Geometric construction of Heisenberg-Weil representations for finite unitary groups and Howe correspondences

TL;DR

This paper presents a geometric approach to constructing Heisenberg-Weil representations for finite unitary groups and realizes the Howe correspondence geometrically, demonstrating unipotency preservation.

Contribution

It introduces a novel geometric construction of Heisenberg-Weil representations and provides a geometric realization of the Howe correspondence over finite fields.

Findings

Unipotency is preserved under the Howe correspondence.

Provides a geometric construction using étale cohomology.

Realizes Howe correspondence for $( ext{Sp}_{2n},O_2^-)$ over any finite field.

Abstract

We give a geometric construction of the Heisenberg-Weil representation of a finite unitary group by the middle \'{e}tale cohomology of an algebraic variety over a finite field, whose rational points give a unitary Heisenberg group. Using also a Frobenius action, we give a geometric realization of the Howe correspondence for $(\mathit{Sp}_{2n},O_2^-)$ over any finite field including characteristic two. As an application, we show that unipotency is preserved under the Howe correspondence.

Equations

$$H_{\mathrm{c}}^n(X_{n,{\overline{\mathbb{F}}}_q},{\overline{\mathbb{Q}}}_{\ell })[\psi ]\simeq {\rho }_{\mathrm{HU}(h_n),\psi }$$

$${\mathrm{Sp}}_{2n}({\mathbb{F}}_q)\times O_2^{-}({\mathbb{F}}_q)\to U_{2n}({\mathbb{F}}_q)⋊\mathrm{Gal}({\mathbb{F}}_{q^2}/{\mathbb{F}}_q).$$

$$H(V,h)=\{(v,a)\in V\times {\mathbb{F}}_{q^2}\mid a+\epsilon a^q=h(v,v)\}.$$

$$(v,a)(v^{\prime },a^{\prime })=(v+v^{\prime },a+a^{\prime }+h(v,v^{\prime })).$$

$$\mathrm{HU}(h)=H(V,h)⋊U(V,h).$$

$$\xi h:V\times V\to {\mathbb{F}}_{q^2};(v,v^{\prime })\mapsto \xi h(v,v^{\prime })$$

$${\omega }_{U(V,h)}={\rho }_{\mathrm{HU}(h),\psi }{|}_{U(V,h)},$$

$${\mu }_{q+1}=\{\zeta \in {\mathbb{F}}_{q^2}\mid {\zeta }^{q+1}=1\}.$$

$$H_{\mathrm{c}}^1(C_{{\overline{\mathbb{F}}}_q},{\overline{\mathbb{Q}}}_{\ell })[\psi ]\simeq \underset{\chi \in {\mu }_{q+1}^{\vee }\setminus \{1\}}{⨁}{\overline{\mathbb{Q}}}_{\ell }(\chi )$$

$$H_{\mathrm{c}}^1(C_{{\overline{\mathbb{F}}}_q},{\overline{\mathbb{Q}}}_{\ell })[\psi ]\simeq H_{\mathrm{c}}^1({\mathbb{A}}^1,{\mathcal{L}}_{\psi }(x^{q+1}))\simeq \underset{\chi \in {\mu }_{q+1}^{\vee }\setminus \{1\}}{⨁}{H}_{\mathrm{c}}^1({\mathbb{G}}_{\mathrm{m}},{\mathcal{L}}_{\psi }\otimes {\mathcal{K}}_{{\mathbb{G}}_{\mathrm{m}},\chi })$$

$$H_{\mathrm{c}}^i({\mathbb{G}}_{\mathrm{m}},{\mathcal{L}}_{\psi }\otimes {\mathcal{K}}_{{\mathbb{G}}_{\mathrm{m}},\chi })=0\text{for i=1},\dim H_{\mathrm{c}}^1({\mathbb{G}}_{\mathrm{m}},{\mathcal{L}}_{\psi }\otimes {\mathcal{K}}_{{\mathbb{G}}_{\mathrm{m}},\chi })=1$$

$$h_n:{\mathbb{F}}_{q^2}^n\times {\mathbb{F}}_{q^2}^n\to {\mathbb{F}}_{q^2};((v_k),(v_k^{\prime }))\mapsto \sum _{k=1}^n{v}_k^q{v}_k^{\prime }.$$

$$H(h_n)=H({\mathbb{F}}_{q^2}^n,h_n),U(h_n)=U({\mathbb{F}}_{q^2}^n,h_n).$$

$$z^q+z=\sum _{k=1}^n{x}_k^{q+1}$$

$$X_{n}\to X_{n};\ ((x_{k}),z)\mapsto\Bigl{(}(x_{k}+v_{k}),z+a+\sum_{k=1}^{n}v_{k}^{q}x_{k}\Bigr{)}\quad\textrm{for $((v_{k}),a)\in\operatorname{H}(h_{n})$}$$

