On generic supercuspidal representations of $Sp_{2n}$

TL;DR

This paper constructs a broad family of irreducible generic supercuspidal representations for symplectic groups over non-archimedean local fields, using compact induction from finite symplectic group representations.

Contribution

It introduces a new method to explicitly construct supercuspidal representations of symplectic groups via compact induction from finite symplectic groups.

Findings

Explicit construction of supercuspidal representations

Use of compact induction from finite symplectic groups

Representation inflation from finite quotient rings

Abstract

We will construct a family of irreducible generic supercuspidal representations of the symplectic groups over non-archimedian local field $F$ of odd residual characteristic. The supercuspidal representations are compactly induced from irreducible representations of the hyperspecial compact subgroup which are inflated from irreducible representations of finite symplectic groups over the finite quotient ring of the integer ring of $F$ modulo high powers of the prime element.

Equations

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

$$J_n=\left[\begin{array}{cc}0& I_n\\ -I_n& 0\end{array}\right]\text{where}{I}_n=\left[\begin{array}{ccc}& & 1\\ & \text{raise}1.0pt.\text{raise}4.0pt.\text{raise}7.0pt\text{vbox}.& \\ 1& & \end{array}\right],$$

$$G(L)=\{g\in G{L}_{2n}(L)\mid g{J}_n{}^{t}g=J_n\}$$

$$(\text{resp. g(L)={X∈gl2n(L)∣XJn+JntX=0}})$$

$$B:\mathfrak{g}\times \mathfrak{g}\to \text{Spec}(O_F[t])$$

$$l^{\prime }=\{\begin{array}{ll}l& :\text{if r=2l is even,}\\ l-1& :\text{if r=2l−1 is odd.}\end{array}$$

$$T(L)=\left\{g\in L[\overline{\beta }{]}^{\times }\mid g{J}_n{}^{t}g{J}_n^{-1}=1_{2n}\right\}$$

$$T(O_F/{\mathfrak{p}}^r)=\left\{\overline{\epsilon }\in {\left(O_K/{\varpi }^r{O}_K\right)}^{\times }\mid \epsilon \in O_K^{\times }\text{s.t.}{N}_{K/K_{+}}(\epsilon )=1\right\}$$

$$X(\mathrm{mod}{\mathfrak{p}}^{l^{\prime }})\mapsto 1_{2n}+{\varpi }^{l}X(\mathrm{mod}{\mathfrak{p}}^r).$$

$${\psi }_{\beta }\left(1_{2n}+{\varpi }^{l}X(\mathrm{mod}{\mathfrak{p}}^r)\right)=\tau \left({\varpi }^{-l^{\prime }}B(X,\beta )\right).$$

$${\psi }_{\beta }\left(1+{\varpi }^{l}x(\mathrm{mod}{\mathfrak{p}}^r)\right)=\tau \left({\varpi }^{-l^{\prime }}{T}_{K/F}(x\beta )\right)$$

$$\dim {\delta }_{\beta ,\theta }=q^{n^{2}r}(1-q^{-n})\prod _{k=1}^{n-1}\left(1-q^{-2k}\right)$$

$${\psi }_{\beta }(gh{g}^{-1})={\psi }_{\beta }(h)\text{for all}h\in G({\mathfrak{p}}^l/{\mathfrak{p}}^r).$$

$$H(O_F/{\mathfrak{p}}^r)$$

$$=T(O_F/{\mathfrak{p}}^r)\cdot G({\mathfrak{p}}^{l^{\prime }}/{\mathfrak{p}}^r)$$

$$\sigma \mapsto {\text{Ind}}_{H(O_F/{\mathfrak{p}}^r)}^{G(O_F/{\mathfrak{p}}^r)}\sigma$$

$$\theta \mapsto {\delta }_{\beta ,\theta }={\text{Ind}}_H^G{\sigma }_{\beta ,\theta }(G=G(O_F/{\mathfrak{p}}^r),H=H(O_F/{\mathfrak{p}}^r))$$

$$H(O_F/{\mathfrak{p}}^r)=T(O_F/{\mathfrak{p}}^r)\cdot G({\mathfrak{p}}^l/{\mathfrak{p}}^r)$$

$${\sigma }_{\beta ,\theta }(gh)=\theta (g)\cdot {\psi }_{\beta }(h)$$

$$H(O_F/{\mathfrak{p}}^r)=T(O_F/{\mathfrak{p}}^r)\cdot G({\mathfrak{p}}^{l-1}/{\mathfrak{p}}^r).$$

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

$$1_{2n}+{\varpi }^{l-1}X(\mathrm{mod}{\mathfrak{p}}^r)\mapsto X(\mathrm{mod}\mathfrak{p})$$

$$V=\{X\in \mathfrak{g}(\mathbb{F})\mid B(X,\mathfrak{t}(\mathbb{F}))=0\}.$$

$$(u,s)\cdot (v,t)=(u+v,s\cdot t\cdot \hat{\tau }(2^{-1}⟨u,v{⟩}_{\beta }))$$

$$\left({\omega }_{\beta }(u,s)f\right)(w)=s\cdot \hat{\tau }\left(2^{-1}⟨u_{-},u_{+}{⟩}_{\beta }+⟨w,u_{+}{⟩}_{\beta }\right)\cdot f(w+u_{-})$$

$${\omega }_{\beta }(u\sigma ,s)=\Omega (\sigma {)}^{-1}\circ {\omega }_{\beta }(u,s)\circ \Omega (\sigma )$$

$${\omega }_{\beta ,\rho }(h)=\tau \left({\varpi }^{-l}B(T,\beta )-2^{-1}B(T^2,\beta )\right)\cdot \rho (Y)\cdot {\omega }_{\beta }(v,1).$$

$${\psi }_{\beta ,\rho }(h)=\tau \left({\varpi }^{-l}B(T,\beta )-2^{-1}B(T^2,\beta )\right)\cdot \rho (T(\mathrm{mod}\mathfrak{p})),$$

$${\text{Ind}}_{Z({\mathfrak{p}}^{l-1}/{\mathfrak{p}}^r)}^{G({\mathfrak{p}}^{l-1}/{\mathfrak{p}}^r)}{\psi }_{\beta ,\rho }=\stackrel{\dim {\omega }_{\beta ,\rho }}{⨁}{\omega }_{\beta ,\rho }$$

$$\dim {\omega }_{\beta ,\rho }=q^{2^{-1}({\dim }_{\mathbb{F}}\mathfrak{g}(\mathbb{F})-\mathfrak{t}(\mathbb{F}))}=q^{n^2}.$$

$${\sigma }_{\beta ,\rho }(gh)=\Omega ({\sigma }_{\overline{g}})\circ {\omega }_{\beta ,\rho }(h)$$

Full text

1 Introduction

The supercuspidal representations of a classical group $\displaystyle G$ (that is a unitary, symplectic or special orthogonal group) over a non-arcimedian local field $\displaystyle F$ are realized as compactly induced representations $\displaystyle\text{\rm ind}_{J}^{G(F)}\delta$ where $\displaystyle\delta$ is an irreducible finite dimensional complex representation of an open compact subgroup $\displaystyle J$ of $\displaystyle G(F)$ (see [5]).

