Regular irreducible represntations of classical groups over finite quotient rings

TL;DR

This paper provides a new parametrization of irreducible representations of classical groups over finite quotient rings, using Weil representations and character groups of centralizers, with explicit cases for several classical groups.

Contribution

It introduces a novel parametrization method for regular irreducible representations over finite quotient rings of classical groups, based on Weil representations and centralizer character groups.

Findings

Parametrization of irreducible representations via centralizer character groups.

Explicit descriptions for general linear, symplectic, orthogonal, and special linear groups.

Application of Weil representations over finite fields to representation theory.

Abstract

A parametrization of irreducible representations associated with a regular adjoint orbit of a classical group over finite quotient rings of the ring of integer of a non-dyadic non-archimedean local field is presented. The parametrization is given by means of (a subset of) the character group of the centralizer of a representative of the regular adjoint orbit. Our method is based upon Weil representations over finite fields. More explicit parametrization in terms of tamely ramified extensions of the base field is given for the general linear group, the special linear group, the symplectic group and the orthogonal group.

Equations

$$K_{r-1}(O_r)\tilde{\to }\mathfrak{g}(\mathbb{F})$$

$$K_l(O_r)\tilde{\to }\mathfrak{g}(O_{l^{\prime }})$$

$$G{L}_n(R)=\{g\in M_n(R)\mid \det g\in R^{\times }\}$$

$$B:{\mathfrak{gl}}_n{\times }_O{\mathfrak{gl}}_n\to {\mathbb{A}}_O^1$$

$$\mathfrak{g}(O)/{\varpi }^r\mathfrak{g}(O)\tilde{\to }\mathfrak{g}(O_r)=\mathfrak{g}(O){\otimes }_O{O}_r$$

$$\mathfrak{g}(O_{l^{\prime }})\tilde{\to }{K}_l(O_r)$$

$$\mathfrak{g}(O)\to K_{l-1}(O_r)$$

$$\{x\in F\mid \tau (xO)=1\}=O.$$

$$l^{\prime }=\{\begin{array}{ll}l& :r=2l,\\ l-1& :r=2l-1.\end{array}$$

$${\psi }_{\beta }((1+{\varpi }^{l}X)(\mathrm{mod}{\mathfrak{p}}^r))=\tau ({\varpi }^{-l^{\prime }}B(X,\beta ))(X\in \mathfrak{g}(O)).$$

$${\psi }_{\beta }(g_r^{-1}h{g}_r)={\psi }_{\text{Ad}(g)\beta }(h)(h\in K_l(O_r)).$$

$$\displaystyle G(O_{r},\beta)=\left\{g_{r}\in G(O_{r})\bigm{|}\text{\rm Ad}(g)\beta\equiv\beta\pmod{\mathfrak{p}^{l^{\prime}}}\right\}$$

$$⟨{\psi }_{\beta },\pi {⟩}_{K_l(O_r)}={\dim }_{\mathbb{C}}{\text{Hom}}_{K_l(O_r)}({\psi }_{\beta },\pi )>0$$

$$\sigma \mapsto {\text{Ind}}_{G(O_r,\beta )}^{G(O_r)}\sigma .$$

$$\text{Irr}(G(O_r,\beta )\mid {\psi }_{\beta }).$$

$${\mathfrak{g}}_{\beta }(R)=\{X\in \mathfrak{g}(R)\mid [X,\overline{\beta }]=0\}$$

$$\displaystyle\left\{\theta\in G_{\beta}(O_{r})^{^}\;\;\;\text{\rm s.t. $\displaystyle\theta=\psi_{\beta}$ on $\displaystyle G_{\beta}(O_{r})\cap K_{l}(O_{r})$}\right\}$$

$$\theta \mapsto {\delta }_{\beta ,\theta }={\text{Ind}}_{G(O_r,\beta )}^{G(O_r)}{\sigma }_{\beta ,\theta }$$

$$\displaystyle\left\{\theta\in G_{\beta}(O_{r})^{^}\;\;\;\text{\rm s.t. $\displaystyle\theta=\psi_{\beta}$ on $\displaystyle G_{\beta}(O_{r})\cap K_{l}(O_{r})$}\right\}$$

$$G(O_r,\beta )=G_{\beta }(O_r)\cdot K_l(O_r)$$

$${\sigma }_{\beta ,\theta }(gh)=\theta (g)\cdot {\psi }_{\beta }(h)(g\in G_{\beta }(O_r),h\in K_l(O_r)).$$

$$\displaystyle\left\{\theta\in G_{\beta}(O_{r})^{^}\;\;\;\text{\rm s.t. $\displaystyle\theta=\psi_{\beta}$ on $\displaystyle G_{\beta}(O_{r})\cap K_{l}(O_{r})$}\right\}$$

$$V_{\sigma }({\psi }_{\beta })=\{v\in V_{\sigma }\mid \sigma (g)v={\psi }_{\beta }(g)v\text{for ∀g∈Kl(Or)}\}$$

$${\dim }_K{\mathfrak{g}}_{\beta }(K)=\dim {\mathfrak{g}}_{\beta }{\otimes }_{O}K\ge \dim G_{\beta }{\otimes }_{O}K\ge \text{rank}(G).$$

$${\beta }_s\in \text{Lie}(T)(K)$$

$${\beta }_n=\sum _{\alpha \in {\Phi }^{+}}{c}_{\alpha }\cdot X_{\alpha }$$

$$J_{n_1}({\alpha }_1)⊞\cdots ⊞J_{n_r}({\alpha }_r)=\left[\begin{array}{ccc}{J}_{n_1}({\alpha }_1)& & \\ & \ddots & \\ & & J_{n_r}({\alpha }_r)\end{array}\right]$$

$$J_m(\alpha )=\left[\begin{array}{cccc}\alpha & 1& & \\ & \alpha & \ddots & \\ & & \ddots & 1\\ & & & \alpha \end{array}\right]$$

$$\{X\in M_n(O)\mid X\beta =\beta X\}=O[\beta ]$$

$$\displaystyle\left\{\begin{array}[]{l}\theta:O_{r}[\beta_{r}]^{\times}\to\mathbb{C}^{1}:\text{\rm group homomorphism}\\ \hphantom{\theta:}\text{\rm s.t. $\displaystyle\theta(x\!\!\pmod{\mathfrak{p}^{r}})=\tau\left(\varpi^{-r}\text{\rm tr}(\beta(x-1_{n}))\right)$}\\ \hphantom{\theta:s.t.}\text{\rm for $\displaystyle\forall x\in 1_{n}+\varpi^{l}O[\beta]$}\end{array}\right\}$$

Full text

1 Introduction

Let $\displaystyle F$ be a non-dyadic non-archimedean local field. The integer ring of $\displaystyle F$ is denoted by $\displaystyle O=O_{F}$ with the maximal ideal $\displaystyle\mathfrak{p}=\mathfrak{p}_{F}$ generated by $\displaystyle\varpi$. The residue class field $\displaystyle\mathbb{F}=O/\mathfrak{p}$ is a finite field of odd characteristic with $\displaystyle q$ elements. For an integer $\displaystyle r>0$ put $\displaystyle O_{r}=O/\mathfrak{p}^{r}$ so that $\displaystyle\mathbb{F}=O_{1}$.

Let $\displaystyle G$ be a connected reductive group scheme over $\displaystyle O$. The problem which we will consider in this paper is to determine the set $\displaystyle\text{\rm Irr}(G(O_{r}))$ of the equivalence classes of the irreducible complex representations of the finite group $\displaystyle G(O_{r})$.

This problem in the case $\displaystyle r=1$, that is the representation theory of the finite reductive group $\displaystyle G(\mathbb{F})$, has been studied extensively, starting from Green [7] concerned with $\displaystyle GL_{n}(\mathbb{F})$ to the decisive paper of Deligne-Lusztig [3].

On the other hand, the study of the representation theory of the finite group $\displaystyle G(O_{r})$ with $\displaystyle r>1$ is less complete. The systematic studies are done mainly in the case of $\displaystyle G=GL_{n}$ [8, 9, 10, 11], [13], [17], [18].

Shechter [14] studies the problem when $\displaystyle G$ is a symplectic group or a special orthogonal group with $\displaystyle r>1$. We have a canonical isomorphism

[TABLE]

where $\displaystyle K_{r-1}(O_{r})$ is the kernel of the canonical group homomorphism $\displaystyle G(O_{r})\to G(O_{r-1})$ and $\displaystyle\mathfrak{g}$ is the Lie algebra scheme of $\displaystyle G$. Take an irreducible representation $\displaystyle\pi$ of $\displaystyle G(O_{r})$. Then, by Clifford’s theorem, the restriction $\displaystyle\pi|_{K_{r-1}(O_{r})}$ is a multiple of the sum over a single $\displaystyle G(\mathbb{F})$-conjugacy class of characters of $\displaystyle K_{r-1}(O_{r})$ which determines, by the isomorphism (1.0.1) combined with the invariant bilinear form on $\displaystyle\mathfrak{g}(\mathbb{F})$, an adjoint $\displaystyle G(\mathbb{F})$-orbit $\displaystyle\Omega$ in $\displaystyle\mathfrak{g}(\mathbb{F})$. So we have a correspondence $\displaystyle\pi\mapsto\Omega$ of the set of the equivalence classes of the irreducible representations of $\displaystyle G(O_{r})$ to the set of the adjoint $\displaystyle G(\mathbb{F})$-orbits in $\displaystyle\mathfrak{g}(\mathbb{F})$. Shechter [14] constructs (when $\displaystyle G$ is a symplectic group or a special orthogonal group) the irreducible representations of $\displaystyle G(O_{r})$ ($\displaystyle r>1$) which correspond to an adjoint $\displaystyle G(\mathbb{F})$-orbit $\displaystyle\Omega$ consisting of regular elements of $\displaystyle\mathfrak{g}(\mathbb{F})$ (see section 3 for the definition of a regular Lie element).

In this paper, we will treat more generally any smooth $\displaystyle O$-group scheme $\displaystyle G$ with Lie algebra scheme $\displaystyle\mathfrak{g}$ which satisfies three fundamental conditions I), II) and III) presented in subsection 2.1. We will also assume that $\displaystyle G$ is an $\displaystyle O$-group subscheme of $\displaystyle GL_{n}$ suitably and hence $\displaystyle\mathfrak{g}$ is a closed $\displaystyle O$-subscheme of $\displaystyle\mathfrak{gl}_{n}$ the Lie algebra scheme of $\displaystyle GL_{n}$. The condition II) gives explicitly the isomorphism

[TABLE]

where $\displaystyle l^{\prime}=l$ if $\displaystyle r=2l$ is even and $\displaystyle l^{\prime}=l-1$ if $\displaystyle r=2l-1>1$ is odd, and $\displaystyle K_{l}(O_{r})$ is the kernel of the canonical group homomorphism $\displaystyle G(O_{r})\to G(O_{l})$ which is surjective due to the formal smoothness [4, p.111, Cor. 4.6]. The condition I) guarantees the existence of an invariant bilinear form on $\displaystyle\mathfrak{g}(O_{l^{\prime}})$. Then the restriction of an irreducible representation of $\displaystyle G(O_{r})$ to $\displaystyle K_{l}(O_{r})$ determines an adjoint $\displaystyle G(O_{l^{\prime}})$-orbit in $\displaystyle\mathfrak{g}(O_{l^{\prime}})$ by Clifford’s theorem as above. Let $\displaystyle\Omega^{\prime}\subset\mathfrak{g}(O_{l^{\prime}})$ be the adjoint $\displaystyle G(O_{l^{\prime}})$-orbit of $\displaystyle\beta\!\!\pmod{\mathfrak{p}^{l^{\prime}}}\in\mathfrak{g}(O_{l^{\prime}})$ with $\displaystyle\beta\in\mathfrak{g}(O)$. Our main result establishes a bijection between the set of the equivalence classes of the irreducible representations of $\displaystyle G(O_{r})$ corresponding to $\displaystyle\Omega^{\prime}$ and a subset of the character group of $\displaystyle G_{\beta}(O_{r})$, where $\displaystyle G_{\beta}$ is the centralizer of $\displaystyle\beta$ in $\displaystyle G$, under the assumptions

$\displaystyle G_{\beta}$ is smooth over $\displaystyle O$, and 2. 2)

the characteristic polynomial of $\displaystyle\beta\!\!\pmod{\mathfrak{p}}\in M_{n}(\mathbb{F})$ is its minimal polynomial where we put $\displaystyle\mathfrak{g}(\mathbb{F})\subset\mathfrak{gl}_{n}(\mathbb{F})=M_{n}(\mathbb{F})$.

Due to the second condition, the centralizer $\displaystyle GL_{n,\beta}$ of $\displaystyle\beta\in\mathfrak{gl}_{n}(O)$ in $\displaystyle GL_{n}$ is commutative (see subsection 3.2), and hence $\displaystyle G_{\beta}$ is a commutative $\displaystyle O$-group scheme. See section 3 for relations between the second condition and the regularity (or the smooth regularity, according to Springer [16]) of Lie elements.

Let us consider the inverse image of an adjoint $\displaystyle G(\mathbb{F})$-orbit $\displaystyle\Omega$ in $\displaystyle\mathfrak{g}(\mathbb{F})$ under the canonical surjection $\displaystyle\mathfrak{g}(O_{l^{\prime}})\to\mathfrak{g}(\mathbb{F})$. The inverse image decomposes into several adjoint $\displaystyle G(O_{l^{\prime}})$-orbits consisting of the same number of elements. Then the set of the equivalence classes of irreducible representations of $\displaystyle G(O_{r})$ corresponding to $\displaystyle\Omega$ are divided into several subclasses corresponding to these adjoint $\displaystyle G(O_{l^{\prime}})$-orbits. In other words, our main result, applied to the case where $\displaystyle G$ is a symplectic group or a special orthogonal group, divides Shechter’s irreducible representations into several subclasses and gives a parametrization of the irreducible representations in each subclasses. In particular we will construct the irreducible representations of Shechter [14] (see subsection 3.4).

The main results of this paper are Theorem 2.3.1 and its applications to the general linear group $\displaystyle GL_{n}$ (Theorem 3.2.1, Theorem 3.2.3), to the special linear group $\displaystyle SL_{n}$ (Theorem 6.1.2), to the symplectic group $\displaystyle Sp_{2n}$ (Theorem 6.2.2) and to the orthogonal groups $\displaystyle SO_{2n}$ (Theorem 6.3.2) and $\displaystyle SO_{2n+1}$ (Theorem 6.3.3).

The situation is quite simple and well known when $\displaystyle r$ is even, and almost all of this paper is devoted to study the case of $\displaystyle r=2l-1$ being odd. In this case Clifford theory requires us to construct an irreducible representation of $\displaystyle G_{\beta}(O_{r})\cdot K_{l-1}(O_{r})$. To construct an irreducible representation of $\displaystyle K_{l-1}(O_{r})$, we will use Schrödinger representation of the Heisenberg group associated with a symplectic space over finite field which is associated with $\displaystyle\beta$ (Proposition 5.4.1). Then we will use Weil representation to extend the irreducible representation of $\displaystyle K_{l-1}(O_{r})$ to a projective representation of $\displaystyle G_{\beta}(O_{r})\cdot K_{l-1}(O_{r})$. Finally we will prove that the Schur multiplier associated with the projective representation is trivial (Proposition 5.6.1) to get the required irreducible linear representation of $\displaystyle G_{\beta}(O_{r})\cdot K_{l-1}(O_{r})$.

