\text{\rm Irr}(G(O_r,\beta)\mid\psi_{\beta})$ onto 
      $\text{\rm Irr}(G(O_r)\mid\psi_{\beta})$ is given by 
$$
 \sigma\mapsto\text{\rm Ind}_{G(O_r,\beta)}^{G(O_r)}\sigma.
$$
\end{enumerate}
So our problem is to give a good parametrization of the 
   set $\text{\rm Irr}(G(O_r,\beta)\mid\psi_{\beta})$ for 
$\beta\in\frak{g}(O)$ which is {\it regular} enough. 

For any $\beta\in\frak{g}(O)$, let us denote
by $G_{\beta}=Z_G(\beta)$ the centralizer of
$\beta$ in $G$  which is a closed $O$-group subscheme of $G$. The Lie
algebra $\frak{g}_{\beta}=Z_{\frak g}(\beta)$ of $G_{\beta}$ is 
the centralizer of $\beta$ in $\frak{g}$ which is 
a closed $O$-subscheme of $\frak{g}$. 

\subsection[]{}
\label{subsec:def-of-schur-multiplier-associated-with-alg-group}
In this subsection we will define a Schur multiplier which is an
obstruction to our theory.

Take a $\beta\in\frak{g}(O)$ such that 
$\frak{g}_{\beta}(\Bbb F)\lvertneqq\frak{g}(\Bbb F)$. Then non-zero 
$\Bbb F$-vector space 
$\Bbb V_{\beta}=\frak{g}(\Bbb F)/\frak{g}_{\beta}(\Bbb F)$ has a
symplectic form 
$$
 \langle \dot X,\dot Y\rangle_{\beta}=B([X,Y],\overline\beta)
$$
where $\dot X=X\pmod{\frak{g}_{\beta}(\Bbb F)}\in\Bbb V_{\beta}$ with 
$X\in\frak{g}_{\beta}(\Bbb F)$.  
Then $g\in G_{\beta}(\Bbb F)$ gives an element $\sigma_g$ of 
 the symplectic group $Sp(\Bbb V_{\beta})$ defined by 
$$
 X\npmod{\frak{g}_{\beta}(\Bbb F)}\mapsto
 \text{\rm Ad}(g)^{-1}X\npmod{\frak{g}_{\beta}(\Bbb F)}.
$$
Note that the group $Sp(\Bbb V_{\beta})$ acts on $\Bbb V_{\beta}$ from
right. Let 
$v\mapsto[v]$ be a $\Bbb F$-linear section on $\Bbb V_{\beta}$ of the
exact sequence 
\begin{equation}
 0\to\frak{g}_{\beta}(\Bbb F)
  \to\frak{g}(\Bbb F)\to\Bbb V_{\beta}\to 0.
\label{eq:fundamental-exact-sequence-of-lie-alg-and-symplectic-space}
\end{equation}
For any $v\in\Bbb V_{\beta}$ and $g\in G_{\beta}(\Bbb F)$, put
$$
 \gamma(v,g)=\gamma_{\frak g}(v,g)
 =\text{\rm Ad}(g)^{-1}[v]-[v\sigma_g]\in\frak{g}_{\beta}(\Bbb F).
$$
Take a $\rho\in\frak{g}_{\beta}(\Bbb F)\sphat$. Then there exists
uniquely a $v_g\in\Bbb V_{\beta}$ such that
$$
 \rho(\gamma(v,g))=\widehat\tau(\langle v,v_g\rangle_{\beta})
$$
for all $v\in\Bbb V_{\beta}$. Note that $v_g\in\Bbb V_{\beta}$ depends
on $\rho$ as well as the section $v\mapsto[v]$. Let
$$
 G_{\beta}(\Bbb F)^{(c)}
 =\{g\in G(\Bbb F)\mid\text{\rm Ad}(g)Y=Y\;\text{\rm for}\;
                        \forall Y\in\frak{g}_{\beta}(\Bbb F)\}
$$
be the centralizer of $\frak{g}_{\beta}(\Bbb F)$ in $G(\Bbb F)$, which
is a subgroup of $G_{\beta}(\Bbb F)$. Then for any 
$g,h\in G_{\beta}(\Bbb F)^{(c)}$, we have
\begin{equation}
 v_{gh}=v_h\sigma_g^{-1}+v_g
\label{eq:multiplication-formula-of-v-epsilon}
\end{equation}
because $\gamma(v,gh)=\gamma(v,g)+\gamma(v\sigma_g^{-1},h)$ for all
$v\in\Bbb V_{\beta}$. Put
$$
 c_{\beta,\rho}(g,h)
 =\widehat\tau(2^{-1}\langle v_g,v_{gh}\rangle_{\beta})
$$
for $g,h\in G_{\beta}(\Bbb F)^{(c)}$. Then the relation 
\eqref{eq:multiplication-formula-of-v-epsilon} shows that 
$c_{\beta,\rho}\in Z^2(G_{\beta}(\Bbb F)^{(c)},\Bbb C^{\times})$ is a
2-cocycle with trivial action of $G_{\beta}(\Bbb F)^{(c)}$ on 
$\Bbb C^{\times}$. Moreover we have

\begin{prop}
\label{prop:schur-multiplier-associated-with-finite-symplectic-space}
The Schur multiplier 
$[c_{\beta,\rho}]\in H^2(G_{\beta}(\Bbb F)^{(c)},\Bbb C^{\times})$ is
independent of the choice of the $\Bbb F$-linear section 
$v\mapsto[v]$.
\end{prop}
\begin{proof}
Take another $\Bbb F$-linear section $v\mapsto[v]^{\prime}$ 
with respect to which
we will define $\gamma^{\prime}(v,g)\in\frak{g}_{\beta}$ 
and $v^{\prime}_g\in\Bbb V_{\beta}$ as above. 
Then there exists a $\delta\in\Bbb V_{\beta}$ such that 
$\rho([v]-[v]^{\prime})
 =\widehat\tau\left(\langle v,\delta\rangle_{\beta}\right)$ 
for all $v\in\Bbb V_{\beta}$. We have 
$v_g^{\prime}=v_g+\delta-\delta\sigma_g$ for all 
$g\in G_{\beta}(\Bbb F)^{(c)}$. So if we put 
$\alpha(g)
 =\widehat\tau\left(2^{-1}\langle 
    v_g^{\prime}-v_{g^{-1}},\delta\rangle_{\beta}
            \right)$ 
for $g\in G_{\beta}(\Bbb F)^{(c)}$, then we have
$$
 \widehat\tau\left(2^{-1}\langle v_g^{\prime},
           v_{gh}^{\prime}\rangle_{\beta}\right)
 =\widehat\tau\left(2^{-1}\langle v_g,
           v_{gh}\rangle_{\beta}\right)\cdot
  \alpha(h)\alpha(gh)^{-1}\alpha(g)
$$
for all $g,h\in G_{\beta}(\Bbb F)^{(c)}$.
\end{proof}

\subsection[]{}\label{subsec:main-result}
Now our main result is

\begin{thm}\label{th:main-result}
Suppose that a $\beta\in\frak{g}(O)$ satisfies the conditions
\begin{enumerate}
\item $G_{\beta}$ is commutative smooth $O$-group scheme, and
\item the Schur multiplier 
      $[c_{\beta,\rho}]\in H^2(G_{\beta}(\Bbb F),\Bbb C^{\times})$ is
      trivial for all characters $\rho\in\frak{g}_{\beta}(\Bbb F)\sphat$.
\end{enumerate}
Then 
we have a bijection $\theta\mapsto\sigma_{\beta,\theta}$ of the set
$$
 \left\{\theta\in G_{\beta}(O_r)\sphat\;\;\;
         \text{\rm s.t. $\theta=\psi_{\beta}$ on 
               $G_{\beta}(O_r)\cap K_l(O_r)$}\right\}
$$
onto $\text{\rm Irr}(G(O_r,\beta)\mid\psi_{\beta})$.
\end{thm}

The proof is given in subsection
\ref{subsec:proof-of-main-result-for-even-r} for even $r$ and 
subsection \ref{subsec:description-of-main-result-for-odd-r} for odd
$r$.

\begin{rem}\label{remark:remarks-on-main-result}
\begin{enumerate}
\item A sufficient condition for the first condition of Theorem
  \ref{th:main-result} is given by Theorem
  \ref{th:sufficient-condition-for-smooth-commutativeness-of-g-beta}.
\item The smoothness of $G_{\beta}$ over $O$ implies that the
      canonical group homomorphism 
      $G_{\beta}(O_r)\to G_{\beta}(O_{l^{\prime}})$ is surjective. So we
      have
\begin{equation}
 G(O_r,\beta)=G_{\beta}(O_r)\cdot K_{l^{\prime}}(O_r),
 \qquad
 l^{\prime}=\begin{cases}
             l&:r=2l,\\
             l-1&:r=2l-1.
            \end{cases}
\label{eq:fundamental-structure-of-stabilizer-of-psi-beta}
\end{equation}
\item As presented in the following two subsections, 
      the second condition in the theorem is required only in the case
      of $r$ being odd.
\item Since $G{\otimes}_O\Bbb F$ and $G_{\beta}{\otimes}_O\Bbb F$ are
      $\Bbb F$-algebraic group, and the former is not commutative
      while the latter is, so we have
$$
 \dim_{\Bbb F}\frak{g}(\Bbb F)=\dim G{\otimes}_O\Bbb F
 >\dim G_{\beta}{\otimes}_O\Bbb F=\dim_{\Bbb F}\frak{g}_{\beta}(\Bbb F).
$$
      That is $\frak{g}_{\beta}(\Bbb F)\lvertneqq\frak{g}(\Bbb F)$.
\item Since $G_{\beta}$ is assumed to be commutative, we have
      $G_{\beta}(\Bbb F)^{(c)}=G_{\beta}(\Bbb F)$ in the definition of
      the Schur multiplier $[c_{\beta,\rho}]$. 
\item Assume that $G_{\beta}(\Bbb F)^{(c)}$ is commutative. Then the
      cohomology class 
      $[c_{\beta,\rho}]
       \in H^2(G_{\beta}(\Bbb F)^{(c)},\Bbb C^{\times})$ 
      is trivial if and only if $c_{\beta,\rho}$ is symmetric, that is 
      $c_{\eta,\rho}(g,h)=c_{\beta,\rho}(h,g)$ for all 
      $g,h\in G_{\beta}(\Bbb F)^{(c)}$. In fact, only if part is
      trivial. Let
\begin{equation}
 1\to\Bbb T\xrightarrow{i}\mathcal{G}
           \xrightarrow{j}G_{\beta}(\Bbb F)^{(c)}\to 1
\label{eq:group-extension-of-c-beta-rho}
\end{equation}
      be the group extension associated with the $2$-cocycle 
      $c_{\beta,\rho}
       \in Z^2(B_{\beta}(\Bbb F)^{(c)},\Bbb T)$ where $\Bbb T$ is the
       subgroup of $z\in\Bbb C^{\times}$ such that $|z|=1$. 
       Then the
       groups are compact commutative group, and we have a group
       extension of the Pontryagin dual groups
\begin{equation}
 1\to{G_{\beta}(\Bbb F)^{(c)}}\sphat
  \xrightarrow{\widehat j}\mathcal{G}\sphat
  \xrightarrow{\widehat i}{\Bbb T}\sphat\to 1.
\label{eq:dual-of-group-extension-of-c-beta-rho}
\end{equation}
      Since ${\Bbb T}\sphat\simeq\Bbb Z$ is free the group extension 
      \eqref{eq:dual-of-group-extension-of-c-beta-rho} is trivial and so
      is the group extension 
      \eqref{eq:group-extension-of-c-beta-rho}.
\end{enumerate}
\end{rem}