If $\displaystyle G$ is quasi-split and unramified over $\displaystyle F$, then $\displaystyle G$ has a smooth model over the integer ring $\displaystyle O_{F}$ of $\displaystyle F$. In this case, [1] shows that the supercuspidal representation $\displaystyle\text{\rm ind}_{J}^{G(F)}\delta$ is generic if and only if $\displaystyle J$ is hyperspecial. We will consider the case $\displaystyle J=G(O_{F})$.

The irreducible complex representations of $\displaystyle G(O_{F})$ come from those of the finite group $\displaystyle G(O_{F}/\mathfrak{p}^{r})$ ($\displaystyle\mathfrak{p}$ is the maximal ideal of $\displaystyle O_{F}$) via the canonical surjection $\displaystyle G(O_{F})\to G(O_{F}/\mathfrak{p}^{r})$. If $\displaystyle r=1$, we have well-developed theories on the representation of the finite reductive group $\displaystyle G(\mathbb{F})$ ($\displaystyle\mathbb{F}=O_{F}/\mathfrak{p}$ is the residue class field of $\displaystyle F$). The understanding of the cases $\displaystyle r>1$ is less complete. For example the representation theory of $\displaystyle GL_{n}(O_{F}/\mathfrak{p}^{r})$ with $\displaystyle r>1$ is studied by [4] and then [2] to construct supercuspidal representations of $\displaystyle GL_{n}(F)$, however their descriptions of representations of $\displaystyle GL_{n}(O_{F}/\mathfrak{p}^{r})$ are rather complicated and not suitable for detailed treatment or for generalization to other classical groups.

Recently [6] gives a uniform and quite explicit description of the irreducible representations of $\displaystyle G(O_{F}/\mathfrak{p}^{r})$ ($\displaystyle r>1$) associated with the regular adjoint orbit, under the assumption of triviality of certain Schur multiplier. The Schur multiplier is trivial if the characteristic polynomial of the adjoint orbit is irreducible modulo $\displaystyle\mathfrak{p}$. So starting from this explicit description of irreducible complex representations $\displaystyle\delta$ of $\displaystyle G(O_{F})$, we may be able to give a simple proof of the fact that the compactly induced representation $\displaystyle\text{\rm ind}_{G(O_{F})}^{G(F)}\delta$ is a generic supercuspidal representation. It also gives a good foundation for the further detailed studies on the generic supercuspidal representations of classical groups.

In this paper, we will study this procedure in detail for the case of the symplectic group. Our main results are presented as Theorem 2.2.1. The rest of this paper is devoted to the proofs.

2 Main results

2.1

Let $\displaystyle F$ be a non-dyadic non-archimedian local field. The integer ring of $\displaystyle F$ is denoted by $\displaystyle O_{F}$ with the maximal ideal $\displaystyle\mathfrak{p}$ generated by $\displaystyle\varpi$. The residue class field is denoted by $\displaystyle\mathbb{F}=O_{F}/\mathfrak{p}$. Let $\displaystyle\tau$ be a continuous unitary additive character of $\displaystyle F$ such that

[TABLE]

Let $\displaystyle G=Sp_{2n}$ be the symplectic group and $\displaystyle\mathfrak{g}=\mathfrak{sp}_{2n}$ its Lie algebra with respect to the symplectic form

[TABLE]

that is to say $\displaystyle G$ (resp. $\displaystyle\mathfrak{g}$) is a smooth $\displaystyle O_{F}$-group scheme (resp. affine $\displaystyle O_{F}$-scheme) such that

[TABLE]

[TABLE]

for any commutative $\displaystyle O_{F}$-algebra $\displaystyle L$. Let us denote by

[TABLE]

the trace form, that is $\displaystyle B(X,Y)=\text{\rm tr}(XY)$ for all $\displaystyle X,Y\in\mathfrak{g}(L)$ with any commutative $\displaystyle O_{F}$-algebra $\displaystyle L$.

For any $\displaystyle O_{F}$-group subscheme $\displaystyle H\subset G$ and for any integers $\displaystyle 0<r<l$, let us denote by $\displaystyle H(\mathfrak{p}^{l}/\mathfrak{p}^{r})$ (resp. $\displaystyle H(\mathfrak{p}^{r})$) the kernel of the canonical group homomorphism $\displaystyle H(O_{F}/\mathfrak{p}^{r})\to H(O_{F}/\mathfrak{p}^{l})$ (resp. $\displaystyle H(O_{F})\to H(O_{F}/\mathfrak{p}^{r})$). These group homomorphisms are surjective if $\displaystyle H$ is smooth over $\displaystyle O_{F}$.

2.2

Fix an integer $\displaystyle r>1$ and put $\displaystyle r=l+l^{\prime}$ with the smallest integer $\displaystyle l$ such that $\displaystyle 0<l^{\prime}\leq l$, in other word

[TABLE]

Fix a $\displaystyle\beta\in\mathfrak{g}(O_{F})$ such that the reduction modulo $\displaystyle\mathfrak{p}$ of the characteristic polynomial $\displaystyle\chi_{\beta}(t)$ of $\displaystyle\beta\in\mathfrak{gl}_{2n}(O_{F})$ is irreducible over $\displaystyle\mathbb{F}$. Then the $\displaystyle F$-subalgebra $\displaystyle K=F[\beta]$ of the matrix algebra $\displaystyle M_{2n}(F)$ is an unramified field extension of $\displaystyle F$ of degree $\displaystyle 2n$. The element $\displaystyle\tau\in\text{\rm Gal}(K/F)$ of order $\displaystyle 2$ of the cyclic extension $\displaystyle K/F$ is given by $\displaystyle x^{\tau}=J_{n}\,^{t}xJ_{n}^{-1}$. In particular $\displaystyle\beta^{\tau}=-\beta$. The integer ring and the residue class field of $\displaystyle K$ are denoted by $\displaystyle O_{K}$ and $\displaystyle\mathbb{K}=O_{K}/\varpi O_{K}$ respectively. On the other hand $\displaystyle\beta$ is a regular semisimple element of $\displaystyle\mathfrak{g}(F)$ and its centralizer $\displaystyle T=Z_{G}(\beta)$ in $\displaystyle G$ is a maximal torus of $\displaystyle G$ which splits over $\displaystyle K$. More precisely $\displaystyle T$ is a smooth $\displaystyle O_{F}$-group subscheme of $\displaystyle G$ such that

[TABLE]

for any commutative $\displaystyle O_{F}$-algebra where $\displaystyle\overline{\beta}\in\mathfrak{g}(L)$ is the image of $\displaystyle\beta\in\mathfrak{g}(O_{F})$ by the canonical homomorphism $\displaystyle\mathfrak{g}(O_{F})\to\mathfrak{g}(L)$. In particular

[TABLE]

where $\displaystyle K_{+}\subset K$ is the fixed subfield of $\displaystyle\tau\in\text{\rm Gal}(K/F)$.