In the case of $\displaystyle G=GL_{n}$, the extendability of the irreducible representation of $\displaystyle K_{l-1}(O_{r})$ to that of $\displaystyle G_{\beta}(O_{r})\cdot K_{l-1}(O_{r})$ is proved by [13], [17]. Based upon this result, we will prove the triviality of the Schur multiplier for general $\displaystyle G\subset GL_{n}$ under the condition that the reduction modulo $\displaystyle\mathfrak{p}$ of the characteristic polynomial of $\displaystyle\beta\in\mathfrak{g}(O)\subset\mathfrak{gl}_{n}(O)$ is the minimal polynomial of $\displaystyle\beta\!\!\pmod{\mathfrak{p}}\in M_{n}(\mathbb{F})$ (in this case $\displaystyle\beta\!\!\pmod{\mathfrak{p}}\in\mathfrak{gl}_{n}(\mathbb{F})$ is regular with respect to $\displaystyle GL_{n}$ over $\displaystyle\mathbb{F}$, see subsection 3.2).

Note that Weil representation over a finite field is a “genuine” linear representation, and the definition of the Schur multiplier is independent of the theory of Weil representation. The definition and fundamental properties of the Schur multiplier will be discussed in section 4.

Notations The multiplicative group of the complex numbers of absolute value one is denoted by $\displaystyle\mathbb{C}^{1}$. The character group of a finite abelian group $\displaystyle\mathcal{G}$, that is the multiplicative group consisting of the group homomorphisms of $\displaystyle\mathcal{G}$ to $\displaystyle\mathbb{C}^{1}$, is denoted by \displaystyle\mathcal{G}^{^}.

For any $\displaystyle O$-scheme $\displaystyle X$, the image of $\displaystyle x\in X(O)$ under the canonical mapping $\displaystyle X(O)\to X(O_{l})$ is denoted by $\displaystyle x_{l}=x\pmod{\mathfrak{p}^{l}}$. On the other hand, when $\displaystyle K=\mathbb{F}$ or $\displaystyle F$, the image of $\displaystyle x\in X(O)$ under the canonical mapping $\displaystyle X(O)\to X(K)$ is denoted by $\displaystyle\overline{x}$. Any notational conflict between the cases $\displaystyle K=\mathbb{F}$ and $\displaystyle K=F$ may not occur in this paper.

Acknowledgment The author express his thanks to the referee who read quite carefully the submitted manuscript and gave suggestions to improve greatly this article. Particularly a critical step of the proof of Proposition 5.6.1 is suggested by the referee.

2 Main results

2.1 Fundamental assumptions

Let $\displaystyle G\subset GL_{n}$ be a closed smooth $\displaystyle O$-group subscheme, and $\displaystyle\mathfrak{g}$ the Lie algebra scheme of $\displaystyle G$ which is a closed affine $\displaystyle O$-subscheme of $\displaystyle\mathfrak{gl}_{n}$ the Lie algebra scheme of $\displaystyle GL_{n}$. We assume that the fibers $\displaystyle G{\otimes}_{O}K$ ($\displaystyle K=F$ or $\displaystyle K=\mathbb{F}$) are non-commutative algebraic $\displaystyle K$-groups (that is smooth $\displaystyle K$-group schemes).

For any $\displaystyle O$-algebra $\displaystyle R$ (in this paper, an $\displaystyle O$-algebra means an commutative unital $\displaystyle O$-algebra) the set of the $\displaystyle R$-valued points $\displaystyle\mathfrak{gl}_{n}(R)$ is identified with the $\displaystyle R$-Lie algebra of square matrices $\displaystyle M_{n}(R)$ of size $\displaystyle n$ with Lie bracket $\displaystyle[X,Y]=XY-YX$, and the group of $\displaystyle R$-valued points $\displaystyle GL_{n}(R)$ is identified with the matrix group

[TABLE]

where $\displaystyle R^{\times}$ is the multiplicative group of $\displaystyle R$. Hence $\displaystyle\mathfrak{g}(R)$ is identified with a matrix Lie subalgebra of $\displaystyle\mathfrak{gl}_{n}(R)$ and $\displaystyle G(R)$ is identified with a matrix subgroup of $\displaystyle GL_{n}(R)$. Let

[TABLE]

be the trace form on $\displaystyle\mathfrak{gl}_{n}$, that is $\displaystyle B(X,Y)=\text{\rm tr}(XY)$ for all $\displaystyle X,Y\in\mathfrak{gl}_{n}(R)$ with any $\displaystyle O$-algebra $\displaystyle R$. The smoothness of $\displaystyle G$ implies that we have a canonical isomorphism

[TABLE]

([4, Chap.II, $\displaystyle\S 4$, Prop.4.8]) and that the canonical group homomorphism $\displaystyle G(O)\to G(O_{r})$ is surjective due to the formal smoothness [4, p.111, Cor. 4.6] and the canonical isomorphism $\displaystyle O\,\tilde{\to}\,\lim_{\stackrel{{\scriptstyle\longleftarrow}}{{r}}}O_{r}$. For any $\displaystyle 0<l<r$, let us denote by $\displaystyle K_{l}(O_{r})$ the kernel of the canonical group homomorphism $\displaystyle G(O_{r})\to G(O_{l})$ which is surjective.

Throughout this paper, we will assume the following three conditions;

• I)

$\displaystyle B:\mathfrak{g}(\mathbb{F})\times\mathfrak{g}(\mathbb{F})\to\mathbb{F}$ is non-degenerate,

• II)

for any integers $\displaystyle r=l+l^{\prime}$ with $\displaystyle 0<l^{\prime}\leq l$, we have a group isomorphism

[TABLE]

defined by $\displaystyle X\!\!\pmod{\mathfrak{p}^{l^{\prime}}}\mapsto 1+\varpi^{l}X\!\!\pmod{\mathfrak{p}^{r}}$,

• III)

if $\displaystyle r=2l-1\geq 3$ is odd, then we have a mapping

[TABLE]

defined by $\displaystyle X\mapsto(1+\varpi^{l-1}X+2^{-1}\varpi^{2l-2}X^{2})\!\!\pmod{\mathfrak{p}^{r}}$.

The condition I) implies that $\displaystyle B:\mathfrak{g}(O_{l})\times\mathfrak{g}(O_{l})\to O_{l}$ is non-degenerate for all $\displaystyle l>0$, and so $\displaystyle B:\mathfrak{g}(O)\times\mathfrak{g}(O)\to O$ 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. Note that the mapping of the condition III) is not a group homomorphism. This mapping plays an important role when we will analyze in subsection 5.2 the structure of $\displaystyle K_{l-1}(O_{r})$ where $\displaystyle r=2l-1$ is odd. We will use it also in the proof of Proposition 5.1.4.

2.2 Clifford theory

Fix a continuous unitary character $\displaystyle\tau$ of the additive group $\displaystyle F$ such that

[TABLE]

We will fix an integer $\displaystyle r\geq 2$ and put $\displaystyle r=l+l^{\prime}$ with the smallest integer $\displaystyle l$ such that $\displaystyle 0<l^{\prime}\leq l$. In other words

[TABLE]

Take a $\displaystyle\beta\in\mathfrak{g}(O)$ and define a character $\displaystyle\psi_{\beta}$ of the finite abelian group $\displaystyle K_{l}(O_{r})$ by

[TABLE]

Then $\displaystyle\beta\!\!\pmod{\mathfrak{p}^{l^{\prime}}}\mapsto\psi_{\beta}$ gives an isomorphism of the additive group $\displaystyle\mathfrak{g}(O_{l^{\prime}})$ onto the character group \displaystyle K_{l}(O_{r})^{^}. For any $\displaystyle g_{r}=g\!\!\pmod{\mathfrak{p}^{r}}\in G(O_{r})$ with $\displaystyle g\in G(O)$, we have

[TABLE]

So the stabilizer of $\displaystyle\psi_{\beta}$ in $\displaystyle G(O_{r})$ is

[TABLE]

which is a subgroup of $\displaystyle G(O_{r})$ containing $\displaystyle K_{l^{\prime}}(O_{r})$.

Now let us denote by $\displaystyle\text{\rm Irr}(G(O_{r})\mid\psi_{\beta})$ (resp. $\displaystyle\text{\rm Irr}(G(O_{r},\beta)\mid\psi_{\beta})$) the set of the isomorphism classes of the irreducible complex representations $\displaystyle\pi$ of $\displaystyle G(O_{r})$ (resp. $\displaystyle\sigma$ of $\displaystyle G(O_{r},\beta)$) such that

[TABLE]

(resp. $\displaystyle\langle\psi_{\beta},\sigma\rangle_{K_{l}(O_{r})}>0$). Then we have

$\displaystyle\text{\rm Irr}(G(O_{r}))=\bigsqcup_{\beta\!\!\pmod{\mathfrak{p}^{l^{\prime}}}}\text{\rm Irr}(G(O_{r})\mid\psi_{\beta})$ where $\displaystyle\bigsqcup_{\beta\!\!\pmod{\mathfrak{p}^{l^{\prime}}}}$ is the disjoint union over the representatives $\displaystyle\beta\!\!\pmod{\mathfrak{p}^{l^{\prime}}}$ of the $\displaystyle\text{\rm Ad}(G(O_{l^{\prime}}))$-orbits in $\displaystyle\mathfrak{g}(O_{l^{\prime}})$, 2. 2)

a bijection of $\displaystyle\text{\rm Irr}(G(O_{r},\beta)\mid\psi_{\beta})$ onto $\displaystyle\text{\rm Irr}(G(O_{r})\mid\psi_{\beta})$ is given by

[TABLE]

The first statement is clear from the fact that $\displaystyle K_{l}(O_{r})\,\tilde{\to}\,\mathfrak{g}(O_{l^{\prime}})$ is an commutative group and the relation (2.2.1). For the second statement, see [12, Th.6.11].

So our problem is reduced to give a good parametrization of the set

[TABLE]

2.3 Main theorem

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

[TABLE]

for any $\displaystyle O$-algebra $\displaystyle R$ where $\displaystyle\overline{\beta}\in\mathfrak{g}(R)$ is the image of $\displaystyle\beta\in\mathfrak{g}(O)$ under the canonical morphism $\displaystyle\mathfrak{g}(O)\to\mathfrak{g}(R)$.

Now our main result is

Theorem 2.3.1

Take a $\displaystyle\beta\in\mathfrak{g}(O)$ such that

$\displaystyle G_{\beta}$* is a smooth $\displaystyle O$-group scheme, and* 2. 2)

the characteristic polynomial $\displaystyle\chi_{\overline{\beta}}(t)=\det(t\cdot 1_{n}-\overline{\beta})$ of $\displaystyle\overline{\beta}\in\mathfrak{g}(\mathbb{F})\subset\mathfrak{gl}_{n}(\mathbb{F})$ is the minimal polynomial of $\displaystyle\overline{\beta}\in M_{n}(\mathbb{F})$.

Then we have a bijection $\displaystyle\theta\mapsto\sigma_{\beta,\theta}$ of the set

[TABLE]

onto $\displaystyle\text{\rm Irr}(G(O_{r},\beta)\mid\psi_{\beta})$.

Corollary 2.3.2

Under the conditions of Theorem 2.3.1, we have a bijection

[TABLE]

of the set

[TABLE]

onto $\displaystyle\text{\rm Irr}(G(O_{r})\mid\psi_{\beta})$.

Note that $\displaystyle G_{\beta}$ in Theorem 2.3.1 is commutative because the centralizer of $\displaystyle\beta$ in $\displaystyle GL_{n}$ is commutative due to the second condition of the theorem. (see subsection 3.2).

The explicit description of the representation $\displaystyle\sigma_{\beta,\theta}$ is given by (2.4.1) if $\displaystyle r$ is even, and by (5.1.3) if $\displaystyle r$ is odd.

The proof of this theorem in the case of even $\displaystyle r$ is quite simple and well known. For the sake of completeness, we will include a proof in the next subsection. The proof for the case of odd $\displaystyle r$ is given in section 5 after studying certain Schur multiplier in section 4.

2.4 Proof of Theorem 2.3.1 for even $\displaystyle r$

Assume that $\displaystyle r=2l$ is even so that $\displaystyle l^{\prime}=l$. Since $\displaystyle G_{\beta}$ is a smooth $\displaystyle O$-group scheme, the canonical map $\displaystyle G_{\beta}(O_{r})\to G_{\beta}(O_{l})$ is surjective. Then we have

[TABLE]

where $\displaystyle G_{\beta}(O_{r})$ is a commutative finite group. Let $\displaystyle\theta$ be a character of $\displaystyle G_{\beta}(O_{r})$ such that $\displaystyle\theta=\psi_{\beta}$ on $\displaystyle G_{\beta}(O_{r})\cap K_{l}(O_{r})$. Then we have an one-dimensional representation $\displaystyle\sigma_{\beta,\theta}$ of $\displaystyle G(O_{r},\beta)$ defined by

[TABLE]

Then $\displaystyle\theta\mapsto\sigma_{\beta,\theta}$ is an injection of the set

[TABLE]

into $\displaystyle\text{\rm Irr}(G(O_{r},\beta)\mid\psi_{\beta})$.

Take any $\displaystyle\sigma\in\text{\rm Irr}(G(O_{r},\beta)\mid\psi_{\beta})$ with representation space $\displaystyle V_{\sigma}$. Then

[TABLE]

is a non-trivial $\displaystyle G(O_{r},\beta)$-subspace of $\displaystyle V_{\sigma}$ so that $\displaystyle V_{\sigma}=V_{\sigma}(\psi_{\beta})$. Then, for any one-dimensional representation $\displaystyle\chi$ of $\displaystyle G(O_{r},\beta)$ such that $\displaystyle\chi=\psi_{\beta}$ on $\displaystyle K_{l}(O_{r})$, we have $\displaystyle K_{l}(O_{r})\subset\text{\rm Ker}(\chi^{-1}\otimes\sigma)$. On the other hand $\displaystyle G(O_{r},\beta)/K_{l}(O_{r})$ is commutative, we have $\displaystyle\dim(\chi^{-1}\otimes\sigma)=1$ and then $\displaystyle\dim\sigma=1$. Then $\displaystyle\theta=\sigma|_{G_{\beta}(O_{r})}$ is a character of $\displaystyle G_{\beta}(O_{r})$ such that $\displaystyle\theta=\psi_{\beta}$ on $\displaystyle G_{\beta}(O_{r})\cap K_{l}(O_{r})$, and we have $\displaystyle\sigma=\sigma_{\beta,\theta}$.

3 Regularity of Lie elements

In this section, we will discuss the relation between the regularity (or smooth regularity, according to [16, p.138]) of a Lie element and the two conditions of Theorem 2.3.1.