\subsection[]{}\label{subsec:proof-of-main-result-for-even-r}
Assume that $r=2l$ is even so that $l^{\prime}=l$. In this case the
proof of Theorem \ref{th:main-result} is quite easy. Let us suppose
more generally that there exists a commutative subgroup $\mathcal{C}$
of $G(O_r,\beta)$ such that
$$
 G(O_r,\beta)=\mathcal{C}\cdot K_l(O_r).
$$
Let us denote by $\mathcal{C}\sphat_{\beta}$ the subset of the
character group $\mathcal{C}\sphat$ consisting of the
$\theta\in\mathcal{C}\sphat$ such 
that $\theta=\psi_{\beta}$ on $\mathcal{C}\cap K_l(O_r)$. Then any 
$\theta\in\mathcal{C}\sphat_{\beta}$ gives an one-dimensional
representation $\sigma_{\beta,\theta}$ of $G(O_r,\beta)$ defined by 
$$
 \sigma_{\beta,\theta}(gh)=\theta(g)\cdot\psi_{\beta}(h)
 \qquad
 (g\in\mathcal{C}, h\in K_l(O_r)).
$$
Then we have a proposition of which our Theorem 
\ref{th:main-result} is a special case;

\begin{prop}\rm\label{prop:generalized-main-result-for-even-r}
$\theta\mapsto\sigma_{\beta,\theta}$ gives a bijection of 
$\mathcal{C}\sphat_{\beta}$ onto 
$\text{\rm Irr}(G(O_r,\beta)\mid\psi_{\beta})$.
\end{prop}
\begin{proof}
Take any $\sigma\in\text{\rm Irr}(G(O_r,\beta)\mid\psi_{\beta})$ with
representation space $V_{\sigma}$. Then
$$
 V_{\sigma}(\psi_{\beta})
 =\{v\in V_{\sigma}\mid \sigma(g)v=\psi_{\beta}(g)v\,
                \text{\rm for $\forall g\in K_l(O_r)$}\}
$$
is a non-trivial $G(O_r,\beta)$-subspace of $V_{\sigma}$ so that 
$V_{\sigma}=V_{\sigma}(\psi_{\beta})$. Then, for any 
one-dimensional representation $\chi$ of $G(O_r,\beta)$ such that 
$\chi=\psi_{\beta}$ on $K_l(O_r)$, we have 
$K_l(O_r)\subset\text{\rm Ker}(\chi^{-1}\otimes\sigma)$. On the other
hand $G(O_r,\beta)/K_l(O_r)$ is commutative, we have 
$\dim(\chi^{-1}\otimes\sigma)=1$ and then $\dim\sigma=1$. 
Put $\theta=\sigma|_{\mathcal{C}}\in\mathcal{C}\sphat_{\beta}$ and 
we have $\sigma=\sigma_{\beta,\theta}$. 
\end{proof}

\subsection[]{}\label{subsec:description-of-main-result-for-odd-r}
Assume that $r=2l-1\geq 3$ is odd so that $l^{\prime}=l-1\geq 1$. 
We have a chain of canonical surjections
\begin{equation}
 \heartsuit:K_{l-1}(O_r)\to K_{l-1}(O_{r-1})\,\tilde{\to}\,
        \frak{g}(O_{l-1})\to\frak{g}(\Bbb F)
\label{eq:canonical-surjection-of-k-l-1-to-g(f)}
\end{equation}
defined by
\begin{align*}
 1+\varpi^{l-1}X\nnpmod{\frak{p}^r}
 &\mapsto
 1+\varpi^{l-1}X\nnpmod{\frak{p}^{r-1}}\\
 &\mapsto
  X\nnpmod{\frak{p}^{l-1}}\mapsto
  \overline X=X\npmod{\frak p}..
\end{align*}
Let us denote by $Z(O_r,\beta)$ the inverse image 
under the surjection $\heartsuit$ of $\frak{g}_{\beta}(\Bbb F)$. Then 
$Z(O_r,\beta)$ is a normal subgroup of $K_{l-1}(O_r)$ containing 
$K_l(O_r)$ as the kernel of $\heartsuit$. 

Let us denote by $Y_{\beta}$ the set of the group
homomorphisms $\psi$ of $Z(O_r,\beta)$ to $\Bbb C^{\times}$ 
such that $\psi=\psi_{\beta}$ on $K_l(O_r)$. Then
a bijection of $\frak{g}_{\beta}(\Bbb F)\sphat$ onto $Y_{\beta}$ is
given by
$$
 \rho\mapsto\psi_{\beta,\rho}
            =\widetilde{\psi}_{\beta}\cdot(\rho\circ\heartsuit),
$$
where a group homomorphism 
$\widetilde{\psi}_{\beta}:Z(O_r,\beta)\to\Bbb C^{\times}$ 
is defined by 
$$
 1+\varpi^{l-1}X\npmod{\frak{p}^r}\mapsto
 \tau\left(\varpi^{-l}B(X,\beta)-(2\varpi)^{-1}B(X^2,\beta)\right)
$$
with $\overline X=X\npmod{\frak{p}}\in\frak{g}_{\beta}(\Bbb F)$. 

Take a $\psi\in Y_{\beta}$. 
For two elements 
$$
 x=1+\varpi^{l-1}X\npmod{\frak{p}^r},
 \quad
 y=1+\varpi^{l-1}Y\npmod{\frak{p}^r}
$$ 
of $K_{l-1}(O_r)$, we have 
$x^{-1}=1-\varpi^{l-1}X+2^{-1}\varpi^{2l-2}X^2\npmod{\frak{p}^r}$ so
that we have
$$
 xyx^{-1}y^{-1}
 =1+\varpi^{r-1}[X,Y]\npmod{\frak{p}^r}
 \in K_{r-1}(O_r)\subset K_l(O_r)
$$
and so 
$\psi_{\beta}(xyx^{-1}y^{-1})
 =\tau\left(\varpi^{-1}B(X,\text{\rm ad}(Y)\beta)\right)$. 
Hence we have 
$$
 \psi(xyx^{-1}y^{-1})=\psi_{\beta}(xyx^{-1}y^{-1})=1
$$
for all $x\in K_{l-1}(O_r)$ and $y\in Z(O_r,\beta)$ so
that we can define 
$$
 D_{\psi}:K_{l-1}(O_r)/Z(O_r,\beta)\times K_{l-1}(O_r)/Z(O_r,\beta)
          \to\Bbb C^{\times}
$$
by
$$
 D_{\psi}(\dot g,\dot h)
 =\psi(ghg^{-1}h^{-1})
 =\psi_{\beta}(ghg^{-1}h^{-1})
 =\tau\left(\varpi^{-1}B([X,Y],\beta)\right)
$$
for 
$g=(1+\varpi^{l-1}X)\npmod{\frak{p}^r}, 
 h=(1+\varpi^{l-1}Y)\npmod{\frak{p}^r}
 \in K_{l-1}(O_r)$, which is non-degenerate. Then Proposition 3.1.1 of 
\cite{Takase2016} gives

\begin{prop}
\label{prop:existence-of-fundamental-rep-associated-with-psi} 
For any $\psi=\psi_{\beta,\rho}\in Y_{\beta}$ with 
$\rho\in\frak{g}_{\beta}(\Bbb F)\sphat$, 
there exists unique irreducible
representation $\pi_{\beta,\rho}$ of $K_{l-1}(O_r)$ such that 
$\langle\psi,\pi_{\beta,\rho}\rangle_{Z(O_r,\beta)}>0$. Furthermore 
$$
 \text{\rm Ind}_{Z(O_r,\beta)}^{K_{l-1}(O_r)}\psi
 =\bigoplus^{\dim\pi_{\beta,\rho}}\pi_{\beta,\rho}
$$
and $\pi_{\beta,\rho}(x)$ is the homothety $\psi(x)$ for all 
$x\in Z(O_r,\beta)$.
\end{prop}

Fix a $\psi=\psi_{\beta,\rho}\in Y_{\beta}$ with 
$\rho\in\frak{g}_{\beta}(\Bbb F)\sphat$. 
Let $G_l(O_r,\beta)^{(c)}$ be a subgroup of $G(O_r,\beta)$ defined
by
$$
 G_l(O_r,\beta)^{(c)}
 =\left\{g\npmod{\frak{p}^r}\in G(O_r)\biggm|
    \begin{array}{l}
     \text{\rm Ad}(g)\beta\equiv\beta\npmod{\frak{p}^l},\\
     \text{\rm Ad}(\overline g)X=X\,\text{\rm for}\,
      \forall X\in\frak{g}_{\beta}(\Bbb F)
    \end{array}\right\}.
$$
Then, for any $g_r=g\npmod{\frak{p}^r}\in G_l(O_r,\beta)^{(c)}$ and 
$x=(1+\varpi^{l-1}X)\npmod{\frak{p}^r}\in Z(O_r,\beta)$, we have
\begin{align*}
 g_r^{-1}xg_rx^{-1}
 &=\left(1+\varpi^{l-1}g^{-1}Xg\right)
  \left(1-\varpi^{l-1}X+2^{-1}\varpi^{2l-2}X^2\right)\npmod{\frak{p}^r}\\
 &=1+\varpi^{l-1}\left(\text{\rm Ad}(g)^{-1}X-X\right)
    \npmod{\frak{p}^r}
  \in K_l(O_r),
\end{align*}
and
$$
 \psi(g_r^{-1}xg_rx^{-1})
 =\psi_{\beta}(g_r^{-1}xg_rx^{-1})
 =\tau\left(\varpi^{-l}B(X,\text{\rm Ad}(g)\beta-\beta)\right)=1,
$$
that is $\psi(g_r^{-1}xg_r)=\psi(x)$ for all $x\in Z(O_r,\beta)$. This
means that, for any $g\in G_l(O_r,\beta)^{(c)}$, the $g$-conjugate of
$\pi_{\beta,\rho}$ is isomorphic to $\pi_{\beta,\rho}$, that is, there
exists a $U(g)\in GL_{\Bbb C}(V_{\beta,\rho})$ ($V_{\beta,\rho}$ is the
representation space of $\pi_{\beta,\rho}$) such that
$$
 \pi_{\beta,\rho}(g^{-1}xg)
 =U(g)^{-1}\circ\pi_{\beta,\rho}(x)\circ U(g)
$$
for all $x\in K_{l-1}(O_r)$, and moreover, for any 
$g,h\in G_l(O_r,\beta)^{(c)}$, there exists a 
$c_U(g,h)\in\Bbb C^{\times}$ such that
$$
 U(g)\circ U(h)=c_U(g,h)\cdot U(gh).
$$
Then $c_U\in Z^2(G_l(O_r,\beta)^{(c)},\Bbb C^{\times})$ is a 
$\Bbb C^{\times}$-valued 2-cocycle on $G_l(O_r,\beta)^{(c)}$ with
trivial action on $\Bbb C^{\times}$, and the cohomology class 
$[c_U]\in H^2(G_l(O_r,\beta)^{(c)},\Bbb C^{\times})$ is independent of
the choice of each $U(g)$. 

In the next section, we will construct $\pi_{\beta,\rho}$ by means of
Weil representations over the finite field $\Bbb F$ (see Proposition
\ref{prop:another-expression-of-pi-beta-psi}), and will show
that we can choose $U(g)$ so that we have 
$$
 c_U(g,h)=c_{\beta,\rho}(\overline g,\overline h)
$$ 
for all $g,h\in G_l(O_r,\beta)^{(c)}$, where 
$\overline g\in G_{\beta}(\Bbb F)^{(c)}$ is the image of 
$g\in G_l(O_r,\beta)^{(c)}$ under the canonical surjection 
$G(O_r)\to G(\Bbb F)$ (see subsection \ref{subsec:action-of-g-o-f-beta}).