A group isomorphism $\displaystyle\mathfrak{g}(O_{F}/\mathfrak{p}^{l^{\prime}})\,\tilde{\to}\,G(\mathfrak{p}^{l}/\mathfrak{p}^{r})$ is given by

[TABLE]

Let us denote by $\displaystyle\psi_{\beta}$ the unitary character of $\displaystyle G(\mathfrak{p}^{l}/\mathfrak{p}^{r})$ defined by

[TABLE]

On $\displaystyle T(O_{F}/\mathfrak{p}^{r})\cap G(\mathfrak{p}^{l}/\mathfrak{p}^{r})$, the character $\displaystyle\psi_{\beta}$ is described by

[TABLE]

with $\displaystyle x\in O_{K}=O_{F}[\beta]$ such that $\displaystyle x+x^{\tau}\equiv 0\!\!\pmod{\mathfrak{p}^{l^{\prime}}}$.

The equivalence classes of the irreducible complex representations of the finite group $\displaystyle G(O_{F}/\mathfrak{p}^{r})$ is denoted by $\displaystyle\text{\rm Irr}(G(O_{F}/\mathfrak{p}^{r}))$ which is identified with the equivalence classes of the representations of the compact group $\displaystyle G(O_{F})$ trivial on $\displaystyle G(\mathfrak{p}^{r})$.

Let us denote by $\displaystyle\text{\rm Irr}(G(O_{F}/\mathfrak{p}^{r}),\psi_{\beta})$ the equivalence classes of the irreducible complex representations $\displaystyle\delta$ of $\displaystyle G(O_{F}/\mathfrak{p}^{r})$ such that the restriction $\displaystyle\delta|_{G(\mathfrak{p}^{l}/\mathfrak{p}^{r})}$ contains the character $\displaystyle\psi_{\beta}$.

The set of the group homomorphisms $\displaystyle\theta:T(O_{F}/\mathfrak{p}^{r})\to\mathbb{C}^{\times}$ such that $\displaystyle\theta=\psi_{\beta}$ on $\displaystyle T(O_{F}/\mathfrak{p})\cap G(\mathfrak{p}^{l}/\mathfrak{p}^{r})$ is denoted by $\displaystyle\Theta(T(O_{F}/\mathfrak{p}^{r}),\psi_{\beta})$.

Then a bijection $\displaystyle\theta\mapsto\delta_{\beta,\theta}$ of $\displaystyle\Theta(T(O_{F}/\mathfrak{p}^{r}),\psi_{\beta})$ onto $\displaystyle\text{\rm Irr}(G(O_{F}/\mathfrak{p}^{r}),\psi_{\beta})$ is given as an application of the general theory developed by [6]. See section 3 for some details.

Now our main result is

Theorem 2.2.1

For any $\displaystyle\theta\in\Theta(T(O_{F}/\mathfrak{p}^{r}),\psi_{\beta})$, the compactly induced representation $\displaystyle\pi_{\beta,\theta}=\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}$ is

irreducible generic supercuspidal representation of $\displaystyle G(F)$, 2. 2)

with the formal degree

[TABLE]

if the Haar measure of $\displaystyle G(F)$ is normalized so that the volume of $\displaystyle G(O_{F})$ is one. 3. 3)

The irreducible representation $\displaystyle\delta_{\beta,\theta}$ is contained in $\displaystyle\pi_{\beta,\theta}|_{G(O_{F})}$ with multiplicity one, and 4. 4)

if an irreducible complex representation $\displaystyle\delta$ of $\displaystyle G(O_{F}/\mathfrak{p}^{r})$ is contained in $\displaystyle\pi_{\beta,\theta}|_{G(O_{F})}$, then $\displaystyle\delta=\delta_{\beta,\theta}$.

The explicit description of $\displaystyle\delta_{\beta,\theta}$ will be given in the next section. The proof of Theorem 2.2.1, except the genericity of $\displaystyle\pi_{\beta,\theta}$, is given in Section 4. The genericity is proved in Section 5.

3 Representations of hyperspecial compact subgroup

In this section, we will recall the explicit descriptions of the bijection $\displaystyle\theta\mapsto\delta_{\beta,\theta}$ of $\displaystyle\Theta(T(O_{F}/\mathfrak{p}^{r}),\psi_{\beta})$ onto $\displaystyle\text{\rm Irr}(G(O_{F}/\mathfrak{p}^{r}),\psi_{\beta})$ given by [6].

We will keep the notations of section 2.

3.1

Let us denote by $\displaystyle H(O_{F}/\mathfrak{p}^{r})$ the subgroup of $\displaystyle g\in G(O_{F}/\mathfrak{p}^{r})$ such that

[TABLE]

In other word

[TABLE]

because $\displaystyle T$ is a smooth $\displaystyle O_{F}$-group scheme, and hence the canonical group homomorphism $\displaystyle T(O_{F}/\mathfrak{p}^{r})\to T(O_{F}/\mathfrak{p}^{l^{\prime}})$ is surjective.

Let us denote by $\displaystyle\text{\rm Irr}(H(O_{F}/\mathfrak{p}^{r}),\psi_{\beta})$ the set of the equivalence classes of the irreducible complex representations $\displaystyle\sigma$ of $\displaystyle H(O_{F}/\mathfrak{p}^{r})$ such that the restriction $\displaystyle\sigma|_{G(\mathfrak{p}^{l}/\mathfrak{p}^{r})}$ contains the character $\displaystyle\psi_{\beta}$. Then Clifford’s theory says that

[TABLE]

gives a bijection of $\displaystyle\text{\rm Irr}(H(O_{F}/\mathfrak{p}),\psi_{\beta})$ onto $\displaystyle\text{\rm Irr}(G(O_{F}/\mathfrak{p}^{r}),\psi_{\beta})$. We have a bijection $\displaystyle\theta\mapsto\sigma_{\beta,\theta}$ of $\displaystyle\Theta(T(O_{F}/\mathfrak{p}^{r}),\psi_{\beta})$ onto $\displaystyle\text{\rm Irr}(H(O_{F}/\mathfrak{p}^{r}),\psi_{\beta})$ described as in the following two subsections. Then

[TABLE]

gives the required bijection of $\displaystyle\Theta(T(O_{F}/\mathfrak{p}^{r}),\psi_{\beta})$ onto $\displaystyle\text{\rm Irr}(G(O_{F}/\mathfrak{p}^{r}),\psi_{\beta})$.