3.1 Regularity and smooth regularity

Let us assume that the connected $\displaystyle O$-group scheme $\displaystyle G$ is reductive, that is, the fibers $\displaystyle G{\otimes}_{O}K$ ($\displaystyle K=F,\mathbb{F}$) are reductive $\displaystyle K$-algebraic groups. In this case the dimension of a maximal torus in $\displaystyle G{\otimes}_{O}K$ is independent of $\displaystyle K$ which is denoted by $\displaystyle\text{\rm rank}(G)$. For any $\displaystyle\beta\in\mathfrak{g}(O)$ we have

[TABLE]

We say $\displaystyle\beta$ is smoothly regular (resp. regular) with respect to $\displaystyle G$ over $\displaystyle K$ (or simply with respect to $\displaystyle G{\otimes}_{O}K$) if $\displaystyle\dim_{K}\mathfrak{g}_{\beta}(K)=\text{\rm rank}(G)$ (resp. $\displaystyle\dim G_{\beta}{\otimes}_{O}K=\text{\rm rank}(G)$) (see [16, (5.7)]). The relation (3.1.1) shows that if $\displaystyle\beta$ is smoothly regular with respect to $\displaystyle G{\otimes}_{O}K$ then $\displaystyle\beta$ is regular with respect to $\displaystyle G{\otimes}_{O}K$. If $\displaystyle\beta$ is smoothly regular with respect to $\displaystyle G{\otimes}_{O}K$ then $\displaystyle G_{\beta}{\otimes}_{O}K$ is smooth over $\displaystyle K$.

If $\displaystyle\beta$ is smoothly regular with respect to $\displaystyle G$ over $\displaystyle F$ and over $\displaystyle\mathbb{F}$, then we say $\displaystyle\beta$ is smoothly regular with respect to $\displaystyle G$.

We say $\displaystyle\beta$ is connected with respect to $\displaystyle G$ if the fibers $\displaystyle G_{\beta}{\otimes}_{O}K$ ($\displaystyle K=F,\mathbb{F}$) are connected. See Remark 3.1.3 for a sufficient condition for the connectedness of $\displaystyle G_{\beta}{\otimes}_{O}K$. Then we have

Proposition 3.1.1

If $\displaystyle\beta\in\mathfrak{g}(O)$ is smoothly regular and connected with respect to $\displaystyle G$, then $\displaystyle G_{\beta}$ is smooth over $\displaystyle O$.

[Proof] Let $\displaystyle G_{\beta}^{o}$ be the neutral component of $\displaystyle O$-group scheme $\displaystyle G_{\beta}$ which is a group functor of the category of $\displaystyle O$-scheme (see $\displaystyle\S 3$ of Exposé $\displaystyle\text{\rm VI}_{B}$ in [5]). The following statements are equivalent;

$\displaystyle G_{\beta}^{o}$ is representable as an smooth open $\displaystyle O$-group subscheme of $\displaystyle G_{\beta}$, 2. 2)

$\displaystyle G_{\beta}$ is smooth at the points of unit section, 3. 3)

each fibers $\displaystyle G_{\beta}{\otimes}_{O}K$ ($\displaystyle K=F,\mathbb{F}$) are smooth over $\displaystyle K$ and their dimensions are constant

(see Th. 3.10 and Cor. 4.4 of [5]). So if $\displaystyle\beta$ is smoothly regular with respect to $\displaystyle G$, then $\displaystyle G_{\beta}^{o}$ is smooth open $\displaystyle O$-group subscheme of $\displaystyle G_{\beta}$. If further $\displaystyle\beta$ is connected with respect to $\displaystyle G$, then $\displaystyle G_{\beta}^{o}=G_{\beta}$ is smooth over $\displaystyle O$. $\displaystyle\blacksquare$

Let us present more detail description of the smooth regularity of Lie element. Put $\displaystyle K=F$ or $\displaystyle\mathbb{F}$.

Take a $\displaystyle\beta\in\mathfrak{g}(O)$ and let $\displaystyle\overline{\beta}=\beta_{s}+\beta_{n}$ be the Jordan decomposition of $\displaystyle\overline{\beta}\in\mathfrak{g}(K)$ into the semi-simple part $\displaystyle\beta_{s}\in\mathfrak{g}(K)$ and the nilpotent part $\displaystyle\beta_{n}\in\mathfrak{g}(K)$ ($\displaystyle\overline{\beta}\in\mathfrak{g}(K)$ is the image of $\displaystyle\beta\in\mathfrak{g}(O)$ under the canonical mapping $\displaystyle\mathfrak{g}(O)\to\mathfrak{g}(K)$). The identity component $\displaystyle L=Z_{G{\otimes}_{O}K}(\beta_{s})^{o}$ of the centralizer of $\displaystyle\beta_{s}$ in $\displaystyle G{\otimes}_{O}K$ is a reductive group over $\displaystyle K$ and there exists a maximal torus $\displaystyle T$ of $\displaystyle G{\otimes}_{O}K$ such that

[TABLE]

(see [1, Prop.13.19] and its proof). Then $\displaystyle T\subset L$ and $\displaystyle\text{\rm rank}(L)=\text{\rm rank}(G)$. Put $\displaystyle\mathfrak{l}=\text{\rm Lie}(L)$, then $\displaystyle\mathfrak{l}(K)=Z_{\mathfrak{g}(K)}(\beta_{s})$. So $\displaystyle\overline{\beta}\in\mathfrak{g}(K)$ is smoothly regular with respect to $\displaystyle G{\otimes}_{O}K$ if and only if $\displaystyle\beta_{n}\in\mathfrak{l}(K)$ is smoothly regular with respect to $\displaystyle L$.

Now fix a system of positive roots $\displaystyle\Phi^{+}$ in the root system $\displaystyle\Phi(T,L)$ of $\displaystyle L$ with respect to $\displaystyle T$ such that

[TABLE]

where $\displaystyle X_{\alpha}$ is a root vector of the root $\displaystyle\alpha$. Let us denote by $\displaystyle\Gamma$ (resp. $\displaystyle\Gamma^{\prime}$) the root lattice (the weight lattice) of the root system $\displaystyle\Phi(T,L)$. Then the finite group $\displaystyle\Gamma^{\prime}/\Gamma$ is called the fundamental group of $\displaystyle\Phi(T,L)$ (see [16, (1.4)]). Considering the decomposition into the simple factors, we may assume that the root system $\displaystyle\Phi(T,L)$ is simple. Then results of [16, (5.8), (5.9)] and [2, p.228,III-3.5] implies

Proposition 3.1.2

Assume that $\displaystyle G$ is semi-simple, that is, the fibers $\displaystyle G{\otimes}_{O}K$ ($\displaystyle K=F,\mathbb{F}$) are semi-simple algebraic $\displaystyle K$-groups. Assume also that the characteristic of $\displaystyle K$ is not the bad prime and does not divide the order of the fundamental group of $\displaystyle\Phi(T,L)$ tabulated below.

[TABLE]

Then the following three statements are equivalent:

$\displaystyle\overline{\beta}\in\mathfrak{g}(K)$* is smoothly regular with respect to $\displaystyle G$ over $\displaystyle K$,* 2. 2)

$\displaystyle\overline{\beta}\in\mathfrak{g}(K)$* is regular with respect to $\displaystyle G$ over $\displaystyle K$,* 3. 3)

$\displaystyle c_{\alpha}\neq 0$* for all simple $\displaystyle\alpha\in\Phi^{+}$ in (3.1.2).*

Remark 3.1.3

Assume that $\displaystyle\overline{\beta}\in\mathfrak{g}(K)$ is smoothly regular with respect to $\displaystyle G{\otimes}_{O}K$. Then $\displaystyle G_{\beta}{\otimes}_{O}K$ is connected if $\displaystyle Z_{G{\otimes}_{O}K}(\beta_{s})$ and its center are connected (see Theorem 5.9 b) of [16]).

3.2 Case of $\displaystyle GL_{n}$

Let us consider the case of $\displaystyle GL_{n}$ ($\displaystyle n\geq 2$) which is a connected smooth reductive $\displaystyle O$-group scheme. For $\displaystyle\beta\in\mathfrak{gl}_{n}(O)$, the following statements are equivalent ($\displaystyle K=F,\mathbb{F}$);

$\displaystyle c_{\alpha}\neq 0$ for all simple $\displaystyle\alpha\in\Phi^{+}$ in (3.1.2), 2. 2)

the characteristic polynomial $\displaystyle\chi_{\overline{\beta}}(t)=\det(t1_{n}-\overline{\beta})\in K[t]$ is the minimal polynomial of $\displaystyle\overline{\beta}\in M_{n}(K)$, 3. 3)

$\displaystyle\overline{\beta}\in M_{n}(K)$ is $\displaystyle GL_{n}(\overline{K})$-conjugate to

[TABLE]

where $\displaystyle\alpha_{1},\cdots,\alpha_{r}$ are distinct elements of the algebraic closure $\displaystyle\overline{K}$ of $\displaystyle K$ and

[TABLE]

is a Jordan block of size $\displaystyle m$, 4. 4)

$\displaystyle\{X\in M_{n}(K)\mid X\overline{\beta}=\overline{\beta}X\}=K[\overline{\beta}]$.

In these cases, $\displaystyle\overline{\beta}\in\mathfrak{gl}_{n}(K)$ is smoothly regular with respect to $\displaystyle GL_{n}$ over $\displaystyle K$ and the fiber $\displaystyle GL_{n,\beta}{\otimes}_{O}K$ is connected.

Note also that the following statements are equivalent;

the characteristic polynomial of $\displaystyle\overline{\beta}\in M_{n}(\mathbb{F})$ is its minimal polynomial, 2. 2)

$\displaystyle\{X\in M_{n}(O_{l})\mid X\beta\equiv\beta X\pmod{\mathfrak{p}^{l}}\}=O_{l}[\beta_{l}]$ for all $\displaystyle l>0$, with $\displaystyle\beta_{l}=\beta\pmod{\mathfrak{p}^{l}}$ 3. 3)

$\displaystyle O^{n}$ is a cyclic $\displaystyle O[\beta]$-module, that is, there exists a vector $\displaystyle v\in O^{n}$ such that $\displaystyle O^{n}=O[\beta]v$.

In these cases we have

[TABLE]

and the characteristic polynomial of $\displaystyle\beta\in M_{n}(O)$ is its minimal polynomial. In particular $\displaystyle GL_{n,\beta}$ is a smooth over $\displaystyle O$ by Proposition 3.1.1.

It is easy to show that the conditions I), II) and III) of section 2.1 hold for $\displaystyle GL_{n}$. So Theorem 2.3.1 gives

Theorem 3.2.1

Take a $\displaystyle\beta\in\mathfrak{gl}_{n}(O)$ such that the characteristic polynomial of $\displaystyle\overline{\beta}\in M_{n}(\mathbb{F})$ is its minimal polynomial. Then we have a bijection $\displaystyle\theta\mapsto\text{\rm Ind}_{GL_{n}(O_{r},\beta)}^{GL_{n}(O_{r})}\sigma_{\beta,\theta}$ of the set

[TABLE]

onto the set $\displaystyle\text{\rm Irr}(GL_{n}(O_{r})\mid\psi_{\beta})$.

In order to give a more explicit description, let us recall the following results due to Shintani [15].

Let $\displaystyle L/F$ be a tamely ramified separable field extension of degree $\displaystyle n$ and $\displaystyle O_{L}\subset L$ the integer ring with the maximal ideal $\displaystyle\mathfrak{p}_{L}=\varpi_{L}O_{L}$. The residue class field $\displaystyle\mathbb{F}=O/\mathfrak{p}$ is identified with a subfield of $\displaystyle\mathbb{L}=O_{L}/\mathfrak{p}_{L}$. A prime element $\displaystyle\varpi_{L}$ can be chosen so that we have $\displaystyle\varpi_{L}^{e}\in O_{L_{0}}$ where $\displaystyle L_{0}$ is the maximal unramified subextension of $\displaystyle L/F$ and $\displaystyle e=(L:L_{0})$ is the ramification index of $\displaystyle L/F$. Then we have $\displaystyle O_{L}=O_{L_{0}}[\varpi_{L}]$. We will identify $\displaystyle L$ with a $\displaystyle F$-subalgebra of $\displaystyle M_{n}(F)$ by means of the regular representation of $\displaystyle L$ with respect to an $\displaystyle O$-basis of $\displaystyle O_{L}$. Then we have [15, p.545, Lemma 4-7, Cor.1, p.546, Cor.2]

Proposition 3.2.2

For a $\displaystyle\beta=\sum_{i=0}^{e-1}b_{i}\varpi_{L}^{i}\in O_{L}$ with $\displaystyle b_{i}\in O_{L_{0}}$, the following two statements are equivalent;

$\displaystyle O_{L}=O[\beta]$, 2. 2)

$\displaystyle\overline{b_{0}}^{\sigma}\neq\overline{b_{0}}$* for all $\displaystyle 1\neq\sigma\in\text{\rm Gal}(\mathbb{L}/\mathbb{F})$, and $\displaystyle b_{1}\in O_{L_{0}}^{\times}$ if $\displaystyle e>1$.*

In this case, the characteristic polynomial $\displaystyle\chi_{\beta}(t)=\det(t\cdot 1_{n}-\beta)$ of $\displaystyle\beta\in M_{n}(O)$ has the following properties;

$\displaystyle\chi_{\beta}(t)\!\!\pmod{\mathfrak{p}}\in\mathbb{F}[t]$* is the minimal polynomial of $\displaystyle\overline{\beta}\in M_{n}(\mathbb{F})$,* 2. 2)

$\displaystyle\chi_{\beta}(t)\!\!\pmod{\mathfrak{p}}=p(t)^{e}$* with an irreducible polynomial $\displaystyle p(t)\in\mathbb{F}[t]$,* 3. 3)

$\displaystyle\chi_{\beta}(t)\!\!\pmod{\mathfrak{p}^{2}}\in O_{2}[t]$* is irreducible.*

So any $\displaystyle\beta\in O_{L}$ such that $\displaystyle O_{L}=O[\beta]$ gives an example of a $\displaystyle\beta\in\mathfrak{gl}_{n}(O)$ which is smoothly regular with respect to $\displaystyle G=GL_{n}$. Then Theorem 2.3.1 gives the following result of Shintani [15, Prop.4-2, Prop.4-3]

Theorem 3.2.3

There is a bijection $\displaystyle\theta\mapsto\text{\rm Ind}_{GL_{n}(O_{r},\beta)}^{GL_{n}(O_{r})}\sigma_{\beta,\theta}$ of the set

[TABLE]

onto the set $\displaystyle\text{\rm Irr}(GL_{n}(O_{r})\mid\psi_{\beta})$.

3.3 Regularity for classical groups

Let us see the smooth regularity over $\displaystyle K=F$ or $\displaystyle\mathbb{F}$ for the special linear group, the symplectic group and the orthogonal group. Explicit applications, as Theorem 3.2.3 for $\displaystyle GL_{n}$, of Theorem 2.3.1 to these groups are given in section 6.

3.3.1 Case of $\displaystyle SL_{n}$

For the algebraic group $\displaystyle SL_{n}$ over $\displaystyle K$, the situation is almost the same as the case of $\displaystyle GL_{n}$ in subsection 3.2, except that $\displaystyle SL_{n,\beta}{\otimes}_{O}K$ is connected if and only if the characteristic polynomial of $\displaystyle\overline{\beta}\in M_{n}(K)$ is separable, that is $\displaystyle n_{1}=\cdots=n_{r}=1$ in (3.2.1).