Let us assume 

\begin{assu}
\label{hypo:existence-of-good-abelian-subgroup-of-g1(c)} 
There exists a commutative subgroup 
$\mathcal{C}\subset G_l(O_r,\beta)^{(c)}$ such that
\begin{enumerate}
\item $G(O_r,\beta)=\mathcal{C}\cdot K_{l-1}(O_r)$,
\item the cohomology class 
      $[c_{\beta,\rho}|_{\overline{\mathcal{C}}\times\overline{\mathcal{C}}}]
       \in H^2(\overline{\mathcal{C}},\Bbb C^{\times})$ is trivial for all 
      $\rho\in\frak{g}_{\beta}(\Bbb F)\sphat$, where
       $\overline{\mathcal C}\subset G_{\beta}(\Bbb F)^{(c)}$ is the
       image of $\mathcal{C}$ under the canonical surjection 
      $G(O_r)\to G(\Bbb F)$.
\end{enumerate}
\end{assu}

Under this assumption we have

\begin{prop}
\label{prop:existence-of-canonical-intertwiner-of-pi-psi}
For any $\rho\in\frak{g}_{\beta}(\Bbb F)\sphat$, there exists a group
homomorphism 
$U_{\beta,\rho}:\mathcal{C}\to GL_{\Bbb C}(V_{\beta,\rho})$ such that
\begin{enumerate}
\item $\pi_{\beta,\rho}(g^{-1}xg)
       =U_{\beta,\rho}(g)^{-1}\circ\pi_{\beta,\rho}(x)
                              \circ U_{\beta,\rho}(g)$ for all 
      $g\in\mathcal{C}$ and $x\in K_{l-1}(O_r)$ and
\item $U_{\beta,\rho}(h)=1$ for all $h\in\mathcal{C}\cap K_{l-1}(O_r)$.
\end{enumerate}
\end{prop}
\begin{proof}
Because of 2) in Assumption
\ref{hypo:existence-of-good-abelian-subgroup-of-g1(c)} there exists a
group homomorphism $U:\mathcal{C}\to GL_{\Bbb C}(V_{\beta,\rho})$ such that 
$\pi_{\beta,\rho}(g^{-1}xg)=U(g)^{-1}\circ\pi_{\beta,\rho}(x)\circ U(g)$ for all 
$g\in\mathcal{C}$ and $x\in K_{l-1}(O_r)$. Then for any 
$h\in\mathcal{C}\cap K_{l-1}(O_r)$ there exists a 
$c(h)\in\Bbb C^{\times}$ such that $U(h)=c(h)\cdot\pi_{\beta,\rho}(h)$. 
On the other hand we have 
$$
 G_l(O_r,\beta)^{(c)}\cap K_{l-1}(O_r)\subset Z(O_r,\beta)
$$
since 
$(1+\varpi^{l-1}X)_r\in G_l(O_r,\beta)^{(c)}\cap K_{l-1}(O_r)$
means that
\begin{align*}
 \beta
 &\equiv(1+\varpi^{l-1}X)\beta(1+\varpi^{l-1}X)^{-1}\npmod{\frak{p}^l}\\
 &\equiv(\beta+\varpi^{l-1}X\beta)(1-\varpi^{l-1}X)\npmod{\frak{p}^l}\\
 &\equiv\beta+\varpi^{l-1}[X,\beta]\npmod{\frak{p}^l}
\end{align*}
and then $[X,\beta]\equiv 0\npmod{\frak p}$, that is 
$X\npmod{\frak p}\in\frak{g}_{\beta}(\Bbb F)$. Then 
$\pi_{\beta,\rho}(h)$ is the homothety $\psi_{\beta,\rho}(h)$ for all 
$h\in\mathcal{C}\cap K_{l-1}(O_r)$. Extend the group homomorphism 
$h\mapsto c(h)\psi_{\beta,\rho}(h)$ of $\mathcal{C}\cap K_{l-1}(O_r)$ to a group
homomorphism $\theta:\mathcal{C}\to\Bbb C^{\times}$. Then 
$g\mapsto U_{\psi}(g)=\theta(g)^{-1}U(g)$ is the required group homomorphism.
\end{proof}

Let us denote by 
$\mathcal{C}\sphat{\times}_{K_{l-1}(O_r)}\frak{g}_{\beta}(\Bbb F)\sphat$
the set of 
$(\theta,\rho)
 \in\mathcal{C}\sphat\times\frak{g}_{\beta}(\Bbb F)\sphat$ such that 
$\theta=\psi_{\beta,\rho}$ on $\mathcal{C}\cap K_{l-1}(O_r)$. Then 
$(\theta,\rho)\in
 \mathcal{C}\sphat{\times}_{K_{l-1}(O_r)}\frak{g}_{\beta}(\Bbb F)\sphat$ 
defines an irreducible representation $\sigma_{\theta,\rho}$ of 
$G(O_r,\beta)=\mathcal{C}\cdot K_{l-1}(O_r)$ by
$$
 \sigma_{\theta,\rho}(gh)
 =\theta(g)\cdot U_{\beta,\rho}(g)\circ\pi_{\beta,\rho}(h)
$$
for $g\in\mathcal{C}$ and $h\in K_{l-1}(O_r)$. Then we have

\begin{prop}\label{prop:generalized-main-result-for-odd-r}
Under Assumption
\ref{hypo:existence-of-good-abelian-subgroup-of-g1(c)}, a bijection of 
${\mathcal C}\sphat{\times}_{K_{l-1}(O_r)}\frak{g}_{\beta}(\Bbb F)\sphat$
onto $\text{\rm Irr}(G(O_r,\beta)\mid\psi_{\beta})$ is given by 
$(\theta,\rho)\mapsto\sigma_{\theta,\rho}$.
\end{prop}
\begin{proof}
Clearly 
$\pi_{\theta,\rho}\in\text{\rm Irr}(G(O_r,\beta)\mid\psi_{\beta})$ for
all 
$(\theta,\rho)\in
 {\mathcal C}\sphat{\times}_{K_{l-1}(O_r)}\frak{g}_{\beta}(\Bbb F)\sphat$. 
Take a $\sigma\in\text{\rm Irr}(G(O_r,\beta)\mid\psi_{\beta})$. Then 
\begin{align*}
 \sigma\hookrightarrow
 \text{\rm Ind}_{K_l(O_r)}^{G(O_r,\beta)}\psi_{\beta}
 &=\text{\rm Ind}_{Z(O_r,\beta)}^{G(O_r,\beta)}
   \left(\text{\rm Ind}_{K_l(O_r)}^{Z(O_r,\beta)}\psi_{\beta}
         \right)\\
 &=\bigoplus_{\rho\in\frak{g}_{\beta}(\Bbb F)\sphat}
   \text{\rm Ind}_{Z(O_r,\beta)}^{G(O_r,\beta)}\psi_{\beta,\rho}
\end{align*}
so that there exists a $\rho\in\frak{g}_{\beta}(\Bbb F)\sphat$ such
that 
\begin{align*}
 \sigma\hookrightarrow
    \text{\rm Ind}_{Z(O_r,\beta)}^{G(O_r,\beta)}\psi_{\beta,\rho}
 &=\text{\rm Ind}_{K_{l-1}(O_r)}^{G(O_r,\beta)}
   \left(\text{\rm Ind}_{Z(O_r,\beta)}^{K_{l-1}(O_r)}\psi_{\beta,\rho}
         \right)\\
 &=\bigoplus^{\dim\pi_{\beta,\rho}}
   \text{\rm Ind}_{K_{l-1}(O_r)}^{G(O_r,\beta)}\pi_{\beta,\rho}
  =\bigoplus^{\dim\pi_{\beta,\rho}}\bigoplus_{\theta}\sigma_{\theta,\psi},
\end{align*}
where $\bigoplus_{\theta}$ is the direct sum over 
$\theta\in\mathcal{C}\sphat$ such that $\theta=\psi_{\beta,\rho}$ on 
$\mathcal{C}\cap K_{l-1}(O_r)$. 
Then we have $\sigma=\sigma_{\theta,\rho}$ for some 
$(\theta,\rho)\in
 {\mathcal C}\sphat{\times}_{K_{l-1}(O_r)}\frak{g}_{\beta}(\Bbb F)\sphat$.
\end{proof}

Under the conditions of Theorem \ref{th:main-result}, we can put 
$\mathcal{C}=G_{\beta}(O_r)$. We have the following proposition
by which our Theorem \ref{th:main-result} is given as a special case
of Proposition \ref{prop:generalized-main-result-for-odd-r}.

\begin{prop}\label{prop:fundamental-bijection-of-parameter}
If $G_{\beta}$ is commutative smooth over $O$, then 
$(\theta,\rho)\mapsto\theta$ gives a bijection of 
$G_{\beta}(\Bbb F)\sphat{\times}_{K_{l-1}(O_r)}\frak{g}_{\beta}(\Bbb F)\sphat$ 
onto the set
$$
  \left\{\theta\in G_{\beta}(O_r)\sphat\;\;\;
         \text{\rm s.t. $\theta=\psi_{\beta}$ on 
               $G_{\beta}(O_r)\cap K_l(O_r)$}\right\}.
$$
\end{prop}
\begin{proof}
Take a 
$(\theta,\rho)\in 
 G_{\beta}(\Bbb F)\sphat{\times}_{K_{l-1}(O_r)}\frak{g}_{\beta}(\Bbb F)\sphat$.
The smoothness of $G_{\beta}$ over $O$ implies that the canonical
mapping $\frak{g}_{\beta}(O)\to\frak{g}_{\beta}(\Bbb F)$ is
surjective. So Take a $\overline X\in\frak{g}_{\beta}(\Bbb F)$ with 
$X\in\frak{g}_{\beta}(O)$. Then we have
$$
 g=1+\varpi^{l-1}X+2^{-1}\varpi^{2l-2}X^2\npmod{\frak{p}^r}
 \in K_{l-1}(O_r)\cap G_{\beta}(O_r)
$$
so that
\begin{align*}
 \theta(g)=\psi_{\beta,\rho}(g)
 &=\tau\left(\varpi^{-l}B(X+2^{-1}\varpi^{l-1}X^2,\beta)
             -2^{-1}\varpi^{-1}B(X,\beta)\right)\cdot
   \rho(\overline X)\\
 &=\tau\left(\varpi^{-l}B(X,\beta)\right)\cdot\rho(\overline X).
\end{align*}
Hence we have 
$$
 \rho(\overline X)
 =\tau\left(-\varpi^{-l}B(X,\beta)\right)\cdot
  \theta\left(1+\varpi^{l-1}X+2^{-1}\varpi^{2l-2}X^2\npmod{\frak{p}^r}
              \right).
$$
This means that the mapping $(\theta,\rho)\mapsto\theta$ is
injective. Take $X, X^{\prime}\in\frak{g}_{\beta}(O)$ such that 
$X\equiv X^{\prime}\pmod{\frak p}$. Then we have 
$X^{\prime}=X+\varpi T$ with $T\in\frak{g}_{\beta}(O)$ and
\begin{align*}
 &1+\varpi^{l-1}X^{\prime}
                   +2^{-1}\varpi^{2l-2}X^{\prime 2}\pmod{\frak{p}^r}\\
 =&1+\varpi^{-1}X+2^{-1}\varpi^{2l-2}X^2+\varpi^lT\pmod{\frak{p}^r}\\
 =&(1+\varpi^{l-1}X+2^{-1}\varpi^{2l-2}X^2)(1+\varpi^lT)
         \pmod{\frak{p}^r},
\end{align*}
where $1+\varpi^lT\pmod{\frak{p}^r}\in K_l(O_r)$ and hence
$$
 \theta(1+\varpi^lT\npmod{\frak{p}^r})
 =\psi_{\beta}(1+\varpi^lT\npmod{\frak{p}^r})
 =\tau\left(\varpi^{-(l-1)}B(T,\beta)\right).
$$
This and the commutativity of $G_{\beta}$ show that
$$
 \rho(\overline X)
 =\tau\left(-\varpi^{-l}B(X,\beta)\right)\cdot
  \theta\left(1+\varpi^{l-1}X+2^{-1}\varpi^{2l-2}X^2\npmod{\frak{p}^r}
              \right)
$$
with $\overline X\in\frak{g}_{\beta}(\Bbb F)$ with
$X\in\frak{g}_{\beta}(O)$ gives an well-defined group homomorphism of 
$\frak{g}_{\beta}(\Bbb F)$ to $\Bbb C^{\times}$. Then 
$(\theta,\rho)\in 
 G_{\beta}(O_r)\sphat{\times}_{K_{l-1}(O_r)}\frak{g}_{\beta}(\Bbb F)\sphat$ 
and our mapping in question is surjective.
\end{proof}
         
\subsection[]{}
\label{subsec:suff-condition-for-smooth-commutativeness-of-g-beta}
We will give a sufficient condition on $\beta\in\frak{g}(O)$ under which 
$G_{\beta}$ is commutative and smooth over $O$. 