3.2

Suppose that $\displaystyle r=2l$ is even. Then

[TABLE]

and $\displaystyle\theta\in\Theta(T(O_{F}/\mathfrak{p}^{r}),\psi_{\beta})$ define a character $\displaystyle\sigma_{\beta,\theta}$ of $\displaystyle H(O_{F}/\mathfrak{p}^{r})$ by

[TABLE]

for $\displaystyle g\in T(O_{F}/\mathfrak{p}^{r})$ and $\displaystyle h\in G(\mathfrak{p}^{l}/\mathfrak{p}^{r})$.

3.3

Suppose that $\displaystyle r=2l-1\geq 3$ is odd. In this case we have

[TABLE]

We will construct a representation of $\displaystyle G(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})$ by means of Schrödinger representation, and then we will extend it to a representation of $\displaystyle H(O_{F}/\mathfrak{p}^{r})$ by means of Weil representation over the finite field $\displaystyle\mathbb{F}$.

Let $\displaystyle\mathfrak{t}=\text{\rm Lie}(T)$ be the Lie algebra of the $\displaystyle O_{F}$-group scheme $\displaystyle T=Z_{G}(\beta)$. Then, for any commutative $\displaystyle O_{F}$-algebra $\displaystyle L$

[TABLE]

is the centralizer of $\displaystyle\overline{\beta}\in\mathfrak{g}(L)$ (the image of $\displaystyle\beta\in\mathfrak{g}(O_{F})$ by the canonical homomorphism $\displaystyle\mathfrak{g}(O_{F})\to\mathfrak{g}(L)$) in $\displaystyle\mathfrak{g}(L)$.

Let us denote by $\displaystyle Z(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})$ the inverse image of $\displaystyle\mathfrak{t}(\mathbb{F})$ under the surjective group homomorphism $\displaystyle G(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})\to\mathfrak{g}(\mathbb{F})$ defined by

[TABLE]

which is a normal subgroup of $\displaystyle G(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})$ containing $\displaystyle G(\mathfrak{p}^{l}/\mathfrak{p}^{r})$.

Put $\displaystyle\mathbb{V}=\mathfrak{g}(\mathbb{F})/\mathfrak{t}(\mathbb{F})$ which is a symplectic $\displaystyle\mathbb{F}$-space with respect to the symplectic form $\displaystyle\langle\dot{X},\dot{Y}\rangle_{\beta}=B([X,Y],\overline{\beta})$ ($\displaystyle X,Y\in\mathfrak{g}(\mathbb{F})$ and $\displaystyle\overline{\beta}=\beta\!\!\pmod{\mathfrak{p}}\in\mathfrak{g}(\mathbb{F})$). Sice $\displaystyle B|_{\mathfrak{t}(\mathbb{F})\times\mathfrak{f}(\mathbb{F})}$ is non-degenerate, we have $\displaystyle\mathfrak{g}(\mathbb{F})=V\oplus\mathfrak{t}(\mathbb{F})$ with

[TABLE]

Let us denote by $\displaystyle v\mapsto[v]$ the inverse $\displaystyle\mathbb{F}$-linear mapping of the canonical $\displaystyle\mathbb{F}$-linear isomorphism $\displaystyle V\,\tilde{\to}\,\mathbb{V}$.

Let $\displaystyle H_{\beta}$ be the Heisenberg group associated with the symplectic space $\displaystyle(\mathbb{V},\langle\,,\rangle_{\beta})$, that is $\displaystyle H_{\beta}=\mathbb{V}\times\mathbb{C}^{1}$ with group operation

[TABLE]

where $\displaystyle\widehat{\tau}(x\!\!\pmod{\mathfrak{p}})=\tau(\varpi^{-1}x)$ and $\displaystyle\mathbb{C}^{1}$ is the multiplicative group of complex numbers of absolute value one. Fix a polarization $\displaystyle\mathbb{V}=\mathbb{W}^{\prime}\oplus\mathbb{W}$. Then we have an irreducible unitary representation $\displaystyle(\omega_{\beta},L^{2}(\mathbb{W}^{\prime}))$ of $\displaystyle H_{\beta}$ called Schrödinger representation. Here $\displaystyle L^{2}(\mathbb{W}^{\prime})$ is the complex vector space of the complex-valued functions on $\displaystyle\mathbb{W}^{\prime}$, and

[TABLE]

for $\displaystyle f\in L^{2}(\mathbb{W}^{\prime})$ and $\displaystyle(u,s)\in H_{\beta}$ with $\displaystyle u=u_{-}+u_{+}$ ($\displaystyle u_{-}\in\mathbb{W}^{\prime},u_{+}\in\mathbb{W}$). By a general theory of Weil representation over finite field [3], there exists a group homomorphism $\displaystyle\Omega:Sp(\mathbb{V})\to GL_{\mathbb{C}}(L^{2}(\mathbb{W}^{\prime}))$ such that

[TABLE]

for all $\displaystyle\sigma\in Sp(\mathbb{V})$ and $\displaystyle(u,s)\in H_{\beta}$.

Fix additive character $\displaystyle\rho$ of $\displaystyle\mathfrak{t}(\mathbb{F})$. Then an irreducible representation $\displaystyle\omega_{\beta,\rho}$ of $\displaystyle G(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})$ on $\displaystyle L^{2}(\mathbb{W}^{\prime})$ is defined as follows. Take an element $\displaystyle h=1_{2n}+\varpi^{l-1}T\!\!\pmod{\mathfrak{p}^{r}}$ of $\displaystyle G(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})$ and hence $\displaystyle T\!\!\pmod{\mathfrak{p}}\in\mathfrak{g}(\mathbb{F})$. Put $\displaystyle T\!\!\pmod{\mathfrak{p}}=[v]+Y$ with $\displaystyle v\in\mathbb{V}$ and $\displaystyle Y\in\mathfrak{t}(\mathbb{F})$. Then $\displaystyle\omega_{\beta,\rho}(h)\in GL_{\mathbb{C}}(L^{2}(\mathbb{W}^{\prime}))$ is defined by

[TABLE]

If $\displaystyle h\in Z(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})$, then $\displaystyle\omega_{\beta,\rho}(h)$ is the homothety of

[TABLE]

where $\displaystyle\psi_{\beta,\rho}$ is a character of $\displaystyle Z(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})$ which coincides with $\displaystyle\psi_{\beta}$ on $\displaystyle G(\mathfrak{p}^{l}/\mathfrak{p}^{r})$, and all extensions of $\displaystyle\psi_{\beta}$ to $\displaystyle Z(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})$ are given in this way. Furthermore we have