3.3.2 Case of $\displaystyle Sp_{2n}$

The algebraic group $\displaystyle Sp_{2n}$ over $\displaystyle K$ is defined by

[TABLE]

where

[TABLE]

Then a maximal torus $\displaystyle T$ of $\displaystyle Sp_{2n}$ is

[TABLE]

where ${}^{\tau}a=I_{n}\,^{t}aI_{n}$. The Weyl group is generated by the permutations among $\displaystyle\{a_{1},\cdots,a_{n}\}$ and the sign changes $\displaystyle a_{i}\mapsto a_{i}^{\pm 1}$. Any maximal torus of $\displaystyle Sp_{2n}$ is $\displaystyle Sp_{2n}(\overline{K})$-conjugate to $\displaystyle T$ ($\displaystyle\overline{K}$ is the algebraic closure of $\displaystyle K$). Then for any $\displaystyle\beta\in\text{\rm Lie}(Sp_{2n})(K)$, the semi-simple part $\displaystyle\beta_{s}$ of $\displaystyle\beta\in M_{2n}(K)$ is $\displaystyle Sp_{2n}(\overline{K})$-conjugate to an element

[TABLE]

of $\displaystyle\text{\rm Lie}(T)(\overline{K})$, where $\displaystyle a_{i}\neq\pm a_{j}$ if $\displaystyle i\neq j$ and $\displaystyle a_{i}\neq 0$ for $\displaystyle 1\leq i<r$ possibly $\displaystyle a_{r}=0$. Then the centralizer $\displaystyle L=Z_{Sp_{2n}}(\beta_{s})^{o}$ is

[TABLE]

We can choose a system of positive roots so that the nilpotent part $\displaystyle\beta_{n}$ of $\displaystyle\beta\in M_{2n}(K)$ is an upper triangle matrix. Then Proposition 3.1.2 says that the following two statements are equivalent:

$\displaystyle\beta$ is smoothly regular with respect to $\displaystyle Sp_{2n}$ over $\displaystyle K$, 2. 2)

the characteristic polynomial of $\displaystyle\beta\in M_{2n}(K)$ is its minimal polynomial.

Note that the centralizer $\displaystyle Sp_{2n,\beta}$ is connected if and only if $\displaystyle\det\beta\neq 0$.

3.3.3 Case of $\displaystyle SO_{2n+1}$

The algebraic group $\displaystyle SO_{2n+1}$ over $\displaystyle K$ is defined by

[TABLE]

where

[TABLE]

Then a maximal torus $\displaystyle T$ of $\displaystyle SO_{2n+1}$ is

[TABLE]

where ${}^{\tau}a=I_{n}\,^{t}aI_{n}$. The Weyl group is generated by the permutations among $\displaystyle\{a_{1},\cdots,a_{n}\}$ and the sign changes $\displaystyle a_{i}\mapsto a_{i}^{\pm 1}$. Any maximal torus of $\displaystyle SO_{2n+1}$ is $\displaystyle SO_{2n+1}(\overline{K})$-conjugate to $\displaystyle T$ ($\displaystyle\overline{K}$ is the algebraic closure of $\displaystyle K$). Then for any $\displaystyle\beta\in\text{\rm Lie}(SO_{2n+1})(K)$, the semi-simple part $\displaystyle\beta_{s}$ of $\displaystyle\beta\in M_{2n+1}(K)$ is $\displaystyle SO_{2n+1}(\overline{K})$-conjugate to an element

[TABLE]

of $\displaystyle\text{\rm Lie}(T)(\overline{K})$, where $\displaystyle a_{i}\neq\pm a_{j}$ if $\displaystyle i\neq j$ and $\displaystyle a_{i}\neq 0$ for $\displaystyle 1\leq i<r$ possibly $\displaystyle a_{r}=0$. Then the centralizer $\displaystyle L=Z_{SO_{2n+1}}(\beta_{s})^{o}$ is

[TABLE]

We can choose a system of positive roots so that the nilpotent part $\displaystyle\beta_{n}$ of $\displaystyle\beta\in M_{2n+1}(K)$ is an upper triangle matrix. Then Proposition 3.1.2 says that the following two statements are equivalent:

$\displaystyle\beta$ is smoothly regular with respect to $\displaystyle SO_{2n+1}$ over $\displaystyle K$, 2. 2)

the characteristic polynomial of $\displaystyle\beta\in M_{2n+1}(K)$ is its minimal polynomial.

In this case the centralizer $\displaystyle SO_{2n+1,\beta}$ is connected.

3.3.4 Case of $\displaystyle SO_{2n}$

The algebraic group $\displaystyle SO_{2n}$ over $\displaystyle K$ is defined by

[TABLE]

where

[TABLE]

Then a maximal torus $\displaystyle T$ of $\displaystyle SO_{2n}$ is

[TABLE]

where ${}^{\tau}a=I_{n}\,^{t}aI_{n}$. The Weyl group is generated by the permutations among $\displaystyle\{a_{1},\cdots,a_{n}\}$ and the sign changes

[TABLE]

($\displaystyle\varepsilon_{i}=\pm 1$) such that $\displaystyle\varepsilon_{1}\cdots\varepsilon_{n}=1$. Any maximal torus of $\displaystyle SO_{2n}$ is $\displaystyle SO_{2n}(\overline{K})$-conjugate to $\displaystyle T$ ($\displaystyle\overline{K}$ is the algebraic closure of $\displaystyle K$). Then for any $\displaystyle\beta\in\text{\rm Lie}(SO_{2n})(K)$, the semi-simple part $\displaystyle\beta_{s}$ of $\displaystyle\beta\in M_{2n}(K)$ is $\displaystyle SO_{2n}(\overline{K})$-conjugate to an element

[TABLE]

of $\displaystyle\text{\rm Lie}(T)(\overline{K})$, where $\displaystyle a_{i}\neq a_{j}$ if $\displaystyle i\neq j$ and $\displaystyle a_{i}\neq 0$ for $\displaystyle 1\leq i<r$ possibly $\displaystyle a_{r}=0$. If $\displaystyle a_{i}\neq\pm a_{j}$ for all $\displaystyle i\neq j$ and $\displaystyle a_{r}\neq 0$, then the centralizer $\displaystyle L=Z_{SO_{2n}}(\beta_{s})^{o}$ is

[TABLE]

If $\displaystyle a_{i}=-a_{j}$ for some $\displaystyle i\neq j$, then $\displaystyle L$ contains the factor $\displaystyle GL_{n_{1}+n_{j}}$. If $\displaystyle a_{r}=0$, then $\displaystyle L$ contains the factor $\displaystyle SO_{2n_{r}}$ which breaks our argument. For example

[TABLE]

is smoothly regular with respect to $\displaystyle SO_{4}$ over $\displaystyle K$ but the characteristic polynomial is not the minimal polynomial. Any way we can choose a system of positive roots so that the nilpotent part $\displaystyle\beta_{n}$ of $\displaystyle\beta\in M_{2n}(K)$ is an upper triangle matrix. If $\displaystyle\det\beta\neq 0$ then Proposition 3.1.2 says that the following two statements are equivalent:

$\displaystyle\beta$ is smoothly regular with respect to $\displaystyle SO_{2n}$ over $\displaystyle K$, 2. 2)

the characteristic polynomial of $\displaystyle\beta\in M_{2n}(K)$ is its minimal polynomial.

In this case the centralizer $\displaystyle SO_{2n,\beta}$ is connected.

3.4 Some results of Shechter [14]

Let $\displaystyle G$ be a connected reductive $\displaystyle O$-group scheme which satisfies the conditions I), II) and III) of subsection 2.1. Take an adjoint $\displaystyle G(\mathbb{F})$-orbit $\displaystyle\Omega\subset\mathfrak{g}(\mathbb{F})$ whose elements are smoothly regular with respect to $\displaystyle G$ over $\displaystyle\mathbb{F}$. Take a $\displaystyle\beta\in\mathfrak{g}(O)$ such that $\displaystyle\overline{\beta}\in\Omega$ and assume that the two conditions of Theorem 2.3.1 are satisfied with respect to this $\displaystyle\beta$. Since we have canonical isomorphisms

[TABLE]

for any $\displaystyle m>1$, we have

[TABLE]

for all $\displaystyle l>0$, where $\displaystyle\text{\rm rank}\,G$ is the absolute rank of $\displaystyle G$. Since the canonical group homomorphism $\displaystyle G(O_{l^{\prime}})\to G(\mathbb{F})$ is surjective, the inverse image of $\displaystyle\Omega$ under the canonical surjection $\displaystyle\mathfrak{g}(O_{l^{\prime}})\to\mathfrak{g}(\mathbb{F})$ is divided into $\displaystyle q^{(l^{\prime}-1)\text{\rm rank}\,G}$ adjoint $\displaystyle G(O_{l^{\prime}})$-orbits consisting of

[TABLE]

elements. On the other hand, we have

[TABLE]

Then Theorem 2.3.1 implies that the number of the irreducible representations of $\displaystyle G(O_{r})$ corresponding to the adjoint orbit $\displaystyle\Omega\subset\mathfrak{g}(\mathbb{F})$ is

[TABLE]

This is the first statement of Theorem I of Shechter [14]. This means that we have constructed the irreducible representations of Shechter [14].

The construction of $\displaystyle\sigma_{\beta,\theta}$ presented in subsection 2.4 (when $\displaystyle r$ is even) and section 4 (when $\displaystyle r$ is odd) shows that

[TABLE]

On the other hand, we have

[TABLE]

because we have $\displaystyle G(O_{r},\beta)=G_{\beta}(O_{r})\cdot K_{l^{\prime}}(O_{r})$ and

[TABLE]

Then we have

[TABLE]

which is the second statement of Theorem I of Shechter [14].

4 Schur multiplier

Let $\displaystyle G\subset GL_{n}$ be a closed $\displaystyle\mathbb{F}$-algebraic subgroup and $\displaystyle\mathfrak{g}$ the Lie algebra scheme of $\displaystyle G$ which is a closed affine $\displaystyle\mathbb{F}$-subscheme of the Lie algebra scheme $\displaystyle\mathfrak{gl}_{n}$ of $\displaystyle GL_{n}$. Let us assume that the trace form

[TABLE]

is non-degenerate. Fix a $\displaystyle\beta\in\mathfrak{g}(\mathbb{F})$ such that $\displaystyle\mathfrak{g}_{\beta}(\mathbb{F})\lvertneqq\mathfrak{g}(\mathbb{F})$.

We have fixed a continuous unitary character $\displaystyle\tau$ of the additive group $\displaystyle F$ such that

[TABLE]

Define an additive character $\displaystyle\overline{\tau}$ of $\displaystyle\mathbb{F}$ by $\displaystyle\overline{\tau}(\overline{x})=\tau(\varpi^{-1}x)$.

4.1 Definition of a Schur multiplier

The non-zero $\displaystyle\mathbb{F}$-vector space $\displaystyle\mathbb{V}_{\beta}=\mathfrak{g}(\mathbb{F})/\mathfrak{g}_{\beta}(\mathbb{F})$ has a symplectic form

[TABLE]

where $\displaystyle\dot{X}=X\pmod{\mathfrak{g}_{\beta}(\mathbb{F})}\in\mathbb{V}_{\beta}$ with $\displaystyle X\in\mathfrak{g}_{\beta}(\mathbb{F})$. Then $\displaystyle g\in G_{\beta}(\mathbb{F})$ gives an element $\displaystyle\sigma_{g}$ of the symplectic group $\displaystyle Sp(\mathbb{V}_{\beta})$ defined by

[TABLE]

Note that the group $\displaystyle Sp(\mathbb{V}_{\beta})$ acts on $\displaystyle\mathbb{V}_{\beta}$ from right. Let $\displaystyle v\mapsto[v]$ be a $\displaystyle\mathbb{F}$-linear section on $\displaystyle\mathbb{V}_{\beta}$ of the exact sequence

[TABLE]

For any $\displaystyle v\in\mathbb{V}_{\beta}$ and $\displaystyle g\in G_{\beta}(\mathbb{F})$, put

[TABLE]

Take a character \displaystyle\rho\in\mathfrak{g}_{\beta}(\mathbb{F})^{^}. Then there exists uniquely a $\displaystyle v_{g}\in\mathbb{V}_{\beta}$ such that

[TABLE]

for all $\displaystyle v\in\mathbb{V}_{\beta}$. Note that $\displaystyle v_{g}\in\mathbb{V}_{\beta}$ depends on $\displaystyle\rho$ as well as the section $\displaystyle v\mapsto[v]$. Let

[TABLE]

be the centralizer of $\displaystyle\mathfrak{g}_{\beta}(\mathbb{F})$ in $\displaystyle G(\mathbb{F})$, which is a subgroup of $\displaystyle G_{\beta}(\mathbb{F})$. Then for any $\displaystyle g,h\in G_{\beta}(\mathbb{F})^{(c)}$, we have

[TABLE]

because $\displaystyle\gamma(v,gh)=\gamma(v,g)+\gamma(v\sigma_{g}^{-1},h)$ for all $\displaystyle v\in\mathbb{V}_{\beta}$. Put

[TABLE]

for $\displaystyle g,h\in G_{\beta}(\mathbb{F})^{(c)}$. Then the relation (4.1.2) shows that $\displaystyle c_{\beta,\rho}\in Z^{2}(G_{\beta}(\mathbb{F})^{(c)},\mathbb{C}^{\times})$ is a 2-cocycle with trivial action of $\displaystyle G_{\beta}(\mathbb{F})^{(c)}$ on $\displaystyle\mathbb{C}^{\times}$. Moreover we have

Proposition 4.1.1

The cohomology class $\displaystyle[c_{\beta,\rho}]\in H^{2}(G_{\beta}(\mathbb{F})^{(c)},\mathbb{C}^{\times})$ is independent of the choice of the $\displaystyle\mathbb{F}$-linear section $\displaystyle v\mapsto[v]$.

[Proof] Take another $\displaystyle\mathbb{F}$-linear section $\displaystyle v\mapsto[v]^{\prime}$ with respect to which we will define $\displaystyle\gamma^{\prime}(v,g)\in\mathfrak{g}_{\beta}$ and $\displaystyle v^{\prime}_{g}\in\mathbb{V}_{\beta}$ as above. Then there exists a $\displaystyle\delta\in\mathbb{V}_{\beta}$ such that $\displaystyle\rho([v]-[v]^{\prime})=\overline{\tau}\left(\langle v,\delta\rangle_{\beta}\right)$ for all $\displaystyle v\in\mathbb{V}_{\beta}$. We have $\displaystyle v_{g}^{\prime}=v_{g}+\delta-\delta\sigma_{g}$ for all $\displaystyle g\in G_{\beta}(\mathbb{F})^{(c)}$. So if we put $\displaystyle\alpha(g)=\overline{\tau}\left(2^{-1}\langle v_{g}^{\prime}-v_{g^{-1}},\delta\rangle_{\beta}\right)$ for $\displaystyle g\in G_{\beta}(\mathbb{F})^{(c)}$, then we have

[TABLE]

for all $\displaystyle g,h\in G_{\beta}(\mathbb{F})^{(c)}$. $\displaystyle\blacksquare$

4.2 Relation to an over group

Let us assume that there exists a closed smooth $\displaystyle O$-group subscheme $\displaystyle H\subset GL_{n}$ of which our $\displaystyle G$ is a closed $\displaystyle O$-group subscheme and that the trace form

[TABLE]

is non-degenerate where $\displaystyle\mathfrak{h}$ is the Lie algebra scheme of $\displaystyle H$. Then we have

[TABLE]

where $\displaystyle\mathfrak{g}(\mathbb{F})^{\perp}=\{X\in\mathfrak{h}(\mathbb{F})\mid B(X,\mathfrak{g}(\mathbb{F}))=0\}$ is the orthogonal complement of $\displaystyle\mathfrak{g}(\mathbb{F})$ in $\displaystyle\mathfrak{h}(\mathbb{F})$.