Let us assume that the connected $O$-group scheme $G$ is reductive,
that is, the fibers $G{\otimes}_OK$ ($K=F,\Bbb F$) are reductive
$K$-algebraic groups. In this case the dimension of the maximal torus
in $G{\otimes}_OK$ is independent of $K$ which is denoted by 
$\text{\rm rank}(G)$. For any $\beta\in\frak{g}(O)$ we have
$$
 \dim_K\frak{g}_{\beta}(K)=\dim\frak{g}_{\beta}
 \geq\dim G_{\beta}{\otimes}K\geq\text{\rm rank}(G).
$$
We say $\beta$ to be {\it smoothly regular} with respect to $G$ over
$K$ (or $\overline\beta\in\frak{g}(K)$ is smoothly regular with
respect to $G{\otimes}_OK$) 
if $\dim_K\frak{g}_{\beta}(K)=\text{\rm rank}(G)$ 
(see \cite[1.4]{Springer1971}). In this case $G_{\beta}{\otimes}_OK$
is smooth over $K$. If $\beta$ is smoothly regular with respect to $G$
over $F$ and over $\Bbb F$, then $\beta$ is said to be smoothly regular
with respect to $G$. 

We say $\beta$ to be {\it connected} with respect to $G$ if the fibers 
$G_{\beta}{\otimes}_OK$ ($K=F, \Bbb F$) are connected. 

\begin{thm}
\label{th:sufficient-condition-for-smooth-commutativeness-of-g-beta}
If $\beta\in\frak{g}(O)$ is smoothly regular and connected with
respect to $G$, then $G_{\beta}$ is commutative and smooth over $O$.
\end{thm}
\begin{proof}
Let $G_{\beta}^o$ be the neutral component of $O$-group scheme
$G_{\beta}$ which is a group functor of the category of $O$-scheme 
(see $\S 3$ of Expos\'e $\text{\rm VI}_B$ in \cite{SGA-3}).
The following statements are equivalent;
\begin{enumerate}
\item $G_{\beta}^o$ is representable as an smooth open $O$-group subscheme of
      $G_{\beta}$, 
\item $G_{\beta}$ is smooth at the points of unit section,
\item each fibers $G_{\beta}{\otimes}_OK$ ($K=F,\Bbb F$) are smooth
      over $K$ and their dimensions are constant
\end{enumerate}
(see Th. 3.10 and Cor. 4.4 of \cite{SGA-3}). So if $\beta$ is
smoothly regular with respect to $G$, then $G_{\beta}^o$ is smooth
open $O$-group subscheme of $G_{\beta}$. If further $\beta$ is connected
with respect to $G$, then $G_{\beta}^o=G_{\beta}$ is smooth over
$O$. Let $\beta=\beta_s+\beta_n$ be the Jordan decomposition of 
$\beta\in\frak{g}_{\beta}(F)$. Then the identity component 
$\mathcal{G}$ of the centralizer $Z_{G{\otimes}_OF}(\beta_s)$ is a
reductive $F$-algebraic group and 
$$
 G_{\beta}{\otimes}_OF=Z_{\mathcal{G}}(\beta_n)
$$
because $G_{\beta}{\otimes}_OF$ is connected. Then \cite{Lou1968}
shows that $G_{\beta}(\overline F)$ is commutative ($\overline F$ is
the algebraic closure of $F$), ans hence $G_{\beta}$ is a commutative
$O$-scheme. 
\end{proof}

\subsection[]{}\label{subsec:conjectures-on-triviality-of-c-beta-rho}
With the detailed discussion given in the section
\ref{sec:properties-of-c-beta-rho} and the results of 
\cite{Takase2016}, the truth of the following statement is highly
probable;
\begin{quote}
Assume that $G$ is connected smooth reductive $O$-group scheme. If 
$\beta\in\frak{g}(O)$ is smoothly regular with respect to $G$ over
$\Bbb F$, then the Schur multiplier 
$[c_{\beta,\rho}]\in H^2(G_{\beta}(\Bbb F)^{(c)},\Bbb C^{\times})$ is
trivial for all $\rho\in\frak{g}_{\beta}(\Bbb F)\sphat$ 
provided that the characteristic of $\Bbb F$ is big enough.
\end{quote}



\section{Properties of $c_{\beta,\rho}$}
\label{sec:properties-of-c-beta-rho}
In this section we will present several properties of the Schur
multiplier 
$[c_{\beta,\rho}]\in H^2(G_{\beta}(\Bbb F)^{(c)},\Bbb C^{\times})$
defined in subsection
\ref{subsec:def-of-schur-multiplier-associated-with-alg-group}. 
We will keep the notations and conventions of section
\ref{sec:main-results}, while 
 only the structure of algebraic groups over finite field 
$\Bbb F$ is required to define and to discuss the Schur multiplier.

\subsection[]{}
\label{subsec:scalor-extension}
Let $\Bbb K/\Bbb F$ be a finite field extension. More specifically
take the unramified extension $K/F$ with the integer ring 
$O_K\subset K$ such that $\Bbb K=O_K/\varpi O_K$. Then 
$\frak{g}{\otimes}_OO_K$ is the Lie algebra of smooth $O_K$-group
scheme $G{\otimes}_OO_K$ and 
$$
 \frak{g}(\Bbb K)=(\frak{g}{\otimes}_OO_K)(\Bbb K)
                 =\frak{g}(\Bbb F){\otimes}_{\Bbb F}\Bbb K.
$$
So $B:\frak{g}(\Bbb K)\times\frak{g}(\Bbb K)\to\Bbb K$ is
non-degenerate. 

We will assume that the degree of the extension $(\Bbb K:\Bbb F)$ is
not divisible by the characteristic of $\Bbb F$, and put 
$T^{\prime}_{\Bbb K/\Bbb F}
 =(\Bbb K:\Bbb F)^{-1}T_{\Bbb K/\Bbb F}$. Then the additive character
$$
 \widetilde\tau:\Bbb K\xrightarrow{T^{\prime}_{\Bbb K/\Bbb F}}
                \Bbb F\xrightarrow{\widehat\tau}
                \Bbb C^{\times}
$$
is an extension of $\widehat\tau:\Bbb F\to\Bbb C^{\times}$. 

Fix a $\beta\in\frak{g}(O)$ such that 
$\frak{g}_{\beta}(\Bbb F)\lvertneqq\frak{g}(\Bbb F)$. Then for any 
$\rho\in\frak{g}_{\beta}(\Bbb F)$, the additive character
$$
 \widetilde\rho:(\frak{g}{\otimes}_OO_K)_{\beta}(\Bbb K)
                =(\frak{g}_{\beta})(\Bbb F){\otimes}_{\Bbb F}\Bbb K
                \xrightarrow{1\otimes T^{\prime}_{\Bbb K/\Bbb F}}
                \frak{g}_{\beta}(\Bbb F)\xrightarrow{\rho}
                \Bbb C^{\times}
$$
is an extension of $\rho:\frak{g}_{\beta}(\Bbb F)\to\Bbb C^{\times}$. 
Then we have 

\begin{prop}
\label{prop:scalor-extension-and-schur-multiplier-associated-with-alg-group}
The Schur multiplier 
$[c_{\beta,\rho}]\in H^2(G(\Bbb F)_{\beta}^{(c)},\Bbb C^{\times})$ is
the image of 
$[c_{\beta,\widetilde\rho}]
 \in H^2(G(\Bbb K)_{\beta}^{(c)},\Bbb C^{\times})$ under the
 restriction mapping 
$$
 \text{\rm Res}:H^2(G(\Bbb K)_{\beta}^{(c)},\Bbb C^{\times})
                \to
                H^2(G(\Bbb F)_{\beta}^{(c)},\Bbb C^{\times}).
$$
\end{prop}
\begin{proof}
There exists a $\Bbb F$-basis
$\{\alpha_{\lambda}\}_{\lambda\in\Lambda}$ of $\Bbb K$ such that 
$T_{\Bbb K/\Bbb F}(\alpha_{\lambda})=1$ for all $\lambda\in\Lambda$. 
Let $v\mapsto[v]$ be a $\Bbb F$-linear section of the exact sequence 
\eqref{eq:fundamental-exact-sequence-of-lie-alg-and-symplectic-space}. 
Its $\Bbb K$-linear extension gives a $\Bbb K$-linear section of the
exact sequence 
$$
 0\to\frak{g}_{\beta}(\Bbb F){\otimes}_{\Bbb F}\Bbb K
  \to\frak{g}(\Bbb F){\otimes}_{\Bbb F}\Bbb K
  \to\Bbb V_{\beta}{\otimes}_{\Bbb F}\Bbb K\to 0.
$$
Take a 
$g\in G(\Bbb F)_{\beta}^{(c)}\subset G(\Bbb K)_{\beta}^{(c)}$ and
$$
 v=\sum_{\lambda\in\Lambda}v_{\lambda}\otimes\alpha_{\lambda}
 \in\Bbb V_{\beta}{\otimes}_{\Bbb F}\Bbb K
 \qquad
 (v_{\lambda}\in\Bbb V_{\beta}). 
$$
Then we have
\begin{align*}
 \gamma(v,g)
 &=\text{\rm Ad}(g)^{-1}[v]-[v\sigma_g]\\
 &=\sum_{\lambda\in\Lambda}
    \left(\text{\rm Ad}(g)^{-1}[v_{\lambda}]-[v_{\lambda}\sigma_g]
           \right)\otimes\alpha_{\lambda}
  =\sum_{\lambda\in\Lambda}\gamma(v_{\lambda},g)\otimes\alpha_{\lambda},
\end{align*}
hence 
\begin{align*}
 \widetilde\rho\left(\gamma(v,g)\right)
 &=\prod_{\lambda\in\Lambda}\rho\left(
    \gamma((\Bbb K:\Bbb F)^{-1}v_{\lambda},g)\right)
  =\prod_{\lambda\in\Lambda}\widehat\tau\left(
    (\Bbb K:\Bbb F)^{-1}\langle
  v_{\lambda},v_g\rangle_{\beta}\right)\\
 &=\widetilde\tau\left(
    \sum_{\lambda\in\Lambda}\langle v_{\lambda},v_g\rangle_{\beta}
                        \alpha_{\lambda}\right)
  =\widetilde\tau\left(\langle v,v_g\rangle_{\beta}\right).
\end{align*}
So we have $c_{\beta,\rho}(g,h)=c_{\beta,\widetilde\rho}(g,h)$ for all 
$g,h\in G(\Bbb F)_{\beta}^{(c)}\subset G(\Bbb K)_{\beta}^{(c)}$.
\end{proof}

\subsection[]{}
\label{subsec:schur-multiplier-and-over-group}
Let us assume that there exists a closed smooth $O$-group subscheme 
$H\subset GL_n$ of which our $G$ is a closed $O$-group subscheme and
that the trace form
$$
 B:\frak{h}(\Bbb F)\times\frak{h}(\Bbb F)\to\Bbb F
$$ 
is non-degenerate where $\frak{h}$ is the Lie algebra of $H$. 
Then we have 
$\frak{h}(\Bbb F)=\frak{g}(\Bbb F)\oplus\frak{g}(\Bbb F)^{\perp}$ where 
$\frak{g}(\Bbb F)^{\perp}
 =\{X\in\frak{h}(\Bbb F)\mid\widetilde B(X,\frak{g}(\Bbb F))=0\}$ 
is the orthogonal complement of $\frak{g}(\Bbb F)$ in 
$\frak{h}(\Bbb F)$. 

Take a $\beta\in\frak{g}(O)$ such that 
$\frak{g}_{\beta}(\Bbb F)\lvertneqq\frak{g}(\Bbb F)$. Then 
$\beta\in\frak{h}(O)$ and 
$\frak{h}_{\beta}(\Bbb F)\lvertneqq\frak{h}(\Bbb F)$ where 
$\frak{h}_{\beta}=Z_{\frak h}(\beta)$ is the centralizer. We have
decompositions 
$\frak{h}_{\beta}(\Bbb F)
 =\frak{g}_{\beta}(\Bbb F)\oplus
  \left(\frak{g}(\Bbb F)^{\perp}\right)_{\beta}$ where 
$\left(\frak{g}(\Bbb F)^{\perp}\right)_{\beta}
 =\frak{h}_{\beta}(\Bbb F)\cap\frak{g}(\Bbb F)^{\perp}$, and
$$
 \widetilde{\Bbb V}_{\beta}
 =\frak{h}(\Bbb F)/\frak{h}_{\beta}(\Bbb F)
 =\Bbb V_{\beta}\oplus
  \left(\frak{g}(\Bbb F)^{\perp}/
   \left(\frak{g}(\Bbb F)^{\perp}\right)_{\beta}\right)
$$
is an orthogonal decomposition of symplectic spaces. 