[TABLE]

where

[TABLE]

Take a $\displaystyle g\in T(O_{F}/\mathfrak{p}^{r})$ and put $\displaystyle\overline{g}=g\!\!\pmod{\mathfrak{p}}\in T(\mathbb{F})$. Then $\displaystyle\sigma_{\overline{g}}\in Sp(\mathbb{V})$ is defined by $\displaystyle v\sigma_{\overline{g}}=\text{\rm Ad}(\overline{g})^{-1}X\!\!\pmod{\mathfrak{t}(\mathbb{F})}$ for $\displaystyle v=X\!\!\pmod{\mathfrak{t}(\mathbb{F})}\in\mathbb{V}$. Then an irreducible representation $\displaystyle(\sigma_{\beta,\rho},L^{2}(\mathbb{W}^{\prime}))$ of $\displaystyle H(O_{F}/\mathfrak{p}^{r})=T(O_{F}/\mathfrak{p}^{r})\cdot G(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})$ is defined by

[TABLE]

for $\displaystyle g\in T(O_{F}/\mathfrak{p}^{r})$ and $\displaystyle h\in G(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})$. Note that on

[TABLE]

$\displaystyle\sigma_{\beta,\rho}$ is the homothety by $\displaystyle\psi_{\beta,\rho}$.

Now take a $\displaystyle\theta\in\Theta(T(O_{F}/\mathfrak{p}^{r}),\psi_{\beta})$ and restrict $\displaystyle\rho$ by the condition that $\displaystyle\psi_{\beta,\rho}=\theta$ on $\displaystyle T(O_{F}/\mathfrak{p}^{r})\cap G(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})$. In other word $\displaystyle\rho$ is given by

[TABLE]

for $\displaystyle Y\in\mathfrak{t}(O_{F})$ and

[TABLE]

Note that the canonical morphism $\displaystyle\mathfrak{t}(O_{F})\to\mathfrak{t}(\mathbb{F})$ is surjective. Then finally the representation $\displaystyle\sigma_{\beta,\theta}$ of $\displaystyle H(O_{F}/\mathfrak{p}^{r})$ is defined by

[TABLE]

for $\displaystyle g\in T(O_{F}/\mathfrak{p}^{r})$ and $\displaystyle h\in G(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})$.

3.4

The explicit construction of $\displaystyle\beta_{\beta,\theta}$ being given, the dimension of it is

Proposition 3.4.1

For any $\displaystyle\theta\in\Theta(T(O_{F}/\mathfrak{p}^{r}),\psi_{\beta})$, we have

[TABLE]

[Proof] To begin with

[TABLE]

and

[TABLE]

Since $\displaystyle H(O_{F}/\mathfrak{p}^{r})=T(O_{F}/\mathfrak{p}^{r})G(\mathfrak{p}^{l^{\prime}}/\mathfrak{p}^{r})$, we have

[TABLE]

Now

[TABLE]

On the other hand we have an exact sequence

[TABLE]

because of (1), and hence $\displaystyle|T(O_{F}/\mathfrak{p}^{r})|=q^{nr}(1+q^{-n})$. $\displaystyle\blacksquare$

4 Construction of irreducible supercuspidal representations

In this section we will prove Theorem 2.2.1 except the genericity.

We will keep the notations of the preceding sections. Fix a $\displaystyle\theta\in\Theta(T(O_{F}/\mathfrak{p}^{r}),\psi_{\beta})$. Let us denote by $\displaystyle V_{\beta,\theta}$ the representation space of $\displaystyle\delta_{\beta,\theta}$.

4.1

To begin with, we will fix standard parabolic subgroups of $\displaystyle G=Sp_{2n}$.

[TABLE]

is a Borel subgroup where

[TABLE]

We use the notation ${}^{\tau}a=I_{n}\,^{t}aI_{n}$ for a square matrix $\displaystyle a$ of size $\displaystyle n$, that is the matrix transposed with respect to the second diagonal. The standard maximal parabolic subgroups $\displaystyle P_{i}=L_{i}\ltimes U_{i}$ ($\displaystyle 1\leq i\leq n$) are given by

[TABLE]

They are all smooth $\displaystyle O_{F}$-group subscheme of $\displaystyle G=Sp_{2n}$. The Lie algebra $\displaystyle\mathfrak{u}_{i}$ and $\displaystyle\mathfrak{l}_{i}$ of $\displaystyle U_{i}$ and $\displaystyle L_{i}$ are given by

[TABLE]

and

[TABLE]

respectively. The Lie algebra of $\displaystyle P_{i}$ is $\displaystyle\mathfrak{p}_{i}=\mathfrak{l}_{i}\oplus\mathfrak{u}_{i}$.

4.2

We will use the following proposition repeatedly.

Proposition 4.2.1

For all $\displaystyle 1\leq i\leq n$, the space of $\displaystyle U_{i}(\mathfrak{p}^{r-1}/\mathfrak{p}^{r})$-fixed vectors of $\displaystyle V_{\beta,\theta}$ is trivial.

[Proof] $\displaystyle G(\mathfrak{p}^{l}/\mathfrak{p}^{r})$ is a normal subgroup of $\displaystyle G(O_{F}/\mathfrak{p}^{r})$ and $\displaystyle\delta_{\beta,\theta}|_{G(\mathfrak{p}^{l}/\mathfrak{p}^{r})}$ contains the character $\displaystyle\psi_{\beta}$ of $\displaystyle G(\mathfrak{p}^{l}/\mathfrak{p}^{r})$. Then

[TABLE]

where $\displaystyle\bigoplus_{g}$ is the direct sum over the representatives $\displaystyle g$ of $\displaystyle H(O_{F}/\mathfrak{p}^{r})\backslash G(O_{F}/\mathfrak{p}^{r})$ and

[TABLE]

Note that $\displaystyle U_{i}(\mathfrak{p}^{r-1}/\mathfrak{p}^{r})\subset G(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})$. If there exists a non-trivial $\displaystyle U_{i}(\mathfrak{p}^{r-1}/\mathfrak{p}^{r})$-fixed vector in $\displaystyle V_{\beta,\beta}$, then there exists a $\displaystyle g\in G(O_{F})$ such that $\displaystyle\psi_{\text{\rm Ad}(g)\beta}(h)=1$ for all $\displaystyle h\in U_{i}(\mathfrak{p}^{r-1}/\mathfrak{p}^{r})$. This means that

[TABLE]

for all $\displaystyle X\in\mathfrak{u}_{i}(O_{F})$. This implies that $\displaystyle\text{\rm Ad}(g)\beta\!\!\pmod{\mathfrak{p}}\in\mathfrak{p}_{i}(\mathbb{F})$, and this means that the characteristic polynomial $\displaystyle\chi_{\beta}(t)\!\!\pmod{\mathfrak{p}}$ is reducible in $\displaystyle\mathbb{F}[t]$, contradicting the assumption on $\displaystyle\beta$. $\displaystyle\blacksquare$

4.3

In this subsection, we will prove the statements 1), 3) and 4) of Theorem 2.2.1 except the genericity.