Take a $\displaystyle\beta\in\mathfrak{g}(O)$ such that $\displaystyle\mathfrak{g}_{\beta}(\mathbb{F})\lvertneqq\mathfrak{g}(\mathbb{F})$. Then $\displaystyle\beta\in\mathfrak{h}(O)$ and $\displaystyle\mathfrak{h}_{\beta}(\mathbb{F})\lvertneqq\mathfrak{h}(\mathbb{F})$ where $\displaystyle\mathfrak{h}_{\beta}=Z_{\mathfrak{h}}(\beta)$ is the centralizer. We have decompositions

[TABLE]

where $\displaystyle\left(\mathfrak{g}(\mathbb{F})^{\perp}\right)_{\beta}=\mathfrak{h}_{\beta}(\mathbb{F})\cap\mathfrak{g}(\mathbb{F})^{\perp}$, and

[TABLE]

is an orthogonal decomposition of symplectic spaces.

Let $\displaystyle v\mapsto[v]$ be a $\displaystyle\mathbb{F}$-linear section of the exact sequence

[TABLE]

of $\displaystyle\mathbb{F}$-vector space such that $\displaystyle[\mathbb{V}_{\beta}]\subset\mathfrak{g}(\mathbb{F})$ and $\displaystyle[\mathfrak{g}(\mathbb{F})^{\perp}/\left(\mathfrak{g}(\mathbb{F})^{\perp}\right)_{\beta}]\subset\mathfrak{g}(\mathbb{F})^{\perp}$.

Take \displaystyle\rho\in\mathfrak{g}_{\beta}(\mathbb{F})^{^} and put

[TABLE]

For any $\displaystyle g\in G_{\beta}(\mathbb{F})\subset H_{\beta}(\mathbb{F})$, there exists uniquely a $\displaystyle v_{g}\in\mathbb{V}_{\beta}$ such that

[TABLE]

for all $\displaystyle v\in\mathbb{V}_{\beta}$. Then we have

[TABLE]

for all $\displaystyle v\in\widetilde{\mathbb{V}}_{\beta}$. In fact if we put $\displaystyle v=v^{\prime}+v^{\prime\prime}$ with $\displaystyle v^{\prime}\in\mathbb{V}_{\beta}$ and $\displaystyle v^{\prime\prime}\in\mathfrak{g}(\mathbb{F})^{\perp}/\left(\mathfrak{g}(\mathbb{F})^{\perp}\right)_{\beta}$, then we have $\displaystyle\gamma_{\mathfrak{h}}(v,g)=\gamma_{\mathfrak{g}}(v^{\prime},g)+\gamma_{\mathfrak{h}}(v^{\prime\prime},g)$ with $\displaystyle\gamma_{\mathfrak{h}}(v^{\prime\prime},g)\in\left(\mathfrak{g}(\mathbb{F})^{\perp}\right)_{\beta}$, since

[TABLE]

Then we have

[TABLE]

because $\displaystyle\langle v^{\prime\prime},v_{g}\rangle_{\beta}=0$. Hence we have

Proposition 4.2.1

If $\displaystyle G_{\beta}(\mathbb{F})^{(c)}\subset H_{\beta}(\mathbb{F})^{(c)}$ then the Schur multiplier $\displaystyle[c_{\beta,\rho}]\in H^{2}(G_{\beta}(\mathbb{F})^{(c)},\mathbb{C}^{\times})$ is the image under the restriction mapping

[TABLE]

of the Schur multiplier $\displaystyle[c_{\beta,\widetilde{\rho}}]\in H^{2}(H_{\beta}(\mathbb{F})^{(c)},\mathbb{C}^{\times})$.

5 Proof of Theorem 2.3.1 for odd $\displaystyle r$

In this section, we will give a proof of Theorem 2.3.1 in the case of odd $\displaystyle r$.

Let $\displaystyle G\subset GL_{n}$ be a smooth $\displaystyle O$-group scheme which satisfies the conditions described in subsection 2.1. Take a $\displaystyle\beta\in\mathfrak{g}(O)$ such that the centralizer $\displaystyle G_{\beta}$ is commutative and smooth over $\displaystyle O$. Put $\displaystyle r=2l-1$ with an integer $\displaystyle l\geq 2$ and put $\displaystyle l^{\prime}=l-1\geq 1$.

5.1 Construction of irreducible representations

We have a chain of canonical surjections

[TABLE]

defined by

[TABLE]

Here we use the condition II) of subsection 2.1. Let us denote by $\displaystyle Z(O_{r},\beta)$ the inverse image under the surjection $\displaystyle\heartsuit$ of $\displaystyle\mathfrak{g}_{\beta}(\mathbb{F})$. Then $\displaystyle Z(O_{r},\beta)$ is a normal subgroup of $\displaystyle K_{l-1}(O_{r})$ containing $\displaystyle K_{l}(O_{r})$ as the kernel of $\displaystyle\heartsuit$.

Let us denote by $\displaystyle Y_{\beta}$ the set of the group homomorphisms $\displaystyle\psi$ of $\displaystyle Z(O_{r},\beta)$ to $\displaystyle\mathbb{C}^{\times}$ such that $\displaystyle\psi=\psi_{\beta}$ on $\displaystyle K_{l}(O_{r})$. Then a bijection of \displaystyle\mathfrak{g}_{\beta}(\mathbb{F})^{^} onto $\displaystyle Y_{\beta}$ is given by

[TABLE]

where a group homomorphism $\displaystyle\widetilde{\psi}_{\beta}:Z(O_{r},\beta)\to\mathbb{C}^{\times}$ is defined by

[TABLE]

with $\displaystyle\overline{X}=X\!\!\pmod{\mathfrak{p}}\in\mathfrak{g}_{\beta}(\mathbb{F})$.

Take a $\displaystyle\psi\in Y_{\beta}$. For two elements

[TABLE]

of $\displaystyle K_{l-1}(O_{r})$, we have $\displaystyle x^{-1}=1-\varpi^{l-1}X+2^{-1}\varpi^{2l-2}X^{2}\!\!\pmod{\mathfrak{p}^{r}}$ so that we have

[TABLE]

and so $\displaystyle\psi_{\beta}(xyx^{-1}y^{-1})=\tau\left(\varpi^{-1}B(X,\text{\rm ad}(Y)\beta)\right)$. Hence we have

[TABLE]

for all $\displaystyle x\in K_{l-1}(O_{r})$ and $\displaystyle y\in Z(O_{r},\beta)$ so that we can define

[TABLE]

by

[TABLE]

for $\displaystyle g=(1+\varpi^{l-1}X)\!\!\pmod{\mathfrak{p}^{r}},h=(1+\varpi^{l-1}Y)\!\!\pmod{\mathfrak{p}^{r}}\in K_{l-1}(O_{r})$. Note that $\displaystyle D_{\psi}$ is non-degenerate. Then Proposition 3.1.1 of [18] gives

Proposition 5.1.1

For any $\displaystyle\psi=\psi_{\beta,\rho}\in Y_{\beta}$ with \displaystyle\rho\in\mathfrak{g}_{\beta}(\mathbb{F})^{^}, there exists unique irreducible representation $\displaystyle\pi_{\psi}$ of $\displaystyle K_{l-1}(O_{r})$ such that $\displaystyle\langle\psi,\pi_{\psi}\rangle_{Z(O_{r},\beta)}>0$. Furthermore

[TABLE]

and $\displaystyle\pi_{\psi}(x)$ is the homothety $\displaystyle\psi(x)$ for all $\displaystyle x\in Z(O_{r},\beta)$.

Fix a $\displaystyle\psi=\psi_{\beta,\rho}\in Y_{\beta}$ with \displaystyle\rho\in\mathfrak{g}_{\beta}(\mathbb{F})^{^}. Our problem is to extend the representation $\displaystyle\pi_{\psi}$ of $\displaystyle K_{l-1}(O_{r})$ to a representation of $\displaystyle G(O_{r},\beta)=G_{\beta}(O_{r})\cdot K_{l-1}(O_{r})$. Now for any $\displaystyle g_{r}=g\!\!\pmod{\mathfrak{p}^{r}}\in G_{\beta}(O_{r})$ and $\displaystyle x=(1+\varpi^{l-1}X)\!\!\pmod{\mathfrak{p}^{r}}\in Z(O_{r},\beta)$, we have

[TABLE]

and

[TABLE]

that is $\displaystyle\psi(g_{r}^{-1}xg_{r})=\psi(x)$ for all $\displaystyle x\in Z(O_{r},\beta)$. This means that, for any $\displaystyle g\in G_{\beta}(O_{r})$, the $\displaystyle g$-conjugate of $\displaystyle\pi_{\psi}$ is isomorphic to $\displaystyle\pi_{\psi}$, that is, there exists a $\displaystyle U(g)\in GL_{\mathbb{C}}(V_{\psi})$ ($\displaystyle V_{\psi}$ is the representation space of $\displaystyle\pi_{\psi}$) such that

[TABLE]

for all $\displaystyle x\in K_{l-1}(O_{r})$, and moreover, for any $\displaystyle g,h\in G_{\beta}(O_{r})$, there exists a $\displaystyle c_{U}(g,h)\in\mathbb{C}^{\times}$ such that

[TABLE]

Then $\displaystyle c_{U}\in Z^{2}(G_{\beta}(O_{r}),\mathbb{C}^{\times})$ is a $\displaystyle\mathbb{C}^{\times}$-valued 2-cocycle on $\displaystyle G_{\beta}(O_{r})$ with trivial action on $\displaystyle\mathbb{C}^{\times}$, and the cohomology class $\displaystyle[c_{U}]\in H^{2}(G_{\beta}(O_{r}),\mathbb{C}^{\times})$ is independent of the choice of each $\displaystyle U(g)$.

We will construct $\displaystyle\pi_{\psi}$ by means of Schrödinger representations over the finite field $\displaystyle\mathbb{F}$ in subsection 5.4 (see Proposition 5.4.1), and will show in subsection 5.5 that we can construct $\displaystyle U(g)$ by means of Weil representation so that we have

[TABLE]

for all $\displaystyle g,h\in G_{\beta}(O_{r})$, where $\displaystyle\overline{g}\in G_{\beta}(\mathbb{F})$ is the image of $\displaystyle g\in G_{\beta}(O_{r})$ under the canonical surjection $\displaystyle G(O_{r})\to G(\mathbb{F})$ (see subsection 5.5), and $\displaystyle c_{\beta,\rho}$ is the Schur multiplier defined in section 4. Furthermore, Proposition 5.6.1 tells us that the Schur multiplier $\displaystyle[c_{\beta,\rho}]\in H^{2}(G_{\beta}(\mathbb{F}),\mathbb{C}^{\times})$ is trivial if the characteristic polynomial of $\displaystyle\overline{\beta}\in M_{n}(\mathbb{F})$ is its minimal polynomial. So the Schur multiplier $\displaystyle[c_{U}]\in H^{2}(G_{\beta}(O_{r}),\mathbb{C}^{\times})$ is actually trivial under the second condition of our main theorem 2.3.1. Now we have

Proposition 5.1.2

Assume that the Schur multiplier $\displaystyle[c_{U}]\in H^{2}(G_{\beta}(O_{r}),\mathbb{C}^{\times})$ is trivial. Then there exists a group homomorphism $\displaystyle U_{\psi}:G_{\beta}(O_{r})\to GL_{\mathbb{C}}(V_{\psi})$ such that

$\displaystyle\pi_{\psi}(g^{-1}xg)=U_{\psi}(g)^{-1}\circ\pi_{\psi}(x)\circ U_{\psi}(g)$* for all $\displaystyle g\in G_{\beta}(O_{r})$ and $\displaystyle x\in K_{l-1}(O_{r})$ and* 2. 2)

$\displaystyle U_{\psi}(h)=1$* for all $\displaystyle h\in G_{\beta}(O_{r})\cap K_{l-1}(O_{r})$.*

[Proof] Since the Schur multiplier $\displaystyle[c_{U}]\in H^{2}(G_{\beta}(O_{r}),\mathbb{C}^{\times})$ is trivial, there exists a group homomorphism $\displaystyle U:G_{\beta}(O_{r})\to GL_{\mathbb{C}}(V_{\psi})$ such that $\displaystyle\pi_{\psi}(g^{-1}xg)=U(g)^{-1}\circ\pi_{\psi}(x)\circ U(g)$ for all $\displaystyle g\in G_{\beta}(O_{r})$ and $\displaystyle x\in K_{l-1}(O_{r})$. Then for any $\displaystyle h\in G_{\beta}(O_{r})\cap K_{l-1}(O_{r})$ there exists a $\displaystyle c(h)\in\mathbb{C}^{\times}$ such that $\displaystyle U(h)=c(h)\cdot\pi_{\psi}(h)$. On the other hand we have

[TABLE]

since $\displaystyle(1+\varpi^{l-1}X)_{r}\in G_{\beta}(O_{r})\cap K_{l-1}(O_{r})$ means that

[TABLE]

and then $\displaystyle[X,\beta]\equiv 0\!\!\pmod{\mathfrak{p}}$, that is $\displaystyle X\!\!\pmod{\mathfrak{p}}\in\mathfrak{g}_{\beta}(\mathbb{F})$. Then $\displaystyle\pi_{\psi}(h)$ is the homothety $\displaystyle\psi(h)$ for all $\displaystyle h\in G_{\beta}(O_{r})\cap K_{l-1}(O_{r})$. Extend the group homomorphism $\displaystyle h\mapsto c(h)\psi(h)$ of $\displaystyle G_{\beta}(O_{r})\cap K_{l-1}(O_{r})$ to a group homomorphism $\displaystyle\theta:G_{\beta}(O_{r})\to\mathbb{C}^{\times}$. Then $\displaystyle g\mapsto U_{\psi}(g)=\theta(g)^{-1}U(g)$ is the required group homomorphism. $\displaystyle\blacksquare$

Let us denote by \displaystyle G_{\beta}(O_{r})^{^}{\times}_{K_{l-1}(O_{r})}\mathfrak{g}_{\beta}(\mathbb{F})^{^} the set of \displaystyle(\theta,\rho)\in G_{\beta}(O_{r})^{^}\times\mathfrak{g}_{\beta}(\mathbb{F})^{^} such that $\displaystyle\theta=\psi_{\beta,\rho}$ on $\displaystyle G_{\beta}(O_{r})\cap K_{l-1}(O_{r})$. Then, under the assumption of Proposition 5.1.2, \displaystyle(\theta,\rho)\in G_{\beta}(O_{r})^{^}{\times}_{K_{l-1}(O_{r})}\mathfrak{g}_{\beta}(\mathbb{F})^{^} defines an irreducible representation $\displaystyle\sigma_{\theta,\rho}$ of $\displaystyle G(O_{r},\beta)=G_{\beta}(O_{r})\cdot K_{l-1}(O_{r})$ by

[TABLE]

for $\displaystyle g\in G_{\beta}(O_{r})$ and $\displaystyle h\in K_{l-1}(O_{r})$ with $\displaystyle\psi=\psi_{\beta,\rho}$. Then we have

Proposition 5.1.3

Assume that the Schur multiplier $\displaystyle[c_{U}]\in H^{2}(G_{\beta}(O_{r}),\mathbb{C}^{\times})$ is trivial. Then a bijection of \displaystyle G_{\beta}(O_{r})^{^}{\times}_{K_{l-1}(O_{r})}\mathfrak{g}_{\beta}(\mathbb{F})^{^} onto $\displaystyle\text{\rm Irr}(G(O_{r},\beta)\mid\psi_{\beta})$ is given by $\displaystyle(\theta,\rho)\mapsto\sigma_{\theta,\rho}$.