Let $v\mapsto[v]$ be a $\Bbb F$-linear section of the exact sequence 
$$
 0\to\frak{h}_{\beta}(\Bbb F)\to\frak{h}(\Bbb F)
  \to\widetilde{\Bbb V}_{\beta}\to 0
$$
of $\Bbb F$-vector space such that $[\Bbb V_{\beta}]=\frak{g}(\Bbb F)$
and  
$[\frak{g}(\Bbb F)^{\perp}/\left(\frak{g}(\Bbb F)^{\perp}\right)_{\beta}]
 =\frak{g}(\Bbb F)^{\perp}$. 

Take $\rho\in\frak{g}_{\beta}(\Bbb F)\sphat$ and put
$$
 \widetilde\rho:\frak{h}_{\beta}
 =\frak{g}_{\beta}\oplus\left(\frak{g}^{\perp}\right)_{\beta}
 \xrightarrow{\text{\rm projection}}\frak{g}_{\beta}
 \xrightarrow{\rho}\Bbb C^{\times}.
$$
For any $g\in G_{\beta}(\Bbb F)\subset H_{\beta}(\Bbb F)$, there
exists uniquely a $v_g\in\Bbb V_{\beta}$ such that
$$
 \rho\left(\gamma_{\frak g}(v,g)\right)
 =\widehat\tau\left(\langle v,v_g\rangle_{\beta}\right)
$$
for all $v\in\Bbb V_{\beta}$. Then we have
$$
 \widetilde\rho\left(\gamma_{\frak{h}}(v,v_g)\right)
 =\widehat\tau\left(\langle v,v_g\rangle_{\beta}\right)
$$
for all $v\in\widetilde{\Bbb V}_{\beta}$. In fact if we put 
$v=v^{\prime}+v^{\prime\prime}$ with $v^{\prime}\in\Bbb V_{\beta}$ and 
$v^{\prime\prime}\in
 \frak{g}(\Bbb F)^{\perp}/\left(\frak{g}(\Bbb F)^{\perp}\right)_{\beta}$,
then we have 
$\gamma_{\frak{h}}(v,g)
 =\gamma_{\frak g}(v^{\prime},g)
 +\gamma_{\frak{h}}(v^{\prime\prime},g)$ with 
$\gamma_{\frak{h}}(v^{\prime\prime},g)\in
 \left(\frak{g}(\Bbb F)^{\perp}\right)_{\beta}$, since 
$$
 \text{\rm Ad}(g)\frak{g}(\Bbb F)^{\perp}=\frak{g}(\Bbb F)^{\perp},
 \qquad
 \text{\rm Ad}(g)\left(\frak{g}(\Bbb F)^{\perp}\right)_{\beta}
 =\left(\frak{g}(\Bbb F)^{\perp}\right)_{\beta}.
$$
Then we have
\begin{align*}
 \widetilde\rho\left(\gamma_{\frak{h}}(v,g)\right)
 &=\rho\left(\gamma_{\frak g}(v^{\prime},g)\right)
  =\widehat\tau\left(\langle v^{\prime},v_g\rangle_{\beta}\right)\\
 &=\widehat\tau\left(\langle v,v_g\rangle_{\beta}\right)
\end{align*}
because $\langle v^{\prime\prime},v_g\rangle_{\beta}=0$. Hence we have

\begin{prop}
\label{prop:regular-schur-multiplier-is-restriction-from-upper-group}
If $G(\Bbb F)_{\beta}^{(c)}\subset H(\Bbb F)_{\beta}^{(c)}$ then 
the Schur multiplier 
      $[c_{\beta,\rho}]
       \in H^2(G(\Bbb F)_{\beta}^{(c)},\Bbb C^{\times})$ is the image
       under the restriction mapping 
$$
 \text{\rm Res}:H^2(H(\Bbb F)_{\beta}^{(c)},\Bbb C^{\times})\to
                H^2(G(\Bbb F)_{\beta}^{(c)},\Bbb C^{\times})
$$
      of the Schur multiplier 
      $[c_{\beta,\widetilde\rho}]
       \in H^2(H(\Bbb F)_{\beta}^{(c)},\Bbb C^{\times})$.
\end{prop}

\subsection[]{}\label{subsec:schur-multiplier-and-jordan-decomposition}
Take a $\beta\in\frak{g}(O)$ such that 
$\frak{g}_{\beta}(\Bbb F)\lvertneqq\frak{g}(\Bbb F)$. 

Let $\overline\beta=\beta_s+\beta_n$ be the Jordan decomposition of 
$\overline\beta=\beta\npmod{\frak p}\in\frak{g}(\Bbb F)$ 
($\beta_s,\beta_n\in\frak{g}(\Bbb F)$ are repectively the semisimple
part and the nilpotent part of $\overline\beta$). Put
$$
 L=Z_{G{\otimes}_O\Bbb F}(\beta_s),
 \quad
 \frak{l}=\text{\rm Lie}(L)(\Bbb F)
         =Z_{\frak{g}(\Bbb F)}(\beta_s).
$$
Let us assume the following assumption;

\begin{assu}\label{assume:b-over-l-is-non-degenerate}
$B:\frak{l}\times\frak{l}\to\Bbb F$ is non-degenerate.
\end{assu}

Then we have $\frak{g}(\Bbb F)=\frak{l}\oplus\frak{l}^{\perp}$ where
$$
 \frak{l}^{\perp}
 =\{X\in\frak{g}(\Bbb F)\mid B(X,\frak{l})=0\}
$$
is an $\text{\rm Ad}(L(\Bbb F))$-submodule of $\frak{g}(\Bbb F)$. Hence there
exists a $\Bbb F$-vector subspace 
$\frak{l}^{\perp}\subset V\subset\frak{g}(\Bbb F)$ such that 
$\frak{g}(\Bbb F)=V\oplus\frak{g}_{\beta}(\Bbb F)$. We can fix a 
$\Bbb F$-linear section $v\mapsto[v]$ 
on $\Bbb V_{\beta}$ of the exact sequence 
$$
 0\to\frak{g}_{\beta}(\Bbb F)\to\frak{g}(\Bbb F)\to\Bbb V_{\beta}
  \to 0
$$
such that $[v]\in V$ for all $v\in\Bbb V_{\beta}$. Then we have

\begin{prop}\label{prop:schur-multiplier-and-jordan-decomposition}
Fix a $\rho\in\frak{g}_{\beta}(\Bbb F)\sphat$. 
Under the assumption \ref{assume:b-over-l-is-non-degenerate}, we have
\begin{enumerate}
\item $X_g=[v_g]\in\frak{l}$ for any $g\in G_{\beta}(\Bbb F)^{(c)}$,
\item for any extension $\widetilde\rho\in\frak{l}\sphat$ of $\rho$,
  we have
$$
 c_{\beta,\rho}(g,h)
 =\widetilde\rho\left(2^{-1}\text{\rm Ad}(g)X_h\right)\cdot
  \widetilde\rho\left(2^{-1}X_{gh}\right)\cdot
  \widetilde\rho\left(2^{-1}X_g\right)
$$
  for all $g,h\in G_{\beta}(\Bbb F)^{(c)}$.
\end{enumerate}
\end{prop}
\begin{proof}
1) Take any $X\in\frak{l}^{\perp}$ and put 
$v=\dot X\in
 \Bbb V_{\beta}=\frak{g}(\Bbb F)/\frak{g}_{\beta}(\Bbb F)$. Then we
 have
$$
 \gamma(v,g)=\text{\rm Ad}(g)^{-1}[v]-[v\sigma_g]=0
$$
because $[v]=X$ and 
$\text{\rm Ad}(g)^{-1}X\in\frak{l}^{\perp}\subset V$ for any 
$g\in G_{\beta}(\Bbb F)^{(c)}$. Then we have
$$
 \widehat\tau(B([X,X_g],\overline\beta)
 =\widehat\tau(\langle v,v_g\rangle_{\beta})
 =\rho(\gamma(v,g))=1
$$
and hence
$$
 B(X,[X_g,\overline\beta])=B([X,X_g],\overline\beta)=0
$$
for all $X\in\frak{l}^{\perp}$. This means that 
$[X_g,\overline\beta]\in(\frak{l}^{\perp})^{\perp}=\frak{l}$. Since
$[\beta_s,\beta]=0$ and $\frak{l}=Z_{\frak{g}(\Bbb F)}(\beta_s)$, we
have
$$
 [[\beta_s,X_g],\overline\beta]=[\beta_s,[X_g,\overline\beta]]=0
$$
that is 
$[\beta_s,X_g]\in\frak{g}_{\beta}(\Bbb F)\subset\frak{l}$. Finally 
$$
 B(Y,[\beta_s,X_g])=B([Y,\beta_s],X_g)=0
$$
for all $Y\in\frak{l}$, so we have $[\beta_s,X_g]=0$, that is 
$X_g\in\frak{l}$.

2) Take $g,h\in G_{\beta}(\Bbb F)^{(c)}$. The relation 
\ref{eq:multiplication-formula-of-v-epsilon} gives 
$v_g\sigma_g=-v_{g^{-1}}$ and 
$$
 [v_h\sigma_{g^{-1}}]=X_{gh}-X_g\in\frak{l}.
$$
Then we have
\begin{align*}
 c_{\beta,\rho}(g,h)
 &=\widehat\tau\left(2^{-1}\langle v_h,v_{g^{-1}}\rangle_{\beta}\right)
  =\rho\left(2^{-1}\gamma(v_h,g^{-1})\right)\\
 &\widetilde\rho\left(2^{-1}\{\text{\rm Ad}(g)X_h-(X_{gh}-X_g)\}
                       \right).
\end{align*}
\end{proof}

\begin{rem}\label{remark:v-g-is-trivial-if-trace-form-is-non-deg-on-g-beta}
Fix a $\beta\in\frak{g}(O)$. If 
$B:\frak{g}_{\beta}(\Bbb F)\times\frak{g}_{\beta}(\Bbb F)\to\Bbb F$ is
non-degenerate, then we have 
$\frak{g}(\Bbb F)
 =\frak{g}_{\beta}(\Bbb F)^{\perp}\oplus\frak{g}_{\beta}(\Bbb F)$
 where
$$
 \frak{g}_{\beta}(\Bbb F)^{\perp}
 =\{X\in\frak{g}(\Bbb X)\mid B(X,\frak{g}_{\beta}(\Bbb F))=0\}
$$
is a $\text{\rm Ad}(G_{\beta}(\Bbb F))$-invariant 
$\Bbb F$-subspace. So if we choose a $\Bbb F$-linear section 
$v\mapsto[v]$ on 
$\Bbb V_{\beta}=\frak{g}(\Bbb F)/\frak{g}_{\beta}(\Bbb F)$ of the
canonical exact sequence of $\Bbb F$-vector space
$$
 0\to\frak{g}_{\beta}(\Bbb F)\to\frak{g}(\Bbb F)\to\Bbb V\to 0
$$
so that $[v]\in\frak{g}_{\beta}(\Bbb F)^{\perp}$ for all $v\in\Bbb V_{\beta}$,
then we have $v_g=0$ for all $g\in G_{\beta}(\Bbb F)$. 
In this case have $c_{\beta,\rho}(g,h)=1$ for all 
$g,h\in G_{\beta}(\Bbb F)^{(c)}$.
\end{rem}


\subsection[]{}\label{subsec:smoothly-regular-in-reductive-group}
In this subsection, we will consider the relation between the
regularity of $\overline\beta\in\frak{g}(\Bbb F)$ and the triviality
of the Schur multiplier 
$[c_{\beta,\rho}]\in H^2(G_{\beta}(\Bbb F)^{(c)},\Bbb C^{\times})$. 