To begin with

Proposition 4.3.1

$\displaystyle E=\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}$* is admissible $\displaystyle G(F)$-module.*

[Proof] For any $\displaystyle 0<m\in\mathbb{Z}$, the space of the $\displaystyle G(\mathfrak{p}^{m})$-fixed vectors is

[TABLE]

where $\displaystyle\bigoplus_{g}$ is the direct sum over the representatives $\displaystyle g$ of the double cosets $\displaystyle G(\mathfrak{p}^{m})\backslash G(F)/G(\mathfrak{p}^{m})$. Pick up one such representative $\displaystyle g$ which should be of the form $\displaystyle g=k\begin{bmatrix}a&0\\ 0&{}^{\tau}a^{-1}\end{bmatrix}$ with $\displaystyle k\in G(O_{F})$ and

[TABLE]

Assume that

[TABLE]

Then we have

[TABLE]

and hence $\displaystyle U_{i}(O_{F})\subset G(O_{F})\cap g^{-1}G(\mathfrak{p}^{m})g$. Then

[TABLE]

by Proposition 4.2.1. Now the number of the elements $\displaystyle k\begin{bmatrix}a&0\\ 0&{}^{\tau}a^{-1}\end{bmatrix}$ where $\displaystyle k$ is the representative of $\displaystyle G(\mathfrak{p}^{m})\backslash G(O_{F})$ and

[TABLE]

is finite. So $\displaystyle E^{G(\mathfrak{p}^{m})}$ is finite dimensional. $\displaystyle\blacksquare$

Next we will prove

Proposition 4.3.2

As a $\displaystyle G(O_{F})$-module

$\displaystyle\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}$* contains $\displaystyle\delta_{\beta,\theta}$ with multiplicity one,* 2. 2)

if an irreducible representation $\displaystyle\delta$ of $\displaystyle G(O_{F}/\mathfrak{p}^{r})$ is contained in $\displaystyle\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}$, then $\displaystyle\delta=\delta_{\beta,\theta}$.

[Proof] Take an irreducible representation $\displaystyle\delta$ of $\displaystyle G(O_{F}/\mathfrak{p}^{r})$ which is identified with a representation of $\displaystyle G(O_{F})$ via the canonical surjection $\displaystyle G(O_{F})\to G(O_{F}/\mathfrak{p}^{r})$. We have

[TABLE]

where $\displaystyle\bigoplus_{g}$ is the direct sum over the representatives $\displaystyle g$ of the double cosets $\displaystyle G(O_{F})\backslash G(F)/G(O_{F})$ and $\displaystyle{\delta_{\beta,\theta}}^{g}(h)=\delta_{\beta,\theta}(ghg^{-1})$. Then we have

[TABLE]

by Frobenius reciprocity. Take a representative $\displaystyle g$ of $\displaystyle G(O_{F})\backslash G(F)/G(O_{F})$ such that $\displaystyle g\not\in G(O_{F})$. Then we can assume that

[TABLE]

and $\displaystyle e_{1}\geq\cdots\geq e_{i}>0$ ($\displaystyle 1\leq i\leq n$). In this case

[TABLE]

so that

[TABLE]

Now we have $\displaystyle gU_{i}(\mathfrak{p}^{r-1})g^{-1}\subset U_{i}(\mathfrak{p}^{r})$ and $\displaystyle\delta$ factors through $\displaystyle G(O_{F})\to G(O_{F}/\mathfrak{p}^{r})$, hence

[TABLE]

because the space of $\displaystyle U_{i}(\mathfrak{p}^{r-1})$-invariant vectors in $\displaystyle V_{\beta,\theta}$ is trivial. Then we have

[TABLE]

and this complete the proof. $\displaystyle\blacksquare$

Then we have

Proposition 4.3.3

$\displaystyle\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}$* is an irreducible representation of $\displaystyle G(F)$.*

[Proof] Put $\displaystyle\delta=\delta_{\beta,\theta}$ for simplicity. Suppose that $\displaystyle E=\text{\rm ind}_{G(O_{F})}^{G(F)}\delta$ has a non-trivial $\displaystyle G(F)$-submodule $\displaystyle V$. Then we have

[TABLE]

by Frobenius reciprocity so that $\displaystyle V$ contains $\displaystyle\delta$ as a $\displaystyle G(O_{F})$-module. On the other hand $\displaystyle M=E/V$ is an admissible $\displaystyle G(F)$-module and we have

[TABLE]

(here $\displaystyle\vee$ denote the contragredient representation) so that $\displaystyle M$ contains $\displaystyle\delta$ as a $\displaystyle G(O_{F})$-module. Since $\displaystyle E$ is semisimple $\displaystyle G(O_{F})$-module, this means that $\displaystyle E$ contains $\displaystyle\delta$ at least twice, contradicting to Proposition 4.3.2. $\displaystyle\blacksquare$

Now the compactly induced representation $\displaystyle\pi_{\beta,\theta}=\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}$ is an irreducible admissible representation of $\displaystyle G(F)$, it is well known that $\displaystyle\pi_{\beta,\theta}$ is supercuspidal.

4.4

In this subsection, we will prove the statement 2) of Theorem 2.2.1. Put $\displaystyle(\delta_{\beta,\theta},V_{\beta,\theta})=(\delta,V)$ for simplicity.

Let $\displaystyle d_{G(O_{F})}(k)$ be the Haar measure on $\displaystyle G(O_{F})$ such that the volume of $\displaystyle G(O_{F})$ is one, and $\displaystyle d_{G(F)}(x)$ the Haar measure on $\displaystyle G(F)$ such that

[TABLE]

for all compactly supported complex valued continuous functions $\displaystyle\varphi$. Then the volume of $\displaystyle G(O_{F})$ with respect to $\displaystyle d_{G(F)}(x)$ is one.

Now $\displaystyle\text{\rm ind}_{G(O_{F})}^{G(F)}\delta$ consists of the smooth function $\displaystyle\varphi:G(F)\to V$ such that

$\displaystyle\varphi(xk)=\delta(k)^{-1}\varphi(x)$ for all $\displaystyle k\in G(O_{F})$ and 2. 2)

$\displaystyle\text{\rm supp}(\varphi)\pmod{G(O_{F})}$ is compact in $\displaystyle G(F)/G(O_{F})$.