[Proof] Clearly $\displaystyle\sigma_{\theta,\rho}\in\text{\rm Irr}(G(O_{r},\beta)\mid\psi_{\beta})$ for all \displaystyle(\theta,\rho)\in G_{\beta}(O_{r})^{^}{\times}_{K_{l-1}(O_{r})}\mathfrak{g}_{\beta}(\mathbb{F})^{^}. Take a $\displaystyle\sigma\in\text{\rm Irr}(G(O_{r},\beta)\mid\psi_{\beta})$. Then

[TABLE]

so that there exists a $\displaystyle\psi=\psi_{\beta,\rho}\in Y_{\beta}$ with \displaystyle\rho\in\mathfrak{g}_{\beta}(\mathbb{F})^{^} such that

[TABLE]

where $\displaystyle\bigoplus_{\theta}$ is the direct sum over \displaystyle\theta\in G_{\beta}(O_{r})^{^} such that $\displaystyle\theta=\psi_{\beta,\rho}$ on $\displaystyle G_{\beta}(O_{r})\cap K_{l-1}(O_{r})$. Then we have $\displaystyle\sigma=\sigma_{\theta,\rho}$ for some \displaystyle(\theta,\rho)\in G_{\beta}(O_{r})^{^}{\times}_{K_{l-1}(O_{r})}\mathfrak{g}_{\beta}(\mathbb{F})^{^}. $\displaystyle\blacksquare$

We have also

Proposition 5.1.4

$\displaystyle(\theta,\rho)\mapsto\theta$* gives a bijection of \displaystyle G_{\beta}(O_{r})^{^}{\times}_{K_{l-1}(O_{r})}\mathfrak{g}_{\beta}(\mathbb{F})^{^} onto the set*

[TABLE]

[Proof] Take a \displaystyle(\theta,\rho)\in G_{\beta}(O_{r})^{^}{\times}_{K_{l-1}(O_{r})}\mathfrak{g}_{\beta}(\mathbb{F})^{^}. The smoothness of $\displaystyle G_{\beta}$ over $\displaystyle O$ implies that the canonical mapping $\displaystyle\mathfrak{g}_{\beta}(O)\to\mathfrak{g}_{\beta}(\mathbb{F})$ is surjective. So Take a $\displaystyle\overline{X}\in\mathfrak{g}_{\beta}(\mathbb{F})$ with $\displaystyle X\in\mathfrak{g}_{\beta}(O)$. Then we have

[TABLE]

so that

[TABLE]

Hence we have

[TABLE]

This means that the mapping $\displaystyle(\theta,\rho)\mapsto\theta$ is injective. Take $\displaystyle X,X^{\prime}\in\mathfrak{g}_{\beta}(O)$ such that $\displaystyle X\equiv X^{\prime}\pmod{\mathfrak{p}}$. Then we have $\displaystyle X^{\prime}=X+\varpi T$ with $\displaystyle T\in\mathfrak{g}_{\beta}(O)$ and

[TABLE]

where $\displaystyle 1+\varpi^{l}T\pmod{\mathfrak{p}^{r}}\in K_{l}(O_{r})$ and hence

[TABLE]

This and the commutativity of $\displaystyle G_{\beta}$ show that

[TABLE]

with $\displaystyle\overline{X}\in\mathfrak{g}_{\beta}(\mathbb{F})$ with $\displaystyle X\in\mathfrak{g}_{\beta}(O)$ gives an well-defined group homomorphism of $\displaystyle\mathfrak{g}_{\beta}(\mathbb{F})$ to $\displaystyle\mathbb{C}^{\times}$. Then \displaystyle(\theta,\rho)\in G_{\beta}(O_{r})^{^}{\times}_{K_{l-1}(O_{r})}\mathfrak{g}_{\beta}(\mathbb{F})^{^} and our mapping in question is surjective. $\displaystyle\blacksquare$

Proposition 5.1.3 and Proposition 5.1.4 give the bijection presented in our main Theorem 2.3.1 in the case of $\displaystyle r$ being odd. So all that we need is to show the triviality of the Schur multiplier $\displaystyle[c_{U}]\in H^{2}(G_{\beta}(O_{r}),\mathbb{C}^{\times})$ in Proposition 5.1.3. The triviality is proved in subsection 5.6 after

analyzing, in subsections 5.2 and 5.3, the structure of $\displaystyle K_{l-1}(O_{r})$ and 2. 2)

describing, in subsection 5.4 and 5.5, the Schur multiplier

[TABLE]

in terms of the Schur multiplier defined in section 4.

5.2 Structure of $\displaystyle K_{l-1}(O_{r})$

A group extension

[TABLE]

is given by the canonical surjection (5.1.1), whose kernel is $\displaystyle K_{l}(O_{r})$, with the group isomorphism

[TABLE]

defined by $\displaystyle S\!\!\pmod{\mathfrak{p}^{l-1}}\mapsto(1+\varpi^{l}S)\!\!\pmod{\mathfrak{p}^{r}}$ which is assumed as the condition II) in subsection 2.1.

In order to determine the 2-cocycle of the group extension (5.2.1), choose any mapping $\displaystyle\lambda:\mathfrak{g}(\mathbb{F})\to\mathfrak{g}(O)$ such that $\displaystyle X=\lambda(X)\!\!\pmod{\mathfrak{p}}$ for all $\displaystyle X\in\mathfrak{g}(\mathbb{F})$ and $\displaystyle\lambda(0)=0$, and define a section

[TABLE]

of (5.2.1) by $\displaystyle X\mapsto 1+\varpi^{l-1}\lambda(X)+2^{-1}\varpi^{2l-2}\lambda(X)^{2}\!\!\pmod{\mathfrak{p}^{r}}$. Then we have

[TABLE]

for all $\displaystyle X\in\mathfrak{g}(\mathbb{F})$ and

[TABLE]

for all $\displaystyle S_{l-1}\in\mathfrak{g}(O_{l-1})$. Furthermore we have

[TABLE]

for all $\displaystyle X,Y\in\mathfrak{g}(\mathbb{F})$ where $\displaystyle\mu:\mathfrak{g}(\mathbb{F})\times\mathfrak{g}(\mathbb{F})\to\mathfrak{g}(O)$ is defined by

[TABLE]

for all $\displaystyle X,Y\in\mathfrak{g}(\mathbb{F})$. Now we have two elements ($\displaystyle 2$-cocycle)

[TABLE]

of $\displaystyle Z^{2}(\mathfrak{g}(\mathbb{F}),\mathfrak{g}(O_{l-1}))$ with trivial action of $\displaystyle\mathfrak{g}(\mathbb{F})$ on $\displaystyle\mathfrak{g}(O_{l-1})$.

Let us consider two groups $\displaystyle\mathbb{M}$ and $\displaystyle\mathbb{G}$ corresponding to the two 2-cocycles $\displaystyle\mu$ and $\displaystyle c$ respectively. That is the group operation on $\displaystyle\mathbb{M}=\mathfrak{g}(\mathbb{F})\times\mathfrak{g}(O_{l-1})$ is defined by

[TABLE]

and the group operation on $\displaystyle\mathbb{G}=\mathfrak{g}(\mathbb{F})\times\mathfrak{g}(O_{l-1})$ is defined by

[TABLE]

Let $\displaystyle\mathbb{G}{\times}_{\mathfrak{g}(\mathbb{F})}\mathbb{M}$ be the fiber product of $\displaystyle\mathbb{G}$ and $\displaystyle\mathbb{M}$ with respect to the canonical projections onto $\displaystyle\mathfrak{g}(\mathbb{F})$. In other words

[TABLE]

is a subgroup of the direct product $\displaystyle\mathbb{G}\times\mathbb{M}$. We have a surjective group homomorphism

[TABLE]

defined by

[TABLE]

5.3 Reduction to an intermediate group

The group homomorphism $\displaystyle B_{\beta}:\mathfrak{g}(O_{l-1})\to O_{l-1}$ ($\displaystyle X\mapsto B(X,\beta_{l-1})$) induces a group homomorphism

[TABLE]

Let us denote by $\displaystyle\mathcal{H}_{\beta}$ the group associated with the 2-cocycle

[TABLE]

That is $\displaystyle\mathcal{H}_{\beta}=\mathfrak{g}(\mathbb{F})\times O_{l-1}$ with a group operation

[TABLE]

Then the center of $\displaystyle\mathcal{H}_{\beta}$ is $\displaystyle Z(\mathcal{H}_{\beta})=\mathfrak{g}_{\beta}(\mathbb{F})\times O_{l-1}$, the direct product of two additive groups $\displaystyle\mathfrak{g}_{\beta}(\mathbb{F})$ and $\displaystyle O_{l-1}$.

The inverse image of $\displaystyle Z(\mathcal{H}_{\beta})$ with respect to the surjective group homomorphism

[TABLE]

is $\displaystyle\left(\mathbb{G}{\times}_{\mathfrak{g}(\mathbb{F})}\mathbb{M}\right)_{\beta}=\left\{(X;S,T)\in\mathbb{G}{\times}_{\mathfrak{g}(\mathbb{F})}\mathbb{M}\mid X\in\mathfrak{g}(\mathbb{F})_{\overline{\beta}}\right\}$ which is mapped onto $\displaystyle Z(O_{r},\beta)\subset K_{l-1}(O_{r})$ by the surjection (5.2.2).

Take a \displaystyle\rho\in\mathfrak{g}_{\beta}(\mathbb{F})^{^} which defines group homomorphisms

[TABLE]

and

[TABLE]

On the other hand we have a group homomorphism

[TABLE]

defined by $\displaystyle\widetilde{\psi}_{0}(X;S_{l-1},T_{l-1})=\tau\left(\varpi^{-l}B(\lambda(X)+\varpi T,\beta)\right)$. Then $\displaystyle\widetilde{\psi}_{0}\cdot\widetilde{\chi}_{\beta}$ is trivial on the kernel of the surjection (5.2.2) and it induces a group homomorphism $\displaystyle\psi_{\beta,\rho}\in Y_{\beta}$ defined in subsection 5.1.

5.4 Schrödinger representations over finite fields

Fix a \displaystyle\rho\in\mathfrak{g}_{\beta}(\mathbb{F})^{^}. Let us determine the 2-cocycle of the group extension

[TABLE]

where $\displaystyle\clubsuit:\mathcal{H}_{\beta}\to\mathbb{V}_{\beta}$ is defined by $\displaystyle(X,s)\mapsto\dot{X}\!\!\pmod{\mathfrak{g}_{\beta}(\mathbb{F})}$. Fix a $\displaystyle\mathbb{F}$-linear section $\displaystyle v\mapsto[v]$ of the exact sequence

[TABLE]

of $\displaystyle\mathbb{F}$-vector spaces and define a section $\displaystyle l:\mathbb{V}_{\beta}\to\mathcal{H}_{\beta}$ of the group extension (5.4.1) by $\displaystyle l(v)=([v],0)$. Then we have

[TABLE]

for $\displaystyle u=\dot{\overline{X}},v=\dot{\overline{Y}}\in\mathbb{V}_{\beta}$ so that the 2-cocycle of the group extension (5.4.1) is

[TABLE]

Define a group operation on $\displaystyle\mathbb{H}_{\beta}=\mathbb{V}_{\beta}\times Z(\mathcal{H}_{\beta})$ by

[TABLE]

Then $\displaystyle\mathbb{H}_{\beta}$ is isomorphic to $\displaystyle\mathcal{H}_{\beta}$ by $\displaystyle(v,(Y,s))\mapsto([v]+Y,s)$.

Let $\displaystyle H_{\beta}$ be the Heisenberg group of the symplectic $\displaystyle\mathbb{F}$-space $\displaystyle\mathbb{V}_{\beta}$, that is $\displaystyle H_{\beta}=\mathbb{V}_{\beta}\times\mathbb{C}^{1}$ with a group operation

[TABLE]

Then we have a group homomorphism

[TABLE]

Fix a polarization $\displaystyle\mathbb{V}_{\beta}=\mathbb{W}^{\prime}\oplus\mathbb{W}$ of the symplectic $\displaystyle\mathbb{F}$-space $\displaystyle\mathbb{V}_{\beta}$. Let us denote by $\displaystyle L^{2}(\mathbb{W}^{\prime})$ the complex vector space of the complex-valued functions $\displaystyle f$ on $\displaystyle\mathbb{W}^{\prime}$ with inner product $\displaystyle(f,f^{\prime})=\sum_{w\in\mathbb{W}^{\prime}}f(w)\overline{f^{\prime}(w)}$. The Schrödinger representation $\displaystyle(\pi^{\beta},L^{2}(\mathbb{W}^{\prime}))$ of $\displaystyle H_{\beta}$ associated with the polarization is defined for $\displaystyle(v,s)\in H_{\beta}$ and $\displaystyle f\in L^{2}(\mathbb{W}^{\prime})$ by

[TABLE]

where $\displaystyle v=v_{-}+v_{+}\in\mathbb{V}_{\beta}$ with $\displaystyle v_{-}\in\mathbb{W}^{\prime},v_{+}\in\mathbb{W}^{\prime}$.

Now an irreducible representation $\displaystyle(\pi^{\beta,\rho},L^{2}(\mathbb{W}^{\prime}))$ of $\displaystyle\mathbb{H}_{\beta}$ is defined by $\displaystyle\pi^{\beta,\rho}(v,z)=\pi^{\beta}(v,\chi_{\rho}(z))$, and an irreducible representation $\displaystyle(\widetilde{\pi}^{\beta,\rho},L^{2}(\mathbb{W}^{\prime}))$ of $\displaystyle\mathbb{G}{\times}_{\mathfrak{g}(\mathbb{F})}\mathbb{M}$ is defined by

[TABLE]

Then $\displaystyle\widetilde{\psi}_{0}\cdot\widetilde{\pi}_{\beta,\rho}$ is trivial on the kernel of $\displaystyle(\ast):\mathbb{G}{\times}_{\mathfrak{g}(\mathbb{F})}\mathbb{M}\to K_{l-1}(O_{r})$ so that it induces an irreducible representation $\displaystyle\pi_{\beta,\rho}$ of $\displaystyle K_{l-1}(O_{r})$ on $\displaystyle L^{2}(\mathbb{W}^{\prime})$.

Proposition 5.4.1

Take a $\displaystyle g=1+\varpi^{l-1}T\!\!\pmod{\mathfrak{p}^{r}}\in K_{l-1}(O_{r})$ with $\displaystyle T\in\mathfrak{gl}_{n}(O)$. Then we have $\displaystyle T\!\!\pmod{\mathfrak{p}^{l-1}}\in\mathfrak{g}(O_{l-1})$ and

[TABLE]

where $\displaystyle\overline{T}=[v]+Y\in\mathfrak{g}(\mathbb{F})$ with $\displaystyle v\in\mathbb{V}_{\beta}$ and $\displaystyle Y\in\mathfrak{g}_{\beta}(\mathbb{F})$. In particular $\displaystyle\pi_{\beta,\rho}(h)$ is the homothety $\displaystyle\psi_{\beta,\rho}(h)$ for all $\displaystyle h\in Z(O_{r},\beta)$.