Put $K=F$ or $K=\Bbb F$. 
Let us suppose that $G{\otimes}_O K$ is
connected reductive algebraic $ K$-group and the characteristic of
$ K$ is not bad with respect to $G{\otimes}_O K$. 
The list of the bad primes is 
\begin{center}
\begin{tabular}{c|c|c|c|c|c|c}
type of $G{\otimes}_O K$   
   &$A_r$      &$B_r, D_r$&$C_r$&$E_6, E_7, F_4$&$E_8$&$G_2$\\
\hline
bad prime&$\emptyset$&$2$       &$2$  &$2,3$        &$2,3,5$&$2,3$
\end{tabular}
\end{center}
(see \cite[p.178, I-4.3]{Borel1968}). 

Take a $\beta\in\frak{g}(O)$ and let $\overline\beta=\beta_s+\beta_n$
be the Jordan decomposition of $\overline\beta\in\frak{g}( K)$
into the semi-simple part $\beta_s\in\frak{g}( K)$ and the
nilpotent part $\beta_n\in\frak{g}( K)$
($\overline\beta\in\frak{g}(K)$ is the image of $\beta\in\frak{g}(O)$
under the canonical mapping $\frak{g}(O)\to\frak{g}(K)$). 
The identity component 
$L=Z_{G{\otimes}_O K}(\beta_s)^o$ of the centralizer of $\beta_s$ in 
$G{\otimes}_O K$ is a reductive group over $ K$ and there
exists a maximal torus $T$ of $G{\otimes}_O K$ such that
$$
 \beta_s\in\text{\rm Lie}(T)( K)
$$
(see \cite[Prop.13.19]{Borel1991} and its proof). 
Then $T\subset L$ and 
$\text{\rm rank}(L)=\text{\rm rank}(G)$. Put 
$\frak{l}=\text{\rm Lie}(L)$, then 
$\frak{l}=Z_{\frak{g}{\otimes}_O K}(\beta_s)$. 
So $\overline\beta\in\frak{g}( K)$ is smoothly regular with
respect to $G{\otimes}_O K$ if and only if 
$\beta_n\in\frak{l}( K)$ is smoothly regular with respect to $L$. 

Now fix a system of
positive roots $\Phi^+$ in the root system $\Phi(T,L)$ of
$L$ with respect to $T$ such that 
$$
 \beta_n=\sum_{\alpha\in\Phi^+}c_{\alpha}\cdot X_{\alpha}
$$
where $X_{\alpha}$ is a root vector of the root $\alpha$. Then 
the result of \cite[p.228,III-3.5]{Borel1968} implies 

\begin{prop}
\label{prop:regularity-in-terms-of-coefficient-of-root-vector}
$\overline\beta\in\frak{g}( K)$ is smoothly regular with respect
to $G$ over $K$ if and only if 
$c_{\alpha}\neq 0$ for all simple $\alpha\in\Phi^+$. 
\end{prop}

\begin{rem}\label{remark:smoothly-regular-in-gl-n}
Let us consider the case of $GL_n$ which is a connected smooth
reductive $O$-group scheme. For $\beta\in\frak{gl}_n(O)$, the
following statements are equivalent;
\begin{enumerate}
\item $\beta\in\frak{gl}_n(O)$ is smoothly regular with
      respect to $GL_n$ over $\Bbb F$,
\item $\overline\beta\in M_n(\Bbb F)$ is 
      $GL_n(\overline{\Bbb F})$-conjugate to 
$$
 J_{n_1}(\alpha_1)\boxplus\cdots\boxplus J_{n_r}(\alpha_r)=
 \begin{bmatrix}
  J_{n_1}(\alpha_1)&      &                 \\
                   &\ddots&                 \\
                   &      &J_{n_r}(\alpha_r)
 \end{bmatrix},
$$
      where $\alpha_1,\cdots,\alpha_r$ are distinct elements of the
      algebraic closure $\overline{\Bbb F}$ of $\Bbb F$ and
$$
 J_m(\alpha)=\begin{bmatrix}
              \alpha&   1  &      &      \\
                    &\alpha&\ddots&      \\
                    &      &\ddots&   1  \\
                    &      &      &\alpha\\
             \end{bmatrix}
$$
      is a Jordan block of size $m$,
\item the characteristic polynomial 
      $\chi_{\overline\beta}(t)
       =\det(t1_n-\overline\beta)\in\Bbb F[t]$ is the
      minimal polynomial of $\overline\beta\in M_n(\Bbb F)$,
\item $\{X\in M_n(\Bbb F)\mid X\overline\beta=\overline\beta X\}
       =\Bbb F[\overline\beta]$,
\item $\{X\in M_n(O_l)\mid X\beta\equiv\beta X\pmod{\frak{p}^l}\}
       =O_l[\beta_l]$ with
       $\beta_l=\beta\pmod{\frak{p}^l}$ for all $l>0$,
\item $\{X\in M_n(O)\mid X\beta=\beta X\}=O[\beta]$.
\end{enumerate}
In this case $\beta\in\frak{gl}_n(O)$ is smoothly regular with respect
to $GL_n$ over $F$ and the centralizer $GL_{n,\beta}$ is commutative
and smooth over $O$.
\end{rem}

The remark above and Proposition
\ref{prop:schur-multiplier-and-jordan-decomposition} combined with the
results of \cite[Examples 4.6.2--4]{Takase2016}, we have

\begin{prop}\label{prop:c-beta-rho-is-trivial-for-gl-n}
Take a $\beta\in\frak{gl}_n(O)$ such that 
\begin{enumerate}
\item $\beta$ is smoothly regular with respect to $GL_n$ over $\Bbb F$,
\item the multiplicities of the roots
      of the characteristic polynomial $\chi_{\overline\beta}(t)$ of 
      $\overline\beta\in M_n(\Bbb F)$ are at most $4$. 
\end{enumerate}
Then the Schur multiplier 
$[c_{\beta,\rho}]\in H^2(GL_{n,\beta}(\Bbb F)^{(c)},\Bbb C^{\times})$ is
trivial for all $\rho\in\frak{gl}_{n,\beta}(\Bbb F)\sphat$ provided that
the characteristic of $\Bbb F$ is big enough.
\end{prop}

The author does not hesitate to present the following 

\begin{conj}
\label{conjecture:schur-multiplier-is-trivial-for-all-rho-if-regular}
The Schur multiplier 
$[c_{\beta,\rho}]\in H^2(GL_{n,\beta}(\Bbb F)^{(c)},\Bbb C^{\times})$
is trivial for all $\rho:\frak{gl}_{n,\beta}(\Bbb F)\sphat$, 
if $\beta\in\frak{gl}_n(O)$ is smoothly regular with respect to 
 $GL_n$ over $\Bbb F$ and the characteristic of $\Bbb F$ is big enough. 
\end{conj}

In general if $\beta\in\frak{g}(O)\subset\frak{gl}_n(O)$ is smoothly
regular with respect to $GL_n$ over $\Bbb F$, then $\beta$ is also
smoothly regular with respect to $GL_n$ over $F$, hence 
$GL_{n,\beta}$ is smooth commutative $O$-group scheme. So 
$G_{\beta}\subset GL_{n,\beta}$ is commutative, and so we have
$$
 G_{\beta}(\Bbb F)^{(c)}=G_{\beta}(\Bbb F)
 \subset GL_{n,\beta}(\Bbb F)=GL_{n,\beta}(\Bbb F)^{(c)}.
$$
Then Proposition
\ref{prop:regular-schur-multiplier-is-restriction-from-upper-group}
says that the Schur multiplier 
$[c_{\beta,\rho}]\in H^2(G_{\beta}(\Bbb F)^{(c)},\Bbb C^{\times})$
with $\rho\in\frak{g}_{\beta}(\Bbb F)\sphat$ is the image under the
restriction mapping
$$
 \text{\rm Res}:H^2(GL_{n,\beta}(\Bbb F)^{(c)},\Bbb C^{\times})
                \to
                H^2(G_{\beta}(\Bbb F)^{(c)},\Bbb C^{\times})
$$
of the Schur multiplier 
$[c_{\beta,\widetilde\rho}]\in 
 H^2(GL_{n,\beta}(\Bbb F)^{(c)},\Bbb C^{\times})$. 

If $G\subset GL_n$ is a group symplectic similitude, 
a general orthogonal group with
respect to a quadratic form of odd variables or an unitary group 
(see section \ref{sec:example}), then 
Proposition
\ref{prop:regularity-in-terms-of-coefficient-of-root-vector} implies
the following; 
$\beta\in\frak{g}(O)\in\frak{gl}_n(O)$ is smoothly regular with respect to 
$G$ over $\Bbb F$ if and only if $\beta$ is smoothly regular with 
respect to $GL_n$ over $\Bbb F$. So Conjecture 
\ref{conjecture:schur-multiplier-is-trivial-for-all-rho-if-regular}
implies the following conjectural proposition

\begin{prop}
\label{prop:conjectural-proposition-on-triviality-of-schur-multiplier}
If $G\subset GL_n$ is a group of symplectic similitudes, 
a general orthogonal group with
respect to a quadratic form of odd variables or an unitary group, then 
the Schur multiplier 
$[c_{\beta,\rho}]\in H^2(G(\Bbb F)_{\beta}^{(c)},\Bbb C^{\times})$ is
trivial for all $\rho\in\frak{g}_{\beta}(\Bbb F)\sphat$ 
if $\beta\in\frak{g}(O)$ is regular with respect to $G$ over $\Bbb F$ 
and the characteristic of $\Bbb F$ is big enough. 
\end{prop}

These arguments provide good reasons for the
conjectural statement of subsection 
\ref{subsec:conjectures-on-triviality-of-c-beta-rho}. 


\section{Weil representations over finite field}
\label{sec:weil-representation-over-finite-field}
In this section we will use the notations of the preceding sections
and will suppose $r=2l-1\geq 3$ is odd so that $l^{\prime}=l-1>0$. 
Fix a 
$\beta\in\frak{g}(O)$ such that $\overline\beta\in\frak{g}(\Bbb F)$ is
not in the center of $\frak{g}(\Bbb F)$. 

\subsection[]{}
\label{subsec:k-l-1-as-group-extension}
A group extension
\begin{equation}
 0\to\frak{g}(O_{l-1})\xrightarrow{\diamondsuit}
     K_{l-1}(O_r)\xrightarrow{\heartsuit}
     \frak{g}(\Bbb F)\to 0
\label{eq:k-l-1-as-group-extension}
\end{equation}
is given by the canonical surjection 
\eqref{eq:canonical-surjection-of-k-l-1-to-g(f)}, whose kernel is
$K_l(O_r)$, with the group isomorphism 
$$
 \diamondsuit:\frak{g}(O_{l-1})\,\tilde{\to}\,K_l(O_r)
$$
defined by 
$S\npmod{\frak{p}^{l-1}}\mapsto(1+\varpi^lS)\npmod{\frak{p}^r}$. 

In order to determine the 2-cocycle of the group extension 
\eqref{eq:k-l-1-as-group-extension}, choose any mapping 
$\lambda:\frak{g}(\Bbb F)\to\frak{g}(O)$ such that 
$X=\lambda(X)\npmod{\frak p}$ for all $X\in\frak{g}(\Bbb F)$ and 
$\lambda(0)=0$, and define a section
$$
 l:\frak{g}(\Bbb F)\to K_{l-1}(O_r)
$$
of \eqref{eq:k-l-1-as-group-extension} by 
$X\mapsto 1+\varpi^{l-1}\lambda(X)
           +2^{-1}\varpi^{2l-2}\lambda(X)^2\npmod{\frak{p}^r}$. 
Then we have
$$
 l(X)^{-1}
 =1-\varpi^{l-1}\lambda(X)+2^{-1}\varpi^{2l-2}\lambda(X)^2
  \npmod{\frak{p}^r}
$$
for all $X\in\frak{g}(\Bbb F)$ and
$$
 l(X)(1+\varpi^lS)l(X)^{-1}
 \equiv 1+\varpi^lS\npmod{\frak{p}^r}
$$
for all $S_{l-1}\in\frak{g}(O_{l-1})$. Furthermore we have
$$
 l(X)l(Y)l(X+Y)^{-1}
 =1+\varpi^l\left(\mu(X,Y)+2^{-1}\varpi^{l-2}[\lambda(X),\lambda(Y)]
                  \right)\npmod{\frak{p}^r}
$$
for all $X,Y\in\frak{g}(\Bbb F)$ where 
$\mu:\frak{g}(\Bbb F)\times\frak{g}(\Bbb F)\to\frak{g}(O)$ 
is defined by 
$$
 \lambda(X)+\lambda(Y)-\lambda(X+Y)
 =\varpi\cdot\mu(X,Y)
$$
for all $X,Y\in\frak{g}(\Bbb F)$. Now we have two elements 
($2$-cocycle) 
$$
 \mu=[(X,Y)\mapsto\mu(X,Y)_{l-1}],
 \quad
 c=[(\overline X,\overline Y)\mapsto 2^{-1}\varpi^{l-2}[X,Y]_{l-1}]
$$
of $Z^2(\frak{g}(\Bbb F),\frak{g}(O_{l-1}))$ 
with trivial action of $\frak{g}(\Bbb F)$ on $\frak{g}(O_{l-1})$. 