On the other hand $\displaystyle\text{\rm Ind}_{G(O_{F})}^{G(F)}\delta^{\vee}$ consists of the smooth functions

[TABLE]

such that $\displaystyle\psi(xk)=\psi(x)\circ\delta(k)$ for all $\displaystyle k\in G(O_{F})$. A non-degenerate pairing

[TABLE]

is defined by

[TABLE]

where $\displaystyle\langle\,,\rangle:V\times V^{\ast}\to\mathbb{C}$ is the canonical pairing.

Take any $\displaystyle v\in V$ and $\displaystyle\alpha\in V^{\ast}$ and define $\displaystyle\varphi_{v}\in\text{\rm ind}_{G(O_{F})}^{G(F)}\delta$ and $\displaystyle\psi_{\alpha}\in\text{\rm Ind}_{G(O_{F})}^{G(F)}\delta^{\vee}$ by

[TABLE]

and by

[TABLE]

respectively. Then we have

[TABLE]

for any $\displaystyle g\in G(F)$. Now

[TABLE]

so that the formal degree of $\displaystyle\text{\rm ind}_{G(O_{F})}^{G(F)}\delta$ is

[TABLE]

by Proposition 3.4.1.

5 Genericity of supercuspidal representations

In this section we will show that the irreducible supercuspidal representation $\displaystyle\pi_{\beta,\theta}=\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}$ constructed in the preceding section is generic, that is

[TABLE]

for some generic character $\displaystyle\chi$ of $\displaystyle U(F)$.

5.1

The characters (that is the continuous group homomorphism to $\displaystyle\mathbb{C}^{1}$) of $\displaystyle U(F)$ are given by

[TABLE]

with $\displaystyle u=(u_{1},\cdots,u_{n-1},u_{n})\in F^{n}$ where $\displaystyle a_{ij}$ and $\displaystyle b_{ij}$ denote the $\displaystyle(i,j)$-entry of the square matrices $\displaystyle a$ and $\displaystyle b$ respectively. Put $\displaystyle\chi_{u}^{g}(h)=\chi_{u}(g^{-1}hg)$ for any $\displaystyle g\in G(F)$. If $\displaystyle g=\begin{bmatrix}t&0\\ 0&{}^{\tau}t^{-1}\end{bmatrix}\in T(F)$, then $\displaystyle\chi_{u}^{g}=\chi_{u^{\prime}}$ with

[TABLE]

So $\displaystyle\chi_{u}$ is generic, that is

[TABLE]

if and only if $\displaystyle u_{i}\neq 0$ ($\displaystyle 1\leq i\leq n$). In this case

[TABLE]

with suitable $\displaystyle g\in T(F)$.

5.2

Put $\displaystyle u=(2,2,\cdots,2,\varpi^{e})$. Because of Iwasawa decomposition $\displaystyle G(F)=G(O_{F})T(F)U(F)$, we have

[TABLE]

where the last $\displaystyle\prod_{g}$ is the direct product over the representatives $\displaystyle g\in T(F)$ of $\displaystyle G(O_{F})\backslash G(F)/U(F)$. We can put $\displaystyle g=\begin{bmatrix}\varpi^{m}&0\\ 0&{}^{\tau}\varpi^{-m}\end{bmatrix}$ with

[TABLE]

Then we have

Proposition 5.2.1

$\displaystyle\text{\rm Hom}_{U(O_{F})}(\delta_{\beta,\theta},\chi_{u}^{g})\neq 0$* for some $\displaystyle g$ only if $\displaystyle e\equiv r\!\!\pmod{2}$. In this case*

[TABLE]

[Proof] Put where Then $\displaystyle\chi_{u}^{g}=\chi_{u^{\prime}}$ with

[TABLE]

If $\displaystyle m_{i+1}-m_{i}\geq-(r-1)$ for some $\displaystyle 1\leq i<n$ or $\displaystyle e-2m_{n}\geq-(r-1)$, then $\displaystyle\chi_{u}^{g}(h)=1$ for all $\displaystyle h\in U_{i}(\mathfrak{p}^{r-1})$ for some $\displaystyle 1\leq i\leq n$. In other word $\displaystyle\delta_{\beta,\theta}$ contains non-trivial $\displaystyle U_{i}(\mathfrak{p}^{r-1})$-invariant vectors which contradicts to Proposition 4.2.1. So we have

[TABLE]

On the other hand $\displaystyle\delta_{\beta,\theta}$ is trivial on $\displaystyle G(\mathfrak{p}^{r})$, and hence $\displaystyle\chi_{u}^{g}(h)=1$ for all $\displaystyle h\in U(O_{F})\cap G(\mathfrak{p}^{r})$. This means that

[TABLE]

Hence $\displaystyle e+r=2m_{n}$ and $\displaystyle m_{i}-m_{i+1}=r$ ($\displaystyle 1\leq i<n$). $\displaystyle\blacksquare$

Put $\displaystyle\chi_{r}=\chi_{(2\varpi^{-r},\cdots,2\varpi^{-r},\varpi^{-r})}$ which is regarded as a character of $\displaystyle U(O_{F}/\mathfrak{p}^{r})$. On the other hand we have

[TABLE]

which is considered as a representation of $\displaystyle G(O_{F})$ via the canonical surjection $\displaystyle G(O_{F})\to G(O_{F}/\mathfrak{p}^{r})$. Then, putting $\displaystyle U=U(O_{F}/\mathfrak{p}^{r})$ and $\displaystyle H=H(O_{F}/\mathfrak{p}^{r})$ for the sake of simplicity, we have

[TABLE]

where $\displaystyle\bigoplus_{g}$ is the direct sum over the representatives of $\displaystyle U\backslash G(O_{F}/\mathfrak{p}^{r})/H$ and $\displaystyle\theta^{g}(h)=\theta(g^{-1}hg)$ ($\displaystyle h\in T(O_{F}/\mathfrak{p}^{r})$). Note that

[TABLE]

To show this fact, replacing $\displaystyle\text{\rm Ad}(g)\beta$ with $\displaystyle\beta$, it is sufficient to consider the case $\displaystyle g=1_{2n}$. In this case $\displaystyle U(O_{F}/\mathfrak{p}^{r})\cap H(O_{F}/\mathfrak{p}^{r})$ is the inverse image of $\displaystyle U(O_{F}/\mathfrak{p}^{l^{\prime}})\cap T(O_{F}/\mathfrak{p}^{l^{\prime}})$ under the canonical surjection $\displaystyle U(O_{F}/\mathfrak{p}^{r})\to U(O_{F}/\mathfrak{p}^{l^{\prime}})$. Take a

[TABLE]