[Proof] By the definition we have

[TABLE]

where $\displaystyle X=[v]+Y\in\mathfrak{g}(\mathbb{F})$ with $\displaystyle v\in\mathbb{V}_{\beta}$ and $\displaystyle Y\in\mathfrak{g}_{\beta}(\mathbb{F})$ and put $\displaystyle 1+\varpi^{l-1}T\equiv l(X)(1+\varpi^{l}S)\!\!\pmod{\mathfrak{p}^{r}}$ with $\displaystyle X\in\mathfrak{g}(\mathbb{F})$ and $\displaystyle S\in\mathfrak{g}(O)$. Then we have

[TABLE]

so that we have $\displaystyle T\!\!\pmod{\mathfrak{p}}=X\in\mathfrak{g}(\mathbb{F})$ and

[TABLE]

Then we have

[TABLE]

$\displaystyle\blacksquare$

This proposition shows that the irreducible representation $\displaystyle(\pi_{\beta,\rho},L^{2}(\mathbb{W}^{\prime}))$ $\displaystyle K_{l-1}(O_{r})$ is exactly the irreducible representation $\displaystyle\pi_{\psi}$ with $\displaystyle\psi=\psi_{\beta,\rho}\in Y_{\beta}$ defined in Proposition 5.1.1.

5.5 Description of Schur multiplier

Fix a \displaystyle\rho\in\mathfrak{g}_{\beta}(\mathbb{F})^{^}. In this subsection we will study the conjugate action of $\displaystyle g_{r}=g\!\!\pmod{\mathfrak{p}^{r}}\in G(O_{r},\beta)$ on $\displaystyle K_{l-1}(O_{r})$ and on $\displaystyle\pi_{\beta,\rho}$. For any $\displaystyle X\in\mathfrak{g}(\mathbb{F})$, we have

[TABLE]

with $\displaystyle\nu(X,g)\in\mathfrak{g}(O)$ such that

[TABLE]

Then we have

[TABLE]

and an action of $\displaystyle g_{r}\in G(O_{r},\beta)$ on $\displaystyle(X;S_{l-1},T_{l-1})\in\mathbb{G}{\times}_{\mathfrak{g}(\mathbb{F})}\mathbb{M}$ is defined by

[TABLE]

The action (5.5.1) is compatible with the action

[TABLE]

of $\displaystyle g_{r}\in G(O_{r},\beta)$ on $\displaystyle(X,s)\in\mathcal{H}_{\beta}$ via the surjection (5.3.1). If we put $\displaystyle X=[v]+Y\in\mathfrak{g}(\mathbb{F})$ with $\displaystyle v\in\mathbb{V}_{\beta}$ and $\displaystyle Y\in\mathfrak{g}_{\beta}(\mathbb{F})$, then we have

[TABLE]

in the notations of subsection 4.1. So $\displaystyle g_{r}\in G(O_{r},\beta)$ acts on $\displaystyle(v,(Y,s))\in\mathbb{H}_{\beta}$ by

[TABLE]

In particular $\displaystyle g_{r}\in G_{\beta}(O_{r})$ acts on $\displaystyle(v,z)\in\mathbb{H}_{\beta}$ by

[TABLE]

There exists a group homomorphism $\displaystyle T:Sp(\mathbb{V}_{\beta})\to GL_{\mathbb{C}}(L^{2}(\mathbb{W}^{\prime}))$ such that

[TABLE]

for all $\displaystyle\sigma\in Sp(\mathbb{V}_{\beta})$ and $\displaystyle(v,s)\in H_{\beta}$ (see [6, Th.2.4]). Then we have

[TABLE]

If we put

[TABLE]

for $\displaystyle g_{r}\in G_{\beta}(O_{r})$ then we have

[TABLE]

for all $\displaystyle g_{r},h_{r}\in G_{\beta}(O_{r})$, in fact

[TABLE]

On the other hand

[TABLE]

for all $\displaystyle g_{r}\in G_{\beta}(O_{r})$. That is $\displaystyle\widetilde{\psi}_{0}$ is invariant under the conjugate action of $\displaystyle G_{l}(O_{r},\beta)^{(c)}$. Hence we have

[TABLE]

for all $\displaystyle g_{r}\in G_{\beta}(O_{r})$ and $\displaystyle h\in K_{l-1}(O_{r})$.

5.6 Triviality of Schur multiplier

The following proposition is the keystone of this paper.

Proposition 5.6.1

If the characteristic polynomial of $\displaystyle\overline{\beta}\in\mathfrak{g}(\mathbb{F})\subset\mathfrak{gl}_{n}(\mathbb{F})$ is the minimal polynomial of $\displaystyle\overline{\beta}\in M_{n}(\mathbb{F})$, then the Schur multiplier $\displaystyle[c_{\beta,\rho}]\in H^{2}(G_{\beta}(\mathbb{F}),\mathbb{C}^{\times})$ is trivial for all \displaystyle\rho\in\mathfrak{g}(\mathbb{F})^{^}.

[Proof] We will divide the proof into two parts.

1. The case of $\displaystyle G=GL_{n}$. In this case, Corollary 5.1 of [17] shows that the Schur multiplier $\displaystyle[c_{U}]\in H^{2}(G_{\beta}(O_{r}),\mathbb{C}^{\times})$ is trivial. On the other hand we have the inflation-restriction exact sequence

[TABLE]

induced by the exact sequence

[TABLE]

Since we have

[TABLE]

and $\displaystyle K_{1}(O_{r})\cap G_{\beta}(O_{r})\subset G_{\beta}(O_{r})$ are finite commutative groups, the restriction mapping

[TABLE]

is surjective. Hence the inflation mapping

[TABLE]

is injective. Since the results of the preceding subsections show that the Schur multiplier $\displaystyle[c_{U}]\in H^{2}(G_{\beta}(O_{r}),\mathbb{C}^{\times})$ is the image of $\displaystyle[c_{\beta,\rho}]\in H^{2}(G_{\beta}(\mathbb{F}),\mathbb{C}^{\times})$ under the inflation mapping, the statement of the proposition is established for the group $\displaystyle G=GL_{n}$.

222This argument is presented by the referee.

2. The general case of $\displaystyle G\subset GL_{n}$. We have $\displaystyle G_{\beta}(\mathbb{F})\subset GL_{n,\beta}(\mathbb{F})$. Then Proposition 4.2.1 says that the Schur multiplier $\displaystyle[c_{\beta,\rho}]\in H^{2}(G_{\beta}(\mathbb{F}),\mathbb{C}^{\times})$ is the image of the Schur multiplier $\displaystyle[c_{\beta,\widetilde{\rho}}]\in H^{2}(GL_{n,\beta}(\mathbb{F}),\mathbb{C}^{\times})$ under the restriction mapping

[TABLE]

Since we have shown in the part one of the proof that $\displaystyle[c_{\beta,\widetilde{\rho}}]\in H^{2}(GL_{n,\beta},\mathbb{C}^{\times})$ is trivial, so is $\displaystyle[c_{\beta,\rho}]\in H^{2}(G_{\beta}(\mathbb{F}),\mathbb{C}^{\times})$. $\displaystyle\blacksquare$

Now we have established the triviality of the Schur multiplier

[TABLE]

which implies the triviality of the Schur multiplier $\displaystyle[c_{U}]\in H^{2}(G_{\beta}(O_{r}),\mathbb{C}^{\times})$ due to the relation (5.1.2). Then Proposition 5.1.3 and Proposition 5.1.4 give the bijection presented in our main Theorem 2.3.1 in the case of $\displaystyle r$ being odd.

It may be quite interesting if we can find a counter example to the following statement;

Let $\displaystyle G$ be a connected reductive algebraic group defined over $\displaystyle\mathbb{F}$ and $\displaystyle\mathfrak{g}$ the Lie algebra scheme of $\displaystyle G$. Take a $\displaystyle\beta\in\mathfrak{g}(\mathbb{F})$ which is smoothly regular with respect to $\displaystyle G$ and $\displaystyle G_{\beta}$ is commutative. Then the Schur multiplier $\displaystyle[c_{\beta,\rho}]\in H^{2}(G_{\beta}(\mathbb{F}),\mathbb{C}^{\times})$ is trivial for all \displaystyle\rho\in\mathfrak{g}(\mathbb{F})^{^}.

6 Classical groups

In this section, we will apply Theorem 2.3.1 to the special linear group, the symplectic group and the orthogonal group.

6.1 Special linear group

Let $\displaystyle G=SL_{n}$ be the $\displaystyle O$-group subscheme $\displaystyle G\subset GL_{n}$ defined by

[TABLE]

Its Lie algebra is

[TABLE]

Then

Proposition 6.1.1

$\displaystyle G=SL_{n}$* is smooth over $\displaystyle O$,* 2. 2)

the conditions II) and III) of subsection 2.1 hold for $\displaystyle G=SL_{n}$, 3. 3)

the condition I) of subsection 2.1 holds for $\displaystyle G=SL_{n}$ if and only if $\displaystyle n$ is prime to the characteristic of $\displaystyle\mathbb{F}$.

[Proof]

1. Take any $\displaystyle O$-algebra $\displaystyle R$ and an ideal $\displaystyle\mathfrak{a}\subset R$ such that $\displaystyle\mathfrak{a}^{2}=0$. For any $\displaystyle g\!\!\pmod{\mathfrak{a}}\in G(R/\mathfrak{a})$ ($\displaystyle g\in M_{n}(R)$), we have $\displaystyle\det g=1+a$ with $\displaystyle a\in\mathfrak{a}$. Then $\displaystyle(1-a)(1+a)=1$ because $\displaystyle a^{2}\in\mathfrak{a}^{2}=0$. Now

[TABLE]

and $\displaystyle h\equiv g\!\!\pmod{\mathfrak{a}}$. Hence the canonical mapping $\displaystyle G(R)\to G(R/\mathfrak{a})$ is surjective, which means that $\displaystyle G=SL_{n}$ is smooth over $\displaystyle O$ (see [4, p.111, Cor. 4.6]).

2. Let $\displaystyle r=l+l^{\prime}$ with $\displaystyle l\geq l^{\prime}>0$. Then for any $\displaystyle X\in M_{n}(O)$ with the eigenvalues $\displaystyle\alpha_{i}$ ($\displaystyle 1\leq i\leq n$), we have

[TABLE]

because $\displaystyle 2l\geq r$. Then $\displaystyle\det(1_{n}+\varpi^{l}X)\equiv 1\!\!\pmod{\mathfrak{p}^{r}}$ if and only if $\displaystyle\text{\rm tr}(X)\equiv 0\!\!\pmod{\mathfrak{p}^{l^{\prime}}}$. Hence the condition II) holds. If $\displaystyle r=2l-1>1$ is odd, then we have

[TABLE]

because $\displaystyle 3l-3\geq r$. Then $\displaystyle 1_{n}+\varpi^{l-1}X+2^{-1}\varpi^{2l-2}X^{2}\!\!\pmod{\mathfrak{p}^{r}}\in G(O_{r})$ if $\displaystyle\text{\rm tr}(X)=0$. Hence the condition III) holds.

3. Take a $\displaystyle X\in\mathfrak{g}(\mathbb{F})$ such that $\displaystyle\text{\rm tr}(XY)=0$ for all $\displaystyle Y\in\mathfrak{g}(\mathbb{F})$. Then $\displaystyle X=x\cdot 1_{n}$ with $\displaystyle\text{\rm tr}(X)=0$. Because $\displaystyle n$ is prime to the characteristic of $\displaystyle\mathbb{F}$, we have $\displaystyle X=0$. $\displaystyle\blacksquare$

Take a $\displaystyle\beta\in\mathfrak{g}(O)=\mathfrak{sl}_{n}(O)$ such that the characteristic polynomial of $\displaystyle\overline{\beta}\in M_{n}(\mathbb{F})$ is its minimal polynomial and is separable. Then the fibers $\displaystyle G_{\beta}{\otimes}_{O}K$ ($\displaystyle K=\mathbb{F},F$) are connected and $\displaystyle\beta$ is smoothly regular with respect to $\displaystyle G$ over $\displaystyle K=F,\mathbb{F}$ by the remark in subsubsection 3.3.1. Hence $\displaystyle G_{\beta}$ is commutative and smooth over $\displaystyle O$ by Proposition 3.1.1, and Theorem 2.3.1 is applicable.

Examples of such $\displaystyle\beta\in\mathfrak{g}(O)=\mathfrak{sl}_{n}(O)$ is given by an unramified separable extension $\displaystyle L/F$ of degree $\displaystyle n$. Identify $\displaystyle L$ with a $\displaystyle F$-subalgebra of $\displaystyle M_{n}(F)$ by means of the regular representation with respect to an $\displaystyle O$-basis of $\displaystyle O_{L}$. Take any $\displaystyle\beta\in L$ such that $\displaystyle O_{L}=O[\beta]$ and $\displaystyle T_{L/F}(\beta)=0$. Then $\displaystyle\beta\in\mathfrak{sl}_{n}(O)$ and the reduction modulo $\displaystyle\mathfrak{p}$ of the characteristic polynomial of $\displaystyle\beta\in M_{n}(O)$ is irreducible by Proposition 3.2.2. In this case, we have

[TABLE]

where $\displaystyle e$ is the ramification index of $\displaystyle L/F$ and

[TABLE]

We have also

[TABLE]

and

[TABLE]

for $\displaystyle x\in O_{L}$ such that $\displaystyle T_{L/F}(x)\equiv 0\!\!\pmod{\mathfrak{p}^{l^{\prime}}}$. Then Theorem 2.3.1 gives

Theorem 6.1.2

Let $\displaystyle G=SL_{n}$ with $\displaystyle n$ prime to the characteristic of $\displaystyle\mathbb{F}$. Then there exists a bijection $\displaystyle\theta\mapsto\text{\rm Ind}_{G(O_{r},\beta)}^{G(O_{r})}\sigma_{\beta,\theta}$ of the set

[TABLE]

onto $\displaystyle\text{\rm Irr}(G(O_{r})\mid\psi_{\beta})$.

6.2 Symplectic group

Let $\displaystyle G=Sp_{2n}$ be the $\displaystyle O$-group scheme such that

[TABLE]

($\displaystyle J_{n}=\begin{bmatrix}0&1_{n}\\ -1_{n}&0\end{bmatrix}$) for all $\displaystyle O$-algebra $\displaystyle R$. The Lie algebra $\displaystyle\mathfrak{g}=\mathfrak{sp}_{2n}$ of $\displaystyle G$ is an affine $\displaystyle O$-subscheme of $\displaystyle\mathfrak{gl}_{2n}$ such that

[TABLE]

for all $\displaystyle O$-algebra $\displaystyle R$. Then

Proposition 6.2.1

$\displaystyle G=Sp_{2n}$* is smooth over $\displaystyle O$,* 2. 2)

the conditions I), II) and III) of subsection 2.1 hold for $\displaystyle G=Sp_{2n}$.

[Proof]

1. Take any $\displaystyle O$-algebra $\displaystyle R$ and an ideal $\displaystyle\mathfrak{a}\subset R$ such that $\displaystyle\mathfrak{a}^{2}=0$. For any $\displaystyle g\!\!\pmod{\mathfrak{a}}\in G(R/\mathfrak{a})$ ($\displaystyle g\in M_{2n}(R)$), we have $\displaystyle gJ_{n}\,^{t}g=(1_{2n}+X)J_{n}$ with $\displaystyle X\in M_{2n}(\mathfrak{a})$. Then $\displaystyle(1_{2n}+2^{-1}X)^{2}=1_{2n}+X$ because $\displaystyle X^{2}=0$. Then we have

[TABLE]

because $\displaystyle XJ_{n}=J_{n}\,^{t}X$. We have also $\displaystyle\det(1_{2n}+2^{-1}X)\in R^{\times}$ because $\displaystyle\det(1_{2n}+2^{-1}X)\equiv 1\!\!\pmod{\mathfrak{a}}$. Now

[TABLE]

and $\displaystyle g\equiv h\!\!\pmod{\mathfrak{a}}$. Hence the canonical mapping $\displaystyle G(R)\to G(R/\mathfrak{a})$ is surjective, which means that $\displaystyle G=Sp_{2n}$ is smooth over $\displaystyle O$ (see [4, p.111, Cor. 4.6]).