We will consider two groups $\Bbb M$ and $\Bbb G$ corresponding to the two
2-cocycles $\mu$ and $c$ respectively. That is the group operation on 
$\Bbb M=\frak{g}(\Bbb F)\times\frak{g}(O_{l-1})$ is defined by
$$
 (X,S_{l-1})\cdot(Y,T_{l-1})=(X+Y,(S+T+\mu(X,Y))_{l-1})
$$
and the group operation on 
$\Bbb G=\frak{g}(\Bbb F)\times\frak{g}(O_{l-1})$ is defined by 
$$
 (\overline X,S_{l-1})\cdot(\overline Y,T_{l-1})
 =((X+Y)\sphat,(S+T+2^{-1}\varpi^{l-2}[X,Y])_{l-1}).
$$
Let $\Bbb G{\times}_{\frak{g}(\Bbb F)}\Bbb M$ be the fiber product of 
$\Bbb G$ and $\Bbb M$ with respect to the canonical projections 
onto $\frak{g}(\Bbb F)$. In other word 
$$
 \Bbb G{\times}_{\frak{g}(\Bbb F)}\Bbb M
 =\left\{(X;S,T)=((X,S),(X,T))\in\Bbb G\times\Bbb M\right\}
$$
is a subgroup of the direct product $\Bbb G\times\Bbb M$. 
We have a surjective group homomorphism 
\begin{equation}
 (\ast):\Bbb G{\times}_{\frak{g}(\Bbb F)}\Bbb M\to K_{l-1}(O_r)
\label{eq:fundamental-surjection-for-general-alg-group}
\end{equation}
defined by 
\begin{align*}
 (X;S_{l-1},T_{l-1})\mapsto&
                      l(X)\cdot(1+\varpi^l(S+T)\npmod{\frak{p}^r})\\
         &=1+\varpi^{l-1}\lambda(X)
            +2^{-1}\varpi^{2l-2}\lambda(X)^2
            +\varpi^l(S+T)\npmod{\frak{p}^r}.
\end{align*}

\subsection[]{}\label{subsec:reduction-to-intermediate-group}
A group homomorphism of additive groups 
$B_{\beta}:\frak{g}(O_{l-1})\to O_{l-1}$ ($X\mapsto B(X,\beta_{l-1})$)
induces a group homomorphism 
$B_{\beta}^{\ast}:H^2(\frak{g}(\Bbb F),\frak{g}(O_{l-1}))\to 
                  H^2(\frak{g}(\Bbb F),O_{l-1})$. Let us denote by 
$\mathcal{H}_{\beta}$ the group associated with the 2-cocycle 
$$
 c_{\beta}=B_{\beta}\circ c
 =\left[(\overline X,\overline Y)\mapsto
        2^{-1}\varpi^{l-2}B([X,Y],\beta)_{l-1}\right]
 \in Z^2(\frak{g}(\Bbb F),O_{l-1}).
$$
That is $\mathcal{H}_{\beta}=\frak{g}(\Bbb F)\times O_{l-1}$ with a
group operation 
$$
 (\overline X,s)\cdot(\overline Y,t)
 =((X+Y)\sphat,s+t+2^{-1}\varpi^{l-2}B([X,Y],\beta)_{l-1}).
$$
Then the center of $\mathcal{H}_{\beta}$ is 
$Z(\mathcal{H}_{\beta})=\frak{g}_{\beta}(\Bbb F)\times O_{l-1}$, the
direct product of two additive groups $\frak{g}_{\beta}(\Bbb F)$ and
$O_{l-1}$. 

The inverse
image of $Z(\mathcal{H}_{\beta})$ with respect to the surjective group
homomorphism 
\begin{equation}
 \spadesuit:\Bbb G{\times}_{\frak{g}(\Bbb F)}\Bbb M\to\mathcal{H}_{\beta}
 \qquad
 \left((X;S_{l-1},T_{l-1})\mapsto
       (X,B(S,\beta)_{l-1})\right)
\label{eq:fund-surjection-on-hesenberg-like-group-for-general-alg-group}
\end{equation}
is 
$\left(\Bbb G{\times}_{\frak{g}(\Bbb F)}\Bbb M\right)_{\beta}
 =\left\{(X;S,T)\in\Bbb G{\times}_{\frak{g}(\Bbb F)}\Bbb M\mid
         X\in\frak{g}(\Bbb F)_{\overline\beta}\right\}$ 
which is 
mapped onto $Z(O_r,\beta)\subset K_{l-1}(O_r)$ by the surjection 
\eqref{eq:fundamental-surjection-for-general-alg-group}. 

Take a $\rho\in\frak{g}_{\beta}(\Bbb F)\sphat$ which defines group
homomorphisms
$$
 \chi_{\rho}
 =\rho\otimes\left[x_{l-1}\mapsto\tau(\varpi^{-(l-1)}x)\right]
 :Z(\mathcal{H}_{\beta})
  =\frak{g}(\Bbb F)_{\overline\beta}\times O_{l-1}
 \to\Bbb C^{\times}
$$
and
$$
 \widetilde{\chi}_{\rho}:
  \left(\Bbb G{\times}_{\frak{g}(\Bbb F)}\Bbb M\right)_{\beta}
  \xrightarrow{\spadesuit}Z(\mathcal{H}_{\beta})
  \xrightarrow{\chi_{\beta}}\Bbb C^{\times}.
$$
On the other hand we have a group homomorphism
$$
 \widetilde{\psi}_0:
 \Bbb G{\times}_{\frak{g}(\Bbb F)}\Bbb M\to\Bbb C^{\times}
$$
defined by 
$\widetilde{\psi}_0(X;S_{l-1},T_{l-1})
 =\tau\left(\varpi^{-l}B(\lambda(X)+\varpi T,\beta)\right)$, and 
$\widetilde{\psi}_0\cdot\widetilde{\chi}_{\beta}$ is trivial on the
 kernel of the surjection
 \eqref{eq:fundamental-surjection-for-general-alg-group}
 and induces a group homomorphism $\psi_{\beta,\rho}\in Y_{\beta}$
 defined in subsection
 \ref{subsec:description-of-main-result-for-odd-r}. 

\subsection[]{}
\label{subsec:schrodinger-representation}
Fix a $\rho\in\frak{g}_{\beta}(\Bbb F)\sphat$. 

Let us determine the 2-cocycle of the group extension
\begin{equation}
 0\to Z(\mathcal{H}_{\beta})\to\mathcal{H}_{\beta}
  \xrightarrow{\clubsuit}\Bbb V_{\beta}\to 0
\label{eq:central-extension-of-h-beta}
\end{equation}
where $\clubsuit:\mathcal{H}_{\beta}\to\Bbb V_{\beta}$
is defined by 
$(X,s)\mapsto\dot X\npmod{\frak{g}_{\beta}(\Bbb F)}$. 
Fix a 
$\Bbb F$-linear section $v\mapsto[v]$ of the exact sequence
$$
 0\to\frak{g}_{\beta}(\Bbb F)\to\frak{g}(\Bbb F)
  \to\Bbb V_{\beta}\to 0
$$
$\Bbb V$-vector spaces and define a section 
$l:\Bbb V_{\beta}\to\mathcal{H}_{\beta}$ of the group
  extension \eqref{eq:central-extension-of-h-beta} by 
$l(v)=([v],0)$. Then we have
$$
 l(u)l(v)l(u+v)^{-1}
 =(0,2^{-1}\varpi^{l-2}B([X,Y],\beta)\npmod{\frak{p}^{l-1}})
$$
for 
$u=\dot{\overline X}, v=\dot{\overline Y}
 \in\Bbb V_{\beta}$ so that the 2-cocycle of the group extension 
\eqref{eq:central-extension-of-h-beta} is
$$
 \left[(\dot{\overline X},\dot{\overline Y})\mapsto
       2^{-1}\varpi^{l-2}B([X,Y],\beta)\npmod{\frak{p}^{l-1}}\right]
 \in Z^2(\Bbb V_{\beta},O_{l-1}).
$$
Define a group operation on 
$\Bbb H_{\beta}=\Bbb V_{\beta}\times Z(\mathcal{H}_{\beta})$
by 
$$
 \left(\dot{\overline X},z\right)\cdot\left(\dot{\overline Y},w\right)
 =\left(\dot{\overline{X+Y}},
    z+w+2^{-1}\varpi^{l-2}B([X,Y],\beta)\npmod{\frak{p}^{l-1}}\right).
$$
Then $\Bbb H_{\beta}$ is isomorphic to $\mathcal{H}_{\beta}$ by 
$(v,(Y,s))\mapsto([v]+Y,s)$.

 Let $H_{\beta}$ be the Heisenberg group
of the symplectic $\Bbb F$-space $\Bbb V_{\beta}$, that is 
$H_{\beta}=\Bbb V_{\beta}\times\Bbb C^1$ with a group
operation
$$
 (u,s)\cdot(v,t)
 =\left(u+v,s\cdot t\cdot
            \widehat\tau(2^{-1}\langle u,v\rangle_{\beta})
        \right).
$$
Then we have a surjective group homomorphism 
$$
 \Bbb H_{\beta}=\Bbb V_{\beta}\times Z(\mathcal{H}_{\beta})\to H_{\beta}
 \qquad
 ((v,z)\mapsto(v,\chi_{\rho}(z)).
$$
Fix a polarization 
$\Bbb V_{\beta}=\Bbb W^{\prime}\oplus\Bbb W$ of the symplectic
$\Bbb F$-space $\Bbb V_{\beta}$. Let us denote by 
$L^2(\Bbb W^{\prime})$ 
the complex vector space of the complex-valued functions $f$ on
$\Bbb W^{\prime}$ with inner product 
$(f,f^{\prime})
 =\sum_{w\in\Bbb W^{\prime}}f(w)\overline{f^{\prime}(w)}$.
The Schr\"odinger representation $(\pi^{\beta},L^2(\Bbb W^{\prime}))$ 
of $H_{\beta}$ associated with the polarization is defined for 
$(v,s)\in H_{\beta}$ and $f\in L^2(\Bbb W^{\prime})$ by
$$
 \left(\pi^{\beta}(v,s)f\right)(w)
 =s\cdot\widehat\tau\left(
   2^{-1}\langle v_-,v_+\rangle_{\overline\beta}
   +\langle w,v_+\rangle_{\overline\beta}\right)\cdot f(w+v_-)
$$
where $v=v_-+v_+\in\Bbb V_{\beta}$ with 
$v_-\in\Bbb W^{\prime}, v_+\in\Bbb W^{\prime}$. 