Then $\displaystyle X\!\!\pmod{\mathfrak{p}^{l^{\prime}}}\in O_{K}/\varpi^{l^{\prime}}O_{K}$ is nilpotent, and hence $\displaystyle X\equiv 0\!\!\pmod{\mathfrak{p}^{l^{\prime}}}$. So $\displaystyle U(O_{F}/\mathfrak{p}^{l^{\prime}})\cap T(O_{F}/\mathfrak{p}^{l^{\prime}})=\{1\}$ and we have

[TABLE]

5.3

Suppose that $\displaystyle r=2l$ is even, and hence $\displaystyle l^{\prime}=l$. In this case the character $\displaystyle\sigma_{\beta,\theta}$ of $\displaystyle H(O_{F}/\mathfrak{p}^{r})=T(O_{F}/\mathfrak{p}^{r})G(\mathfrak{p}^{l}/\mathfrak{p}^{r})$ is defined by

[TABLE]

Then we have

[TABLE]

Hence $\displaystyle\text{\rm Hom}_{U(\mathfrak{p}^{l}/\mathfrak{p}^{r})}(\sigma_{\beta,\theta},\chi_{r})\neq 0$ if and only if $\displaystyle\psi_{\beta}=\chi_{r}$ on $\displaystyle U(\mathfrak{p}^{l}/\mathfrak{p}^{r})$.

5.4

Suppose that $\displaystyle r=2l-1$ is odd, and hence $\displaystyle l^{\prime}=l-1$. In this case the irreducible representation $\displaystyle\sigma_{\beta,\theta}$ of $\displaystyle H(O_{F}/\mathfrak{p}^{r})=T(O_{F}/\mathfrak{p}^{r})G(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})$ is defined by

[TABLE]

for $\displaystyle g\in T(O_{F}/\mathfrak{p}^{r})$ and $\displaystyle h\in G(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})$ with the notations of subsection 3.3. Hence $\displaystyle\sigma_{\beta,\theta}=\omega_{\beta,\rho}$ on $\displaystyle U(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})$. On the other hand we have (2), hence

[TABLE]

if and only if

[TABLE]

Note that

[TABLE]

because $\displaystyle 1_{2n}+\varpi^{l-1}X\!\!\pmod{\mathfrak{p}^{r}}\in U(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})\cap Z(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})$ implies that $\displaystyle X\!\!\pmod{\mathfrak{p}}\in\mathfrak{t}(\mathbb{F})$ is nilpotent, that is $\displaystyle X\!\!\pmod{\mathfrak{p}}=0$. Hence we have

[TABLE]

where $\displaystyle U=U(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})$, $\displaystyle Z=Z(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})$ and $\displaystyle\bigoplus_{g}$ is the direct sum over the representatives $\displaystyle g$ of the double cosets $\displaystyle U\backslash G(\mathfrak{p}^{l-1}/\mathfrak{p}^{r})/Z$. So (3) is equivalent to

[TABLE]

5.5

The combination of the results of subsections 5.2, 5.3 and 5.4 implies that

[TABLE]

with $\displaystyle u=(2,2,\cdots,2,\varpi^{e})$ ($\displaystyle e\equiv r\!\!\pmod{2}$) if and only if

[TABLE]

for some $\displaystyle g\in G(O_{F})$. We will show that this is the case by proving the following proposition.

Proposition 5.5.1

There exists a $\displaystyle g\in G(O_{F})$ such that $\displaystyle\text{\rm Ad}(g)\beta=\begin{bmatrix}A&\ast\\ C&-^{\tau}A\end{bmatrix}$ with

[TABLE]

[Proof] $\displaystyle K$ is the splitting field of the characteristic polynomial $\displaystyle\chi_{\beta}(t)$ of $\displaystyle\beta\in\mathfrak{g}(O_{F})\subset\mathfrak{gl}_{2n}(O_{F})$, and we can put

[TABLE]

Let $\displaystyle\langle u,v\rangle=uJ_{n}^{t}v$ be the symplectic form on $\displaystyle K^{2n}$ and $\displaystyle V_{\lambda}\subset K^{2n}$ the eigen space of $\displaystyle\beta$ with eigen value $\displaystyle\lambda\in K$. Then $\displaystyle\langle V_{\lambda},V_{\mu}\rangle\neq 0$ only if $\displaystyle\lambda+\mu=0$, and we have

[TABLE]

This means that there exists a symplectic $\displaystyle K$-basis $\displaystyle\{u_{1},\cdots,u_{n},v_{n},\cdots,v_{1}\}$ of $\displaystyle K^{2n}$ such that the representation matrix of $\displaystyle\beta$ with respect to the basis is

[TABLE]

In other word, there exists a $\displaystyle g\in G(K)$ such that $\displaystyle g\beta g^{-1}=\begin{bmatrix}\Lambda&0\\ 0&-^{\tau}\Lambda\end{bmatrix}$.

We can choose

[TABLE]

such that $\displaystyle\chi_{\beta^{\prime}}(t)=\chi_{\beta}(t)$ with

[TABLE]

and the entries of $\displaystyle\beta_{0}$ are zero except the elements of the first row and of the last column. Then there exists a $\displaystyle g\in G(K)$ such that $\displaystyle g\beta^{\prime}g^{-1}=\beta$. Because $\displaystyle T=Z_{G}(\beta)$ splits over $\displaystyle K$, the first Galois cohomology group $\displaystyle H^{1}(\text{\rm Gal}(K/F),T(K))$ is trivial. So there exists a $\displaystyle g\in G(F)$ such that $\displaystyle g\beta^{\prime}g^{-1}=\beta$. Due to Iwasawa decomposition $\displaystyle G(F)=G(O_{F})T(F)U(F)$, we can put $\displaystyle g=ktu$ with $\displaystyle k\in G(O_{F})$, $\displaystyle u\in U(F)$ and

[TABLE]

Then $\displaystyle tu\beta^{\prime}u^{-1}t^{-1}=k^{-1}\beta k\in\mathfrak{g}(O_{F})$ and

[TABLE]

with

[TABLE]

This implies that $\displaystyle m_{1}\leq m_{2}\leq\cdots\leq m_{n}\leq 0$. On the other hand

[TABLE]

is irreducible, we have $\displaystyle m_{1}=m_{2}=\cdots=m_{n}=0$. $\displaystyle\blacksquare$

5.6

We have proved the followings;

the irreducible supercuspidal representations $\displaystyle\pi_{\beta,\theta}=\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}$ are generic, 2. 2)

$\displaystyle\text{\rm Hom}_{G(F)}\left(\pi_{\beta,\theta},\text{\rm Ind}_{U(F)}^{G(F)}\chi_{u}\right)\neq 0$ with $\displaystyle u=(2,2,\cdots,2,\varpi^{e})$ if and only if $\displaystyle e\equiv r\!\!\pmod{2}$.