2. For any $\displaystyle T\in M_{2n}(\mathbb{F})$, we have $\displaystyle Y=T+J_{n}\,^{t}TJ_{n}\in\mathfrak{g}(\mathbb{F})$ and

[TABLE]

for all $\displaystyle X\in\mathfrak{g}(\mathbb{F})$. Hence the condition I) holds.

For any $\displaystyle X\in M_{2n}(O)$, we have

[TABLE]

which implies that the condition II) holds.

Take any $\displaystyle X\in\mathfrak{g}(O)$. Then we have

[TABLE]

because $\displaystyle 3l-3\geq r$. So the condition III) holds. $\displaystyle\blacksquare$

Take a $\displaystyle\beta\in\mathfrak{g}(O)$ such that the characteristic polynomial of $\displaystyle\overline{\beta}\in M_{2n}(\mathbb{F})$ is its minimal polynomial and $\displaystyle\det\overline{\beta}\neq 0$. Then the fibers $\displaystyle G_{\beta}{\otimes}_{O}K$ ($\displaystyle K=\mathbb{F},F$) are connected and $\displaystyle\beta$ is smoothly regular with respect to $\displaystyle G$ over $\displaystyle K=F,\mathbb{F}$ by the remark in subsubsection 3.3.2. Hence $\displaystyle G_{\beta}$ is commutative and smooth over $\displaystyle O$ by Proposition 3.1.1, and Theorem 2.3.1 is applicable.

Proposition 3.2.2 gives examples of such $\displaystyle\beta\in\mathfrak{g}(O)$.

Let $\displaystyle L_{+}/F$ be a tamely ramified extension of degree $\displaystyle n$ and $\displaystyle L/L_{+}$ a quadratic extension. Take a $\displaystyle\omega\in O_{L}$ such that

[TABLE]

where $\displaystyle\rho\in\text{\rm Gal}(L/L_{+})$ is the non-trivial element. Then

[TABLE]

with the ramification index $\displaystyle e_{+}$ of $\displaystyle L_{+}/F$ and a prime element $\displaystyle\varpi_{L_{+}}$ of $\displaystyle L_{+}$ is a symplectic form on the $\displaystyle F$-vector space $\displaystyle L$. Fix an $\displaystyle O$-basis $\displaystyle\{u_{1},\cdots,u_{n}\}$ of $\displaystyle O_{L_{+}}$. Since $\displaystyle L_{+}/F$ is a tamely ramified extension, there exists $\displaystyle u_{j}^{\ast}\in\mathfrak{p}_{L_{+}}^{1-e_{+}}$ ($\displaystyle 1\leq j\leq n$) such that $\displaystyle T_{L_{+}/F}(u_{i}u_{j}^{\ast})=\delta_{ij}$. If we put $\displaystyle v_{j}=\omega\cdot\varpi_{L_{+}}^{e_{+}-1}\cdot u_{j}^{\ast}\in O_{L}$, then we have

[TABLE]

Identify the $\displaystyle F$-algebra $\displaystyle L$ with a $\displaystyle F$-subalgebra of $\displaystyle M_{2n}(F)$ by means of the $\displaystyle O$-basis $\displaystyle\{u_{1},\cdots,u_{n},v_{1},\cdots,v_{n}\}$ of $\displaystyle O_{L}$.

Take a $\displaystyle\beta\in O_{L}$ such that $\displaystyle O_{L}=O[\beta]$ and $\displaystyle\beta^{\rho}+\beta=0$. Then

$\displaystyle\beta\in\mathfrak{g}(O)$ and the characteristic polynomial of $\displaystyle\overline{\beta}\in M_{2n}(\mathbb{F})$ is its minimal polynomial, and 2. 2)

$\displaystyle\det\overline{\beta}\neq 0$ for $\displaystyle\overline{\beta}\in M_{2n}(\mathbb{F})$ if and only if $\displaystyle L/F$ is not totally ramified

by Proposition 3.2.2. We have

[TABLE]

where $\displaystyle e$ is the ramification index of $\displaystyle L/F$ and

[TABLE]

We have also

[TABLE]

and

[TABLE]

for $\displaystyle x\in O_{L}$ such that $\displaystyle T_{L/L_{+}}(x)\equiv 0\!\!\pmod{\mathfrak{p}_{L_{+}}^{e_{+}l^{\prime}}}$. Then Theorem 2.3.1 gives

Theorem 6.2.2

Assume that $\displaystyle L/F$ is not totally ramified. Then there exists a bijection $\displaystyle\theta\mapsto\text{\rm Ind}_{G(O_{r},\beta)}^{G(O_{r})}\sigma_{\beta,\theta}$ of the set

[TABLE]

onto $\displaystyle\text{\rm Irr}(G(O_{r})\mid\psi_{\beta})$.

6.3 Orthogonal group

Take a $\displaystyle S\in M_{m}(O)$ such that ${}^{t}S=S$ and $\displaystyle\det S\in O^{\times}$. Let $\displaystyle G=SO(S)$ be the $\displaystyle O$-group subscheme of $\displaystyle SL_{m}$ such that

[TABLE]

for all $\displaystyle O$-algebra $\displaystyle R$. The Lie algebra $\displaystyle\mathfrak{g}=\mathfrak{so}(S)$ of $\displaystyle G$ is an affine $\displaystyle O$-subscheme of $\displaystyle\mathfrak{gl}_{m}$ such that

[TABLE]

for all $\displaystyle O$-algebra $\displaystyle R$. Then

Proposition 6.3.1

$\displaystyle G=SO(S)$* is smooth over $\displaystyle O$,* 2. 2)

the conditions I), II) and III) of subsection 2.1 hold for $\displaystyle G=SO(S)$.

[Proof]

1. Take any $\displaystyle O$-algebra $\displaystyle R$ and an ideal $\displaystyle\mathfrak{a}\subset R$ such that $\displaystyle\mathfrak{a}^{2}=0$. For any $\displaystyle g\!\!\pmod{\mathfrak{a}}\in G(R/\mathfrak{a})$, we have $\displaystyle gS\,^{t}g=(1_{m}+X)S$ with $\displaystyle X\in M_{m}(\mathfrak{a})$. Then, as in the proof of Proposition 6.2.1, we have $\displaystyle\det(1_{m}+2^{-1}X)\in R^{\times}$ and put $\displaystyle h=(1_{m}+2^{-1}X)^{-1}g\in GL_{m}(R)$. Then we have $\displaystyle hS\,^{t}h=S$ and $\displaystyle h\equiv g\!\!\pmod{\mathfrak{a}}$. If $\displaystyle\det h=-1$, then $\displaystyle 2\in\mathfrak{a}$ because $\displaystyle\det h\equiv\det g\equiv 1\!\!\pmod{\mathfrak{a}}$, this means $\displaystyle\mathfrak{a}=R$ since $\displaystyle F$ is non-dyadic. Hence the canonical mapping $\displaystyle G(R)\to G(R/\mathfrak{a})$ is surjective, which means that $\displaystyle G=SO(S)$ is smooth over $\displaystyle O$ (see [4, p.111, Cor. 4.6]).

2. For any $\displaystyle T\in M_{2n}(\mathbb{F})$, we have $\displaystyle Y=T-S\,^{t}TS^{-1}\in\mathfrak{g}(\mathbb{F})$ and

[TABLE]

for all $\displaystyle X\in\mathfrak{g}(\mathbb{F})$. Hence the condition I) holds. Similar arguments as in the proof of Proposition 6.2.1 show that the conditions II) and III) hold for $\displaystyle G=SO(S)$. $\displaystyle\blacksquare$

6.3.1 Case of even variables

Let us consider the case of $\displaystyle m=2n$ being even.

Take a $\displaystyle\beta\in\mathfrak{g}(O)$ such that the characteristic polynomial of $\displaystyle\overline{\beta}\in M_{m}(\mathbb{F})$ is its minimal polynomial and $\displaystyle\det\overline{\beta}\neq 0$. Then $\displaystyle\beta\in\mathfrak{g}(O)$ is smoothly regular with respect to $\displaystyle G=SO(S)$ over $\displaystyle O$ by the remark in subsubsection 3.3.4, and the fibers $\displaystyle G_{\beta}{\otimes}_{O}K$ ($\displaystyle K=\mathbb{F},F$) are connected. So $\displaystyle G_{\beta}$ is smooth over $\displaystyle O$ and Theorem 2.3.1 is applicable.

Proposition 3.2.2 gives examples of such $\displaystyle\beta\in\mathfrak{g}(O)$.

Let $\displaystyle L/F$ be a tamely ramified Galois extension of degree $\displaystyle 2n$. Fix an intermediate field $\displaystyle F\subset L_{+}\subset L$ such that $\displaystyle(L:L_{+})=2$, and assume that $\displaystyle L/L_{+}$ is unramified. Take an $\displaystyle\varepsilon\in O_{L_{+}}^{\times}$ and put

[TABLE]

where $\displaystyle\rho\in\text{\rm Gal}(L/L_{+})$ is the non-trivial element, $\displaystyle e$ is the ramification index of $\displaystyle L/F$ and $\displaystyle\varpi_{L_{+}}$ is a prime element of $\displaystyle L_{+}$. Then $\displaystyle S_{\varepsilon}$ is a regular $\displaystyle F$-quadratic form on $\displaystyle L$. Take a $\displaystyle O$-basis $\displaystyle\{u_{1},\cdots,u_{2n}\}$ of $\displaystyle O_{L}$ and put $\displaystyle B=\left(u_{i}^{\sigma_{j}}\right)_{1\leq i,j\leq 2n}$ with $\displaystyle\text{\rm Gal}(L/F)=\{\sigma_{1},\cdots,\sigma_{2n}\}$. Then we have

[TABLE]

so that the discriminant of the quadratic form $\displaystyle S_{\varepsilon}$ is

[TABLE]

Note that $\displaystyle(\det B)^{\sigma}=\pm\det B$ for any $\displaystyle\sigma\in\text{\rm Gal}(L/F)$. Since $\displaystyle L/F$ is tamely ramified, its discriminant is

[TABLE]

where $\displaystyle 2n=ef$. Hence $\displaystyle\det\left(S_{\varepsilon}(u_{i},u_{j})\right)_{1\leq i,j\leq 2n}\in O^{\times}$. So the $\displaystyle O$-group scheme $\displaystyle G=SO(S_{\varepsilon})$ and its Lie algebra $\displaystyle\mathfrak{g}=\mathfrak{so}(S_{\varepsilon})$ is defined by

[TABLE]

and by

[TABLE]

for all $\displaystyle O$-algebra $\displaystyle R$. Note that $\displaystyle\text{\rm End}_{F}(L)$ acts on $\displaystyle L$ from the right side.

Take a $\displaystyle\beta\in O_{L}$ such that $\displaystyle O_{L}=O[\beta]$ and $\displaystyle\beta^{\rho}+\beta=0$. Since $\displaystyle L/F$ is not totally ramified, Proposition 3.2.2 implies that $\displaystyle\beta$ is an unit of $\displaystyle O_{L}$. Identify $\displaystyle\beta\in L$ with the element $\displaystyle x\mapsto x\beta$ of $\displaystyle\mathfrak{g}(O)\subset\text{\rm End}_{O}(O_{L})$. Then we have

[TABLE]

where $\displaystyle e$ is the ramification index of $\displaystyle L/F$ and

[TABLE]

We have also

[TABLE]

and

[TABLE]

for $\displaystyle x\in O_{L}$ such that $\displaystyle T_{L/L_{+}}(x)\equiv 0\!\!\pmod{\mathfrak{p}_{L_{+}}^{e_{+}l^{\prime}}}$. Then Theorem 2.3.1 gives

Theorem 6.3.2

There exists a bijection $\displaystyle\theta\mapsto\text{\rm Ind}_{G(O_{r},\beta)}^{G(O_{r})}\sigma_{\beta,\theta}$ of the set

[TABLE]

onto $\displaystyle\text{\rm Irr}(G(O_{r})\mid\psi_{\beta})$.

6.3.2 Case of odd variables

Let us consider the case of $\displaystyle m=2n+1$ being odd.

Take a $\displaystyle\beta\in\mathfrak{g}(O)$ such that the characteristic polynomial of $\displaystyle\overline{\beta}\in M_{m}(\mathbb{F})$ is its minimal polynomial. Then $\displaystyle\beta\in\mathfrak{g}(O)$ is smoothly regular with respect to $\displaystyle G=SO(S)$ over $\displaystyle O$ by the remark in subsubsection 3.3.3, and the fibers $\displaystyle G_{\beta}{\otimes}_{O}K$ ($\displaystyle K=\mathbb{F},F$) are connected. So $\displaystyle G_{\beta}$ is smooth over $\displaystyle O$ and Theorem 2.3.1 is applicable.

Proposition 3.2.2 gives examples of such $\displaystyle\beta\in\mathfrak{g}(O)$.

Use the notations of the preceding subsubsection. Take a $\displaystyle\eta\in O^{\times}$ and define a $\displaystyle F$-quadratic form $\displaystyle S_{\varepsilon,\eta}$ on the $\displaystyle F$-vector space $\displaystyle L\times F$ by

[TABLE]

Then the $\displaystyle O$-group scheme $\displaystyle G=SO(S_{\varepsilon,\eta})$ and its Lie algebra $\displaystyle\mathfrak{g}=\mathfrak{so}(S_{\varepsilon,\eta})$ is defined by

[TABLE]

and

[TABLE]

for all $\displaystyle O$-algebra $\displaystyle R$. An element $\displaystyle X\in\text{\rm End}_{R}((O_{L}\times O){\otimes}_{O}R)$ is denoted by

[TABLE]

Take an $\displaystyle\alpha\in O_{L}$ such that $\displaystyle O_{L}=O[\alpha]$ and $\displaystyle\alpha^{\rho}+\alpha=0$. Identify $\displaystyle\alpha\in O_{L}$ with the endomorphism $\displaystyle[x\mapsto x\alpha]\in\text{\rm End}_{O}(O_{L})$. Then

[TABLE]

and the characteristic polynomial of $\displaystyle\overline{\beta}=\beta\!\!\pmod{\mathfrak{p}}\in\text{\rm End}_{\mathbb{F}}(\mathbb{L}\times\mathbb{F})$ is its minimal polynomial by Proposition 3.2.2. In this case we have

[TABLE]

where

[TABLE]

We have also $\displaystyle\psi_{\beta}(h)=\tau\left(\varpi^{-l^{\prime}}T_{L/F}(\alpha x)\right)$ for all

[TABLE]

Then Theorem 2.3.1 gives

Theorem 6.3.3

There exists a bijection $\displaystyle\theta\mapsto\text{\rm Ind}_{G(O_{r},\beta)}^{G(O_{r})}\sigma_{\beta,\theta}$ of the set

[TABLE]

onto $\displaystyle\text{\rm Irr}(G(O_{r})\mid\psi_{\beta})$.