Now an irreducible
representation $(\pi^{\beta,\rho},L^2(\Bbb W^{\prime}))$ of 
$\Bbb H_{\beta}$ is defined by 
$\pi^{\beta,\rho}(v,z)=\pi^{\beta}(v,\chi_{\rho}(z))$, and an
irreducible representation 
$(\widetilde\pi^{\beta,\rho},L^2(\Bbb W^{\prime}))$ of 
$\Bbb G{\times}_{\frak{g}(\Bbb F)}\Bbb M$ is defined by
$$
 \widetilde\pi^{\beta,\rho}:
 \Bbb G{\times}_{\frak{g}(\Bbb F)}\Bbb M
 \xrightarrow{\spadesuit}\mathcal{H}_{\beta}
 \,\tilde{\to}\,\Bbb H_{\beta}
 \xrightarrow{\pi^{\beta,\rho}}
 GL_{\Bbb C}(L^2(\Bbb W^{\prime})).
$$
Then $\widetilde{\psi}_0\cdot\widetilde{\pi}_{\beta,\rho}$ is trivial
on the kernel of 
$(\ast):\Bbb G{\times}_{\frak{g}(\Bbb F)}\Bbb M\to K_{l-1}(O_r)$ so
that it induces an irreducible representation 
$\pi^{\beta,\rho}$ of $K_{l-1}(O_r)$ on $L^2(\Bbb W^{\prime})$. 

\begin{prop}
\label{prop:another-expression-of-pi-beta-psi}
Take a $g=1+\varpi^{l-1}T\npmod{\frak{p}^r}\in K_{l-1}(O_r)$ 
with $T\in\frak{gl}_n(O)$. Then we have 
$T\npmod{\frak{p}^{l-1}}\in\frak{g}(O_{l-1})$ and
$$
 \pi^{\beta,\rho}(g)
  =\tau\left(\varpi^{-l}B(T,\beta)
            -2^{-1}\varpi^{-1}B(T^2,\beta)\right)\cdot
   \rho(Y)\cdot\pi^{\beta}(v,1)
$$
where $\overline T=[v]+Y\in\frak{g}(\Bbb F)$ with 
$v\in\Bbb V_{\beta}$ and 
$Y\in\frak{g}_{\beta}(\Bbb F)$. In particular
$\pi^{\beta,\rho}(h)$ is the homothety $\psi_{\beta,\rho}(h)$ for all 
$h\in Z(O_r,\beta)$.
\end{prop}
\begin{proof}
By the definition we have
\begin{align*}
 \pi^{\beta,\psi}(&l(X)(1+\varpi^lS\npmod{\frak{p}^r}))\\
 =&\psi_0(X,0)\cdot\widetilde\pi^{\beta,\rho}([v]+Y;S_{l-1},0)\\
 =&\psi\left(l(Y)(1+\varpi^lS\npmod{\frak{p}^r})\right)\cdot
   \tau\left(\varpi^{-l}B(\lambda(X)-\lambda(Y),\beta)\right)\cdot
   \pi^{\beta}(v,1)\\
 =&\tau\left(
     \varpi^{-(l-1)}B(S,\beta)+\varpi^{-l}B(\lambda(X),\beta)\right)
      \cdot\rho(Y)\cdot\pi^{\beta}(v,1)
\end{align*}
where $X=[v]+Y\in\frak{g}(\Bbb F)$ with $v\in\Bbb V_{\beta}$
and $Y\in\frak{g}_{\beta}(\Bbb F)$. 
Put $1+\varpi^{l-1}T\equiv l(X)(1+\varpi^lS)\npmod{\frak{p}^r}$ with 
$X\in\frak{g}(\Bbb F)$ and $S\in\frak{g}(O)$. Then we have
$$
 1+\varpi^{l-1}T\equiv 1+\varpi^{l-1}\lambda(X)
 \npmod{\frak{p}^l}
$$
so that we have $T\npmod{\frak p}=X\in\frak{g}(\Bbb F)$ and 
$$
 \varpi S\equiv 
 T-\lambda(\overline T)-2^{-1}\varpi^{l-1}\lambda(\overline T)^2
 \npmod{\frak{p}^l}.
$$
Then we have
\begin{align*}
 \pi^{\beta,\rho}(g)
 =&\pi^{\beta,\rho}\left(l(\overline T)(1+\varpi^lS)_r\right)\\
 =&\tau\left(\varpi^{-(l-1)}B(S,\beta)
             +\varpi^{-l}B(\lambda(\overline T),\beta)\right)\cdot
   \rho(Y)\cdot\pi^{\beta}(v,1)\\
 =&\tau\left(\varpi^{-l}B(T,\beta)-2^{-1}\varpi^{-1}B(T^2,\beta)
             \right)\cdot\rho(Y)\cdot\pi^{\beta}(v,1).
\end{align*}
\end{proof}

This proposition shows that the irreducible representation 
$(\pi^{\beta,\rho},L^2(\Bbb W^{\prime}))$ $K_{l-1}(O_r)$ 
is exactly the irreducible representation $\pi_{\beta,\rho}$ defined
in Proposition
\ref{prop:existence-of-fundamental-rep-associated-with-psi}. 

\subsection[]{}
\label{subsec:action-of-g-o-f-beta}
Fix a $\rho\in\frak{g}_{\beta}(\Bbb F)\sphat$. 
In this subsection we will study the conjugate action of 
$g_r=g\npmod{\frak{p}^r}\in G(O_r,\beta)$ on $K_{l-1}(O_r)$ and on 
$\pi^{\beta,\rho}$. For any $X\in\frak{g}(\Bbb F)$, we have
$$
 g_r^{-1}l(X)g_r
 =l\left(\text{\rm Ad}(\overline g)^{-1}X)
         +\varpi^l\nu(X,g)\npmod{\frak{p}^r}\right)
$$
with $\nu(X,g)\in\frak{g}(O)$ such that 
$$
 \text{\rm Ad}(g)^{-1}\lambda(X)-\lambda(\text{\rm Ad}(g)^{-1}X)
 =\varpi\cdot\nu(X,g).
$$
Then we have
\begin{align*}
 &g_r^{-1}l(X)(1+\varpi^l(S+T)_r)g_r\\
 =&l(\text{\rm Ad}(\overline g)^{-1}X)
   \left(1+\varpi^l(\text{\rm Ad}(g)^{-1}S+\text{\rm Ad}(g)^{-1}T
          +\nu(X,g))\right)\npmod{\frak{p}^r}
\end{align*}
and an action of $g_r\in G(O_r,\beta)$ on 
$(X;S_{l-1},T_{l-1})
 \in\Bbb G{\times}_{\frak{g}(\Bbb F)}\Bbb M$ is defined by 
\begin{equation}
 (X;S_{l-1},T_{l-1})^{g_r}
 =\left(\text{\rm Ad}(\overline g)^{-1}X;
         (\text{\rm Ad}(g)^{-1}S)_{l-1},
         (\text{\rm Ad}(g)^{-1}T+\nu(X,g))_{l-1}\right)
\label{eq:action-on-fibre-product}.
\end{equation}
The action \eqref{eq:action-on-fibre-product} is compatible with the
action  
$$
 (X,s)^{g_r}=(\text{\rm Ad}(\overline g)^{-1}X,s)
$$ 
of $g_r\in G(O_r,\beta)$ on $(X,s)\in\mathcal{H}_{\beta}$ via
the surjection
\eqref{eq:fund-surjection-on-hesenberg-like-group-for-general-alg-group}. 
If
we put $X=[v]+Y\in\frak{g}(\Bbb F)$ with $v\in\Bbb V_{\beta}$
and $Y\in\frak{g}_{\beta}(\Bbb F)$, then we have
$$
 \text{\rm Ad}(\overline g)^{-1}X
 =[v\sigma_{\overline g}]+\gamma(v,\overline g)
                        +\text{\rm Ad}(\overline g)^{-1}Y
$$
in the notations of subsection
\ref{subsec:def-of-schur-multiplier-associated-with-alg-group}. So 
$g_r\in G(O_r,\beta)$ acts on $(v,(Y,s))\in\Bbb H_{\beta}$ by 
$$
 (v,(Y,s))^{g_r}
 =\left(v\sigma_{\overline g},
   (\text{\rm Ad}(\overline g)^{-1}Y+\gamma(v,\overline g),s)\right).
$$
In particular $g_r\in G_l(O_r,\beta)^{(c)}$ acts on 
$(v,z)\in\Bbb H_{\beta}$ by 
$$
 (v,z)^{g_r}
 =(v\sigma_{\overline g},(\gamma(v,\overline g),0)\cdot z).
$$
There exists a group homomorphism 
$T:Sp(\Bbb V_{\beta})\to GL_{\Bbb C}(L^2(\Bbb W^{\prime}))$
  such that 
$$
 \pi^{\beta}(v\sigma,s)
 =T(\sigma)^{-1}\circ\pi^{\beta}(v,s)\circ T(\sigma)
$$
for all $\sigma\in Sp(\Bbb V_{\beta})$ and $(v,s)\in
H_{\beta}$ (see \cite[Th.2.4]{Gerardin1977}). 
Then we have
\begin{align*}
 \pi^{\beta,\rho}((v,z)^{g_r})
 =&\pi^{\beta,\rho}
    \left(v\sigma_{\overline g},(\gamma(v,\overline g),0)\cdot z\right)\\
 =&\pi^{\beta}
    \left(v\sigma_{\overline g},
           \rho(\gamma(v,\overline g))\cdot\chi_{\rho}(z)\right)\\
 =&\pi^{\beta}(v\sigma_{\overline g},
    \widehat\tau\left(\langle v,v_{\overline g}\rangle_{\beta}
                       \right)\cdot\chi_{\rho}(z))\\
 =&T(\sigma_{\overline g})^{-1}\circ
   \pi^{\beta}(v,
    \widehat\tau\left(\langle v,v_{\overline g}\rangle_{\beta}
                       \right)\cdot\chi_{\rho}(z))\circ
   T(\sigma_{\overline g})\\
 =&T(\sigma_{\overline g})^{-1}\circ
   \pi^{\beta}
    \left((v_{\overline g},1)^{-1}(v,\chi_{\rho}(z))(v_{\overline g},1)
           \right)\circ
   T(\sigma_{\overline g})\\
 =&T(\sigma_{\overline g})^{-1}\circ
   \pi^{\beta}(v_{\overline g},1)^{-1}\circ
   \pi^{\beta,\rho}(v,z)\circ
   \pi^{\beta}(v_{\overline g},1)\circ
   T(\sigma_{\overline g}).
\end{align*}
If we put
$$
 U(g_r)
 =\pi^{\beta}(v_{\overline g},1)\circ T(\sigma_{\overline g})
 \in GL_{\Bbb C}(L^2(\Bbb W^{\prime}))
$$
for $g_r\in G_l(O_r,\beta)^{(c)}$ then we have
$$
 U(g_r)\circ U(h_r)
 =c_{\beta,\rho}(\overline g,\overline h)\cdot U((gh)_r)
$$
for all $g_r,h_r\in G_l(O_r,\beta)^{(c)}$, in fact 
\begin{align*}
 U(g_r)\circ U(h_r)
 &=\pi^{\beta}(v_{\overline g},1)\circ T(\sigma_{\overline g})\circ 
   \pi^{\beta}(v_{\overline h},1)\circ T(\sigma_{\overline h})\\
 &=\pi^{\beta}(v_{\overline g},1)\circ 
   \pi^{\beta}(v_{\overline h}^{{\overline g}^{-1}},1)\circ
   T(\sigma_{\overline g})\circ T(\sigma_{\overline h})\\
 &=\pi^{\beta}\left(v_{\overline g}+v_{{\overline h}^{{\overline g}^{-1}}},
    \widehat\tau\left(
      2^{-1}\langle v_{\overline g},v_{{\overline h}^{{\overline g}^{-1}}}\rangle_{\beta}
                    \right)\right)\circ
   T(\sigma_{\overline{gh}})\\
 &=c_{\beta,\rho}(\overline g,\overline h)\cdot
   \pi^{\beta}(v_{\overline{gh}},1)\circ T(\sigma_{\overline{gh}}).
\end{align*}
On the other hand 
\begin{align*}
  \widetilde\psi_0\left((X;S_{l-1},T_{l-1})^{g_r^{-1}}
                       \right)
 &=\tau\left(\varpi^{-l}B(\lambda(X)+\varpi T,\text{\rm Ad}(g)\beta)
             \right)\\
 &=\tau\left(\varpi^{-l}B(\lambda(X)+\varpi T,\beta)
             \right)
\end{align*}
for all $g_r\in G(O_r,\beta)^{(c)}$. That is $\widetilde{\psi}_0$
is invariant under the conjugate action of $G_l(O_r,\beta)^{(c)}$. 
Hence we have
$$
 \pi^{\beta,\psi}(g_r^{-1}hg_r)
 =U(g_r)^{-1}\circ\pi^{\beta,\psi}(h)\circ
  U(g_r)
$$
for all $g_r\in G_l(O_r,\beta)^{(c)}$ and $h\in K_{l-1}(O_r)$. 