$$X_n\to X_n;(x,z)\mapsto (gx,z)\text{for g∈U(hn)},$$

$$\pi :{\mathbb{A}}_{{\mathbb{F}}_q}^n\to {\mathbb{A}}_{{\mathbb{F}}_q}^1;(x_i{)}_{1\le i\le n}\mapsto \sum _{i=1}^n{x}_i^{q+1}.$$

$$X_n\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H_{\mathrm{c}}^i(X_{n,{\overline{\mathbb{F}}}_q},\Lambda )[\psi ]\simeq H_{\mathrm{c}}^i({\mathbb{A}}^n,{\pi }^{\ast }{\mathcal{L}}_{\psi })$$

$$\mathrm{Tr}(\iota ;H_{\mathrm{c}}^2(X_{2,{\overline{\mathbb{F}}}_q},{\overline{\mathbb{Q}}}_{\ell })[\psi ])=-2.$$

$${\mathrm{pr}}_i:{\mathbb{A}}_{{\mathbb{F}}_q}^2\to {\mathbb{A}}_{{\mathbb{F}}_q}^1;(x_1,x_2)\mapsto x_i$$

$$\pi^{*}\mathscr{L}_{\psi}\simeq\mathrm{pr}_{1}^{*}\bigl{(}\mathscr{L}_{\psi}(x^{q+1})\bigr{)}\otimes\mathrm{pr}_{2}^{*}\bigl{(}\mathscr{L}_{\psi}(x^{q+1})\bigr{)}$$

$$H_{\mathrm{c}}^2(X_{2,{\overline{\mathbb{F}}}_q},{\overline{\mathbb{Q}}}_{\ell })[\psi ]\simeq H_{\mathrm{c}}^1(C_{{\overline{\mathbb{F}}}_q},{\overline{\mathbb{Q}}}_{\ell })[\psi ]\otimes H_{\mathrm{c}}^1(C_{{\overline{\mathbb{F}}}_q},{\overline{\mathbb{Q}}}_{\ell })[\psi ]$$

$$H_{\mathrm{c}}^n(X_{n,{\overline{\mathbb{F}}}_q},{\overline{\mathbb{Q}}}_{\ell })[\psi ]\simeq {\rho }_{\mathrm{HU}(h_n),\psi }$$

$${\chi }_2:\mathbb{Z}/3\mathbb{Z}\times {\mathfrak{S}}_3\stackrel{{\mathrm{pr}}_2}{\to }{\mathfrak{S}}_3\stackrel{\mathrm{sgn}}{\to }{\overline{\mathbb{Q}}}_{\ell }^{\times }$$

$$H_{\mathrm{c}}^n(X_{n,{\overline{\mathbb{F}}}_q},{\overline{\mathbb{Q}}}_{\ell })[\psi ]\simeq \{\begin{array}{ll}{\rho }_{\mathrm{HU}(h_n),\psi }\otimes (\chi \circ \det )& \text{if (n,q)=(2,2)},\\ {\rho }_{\mathrm{HU}(h_n),\psi }\otimes (\chi \circ \det )\otimes {\chi }_2^j& \text{for some 0≤j≤1 if (n,q)=(2,2)}.\end{array}$$

$$\mathrm{Tr}(g;H_{\mathrm{c}}^n(X_{n,{\overline{\mathbb{F}}}_q},{\overline{\mathbb{Q}}}_{\ell })[\psi ])=(-1{)}^n.$$

$$(-1{)}^n\chi ({\zeta }_1\cdots {\zeta }_n)=\mathrm{Tr}{\rho }_{\mathrm{HU}(h_n),\psi }(g)\chi (\det (g))=\mathrm{Tr}(g;H_{\mathrm{c}}^n(X_{n,{\overline{\mathbb{F}}}_q},{\overline{\mathbb{Q}}}_{\ell })[\psi ])=(-1{)}^n.$$

$$-2(-1{)}^j=\mathrm{Tr}{\rho }_{\mathrm{HU}(h_n),\psi }(\iota )\chi (\det (\iota )){\chi }_2(\iota {)}^j=\mathrm{Tr}(\iota ;H_{\mathrm{c}}^n(X_{n,{\overline{\mathbb{F}}}_q},{\overline{\mathbb{Q}}}_{\ell })[\psi ])=-2,$$

$$(H_{\mathrm{c}}^i(X_{n,{\overline{\mathbb{F}}}_q},{\overline{\mathbb{Z}}}_{\ell })[\psi ]){\otimes }_{{\overline{\mathbb{Z}}}_{\ell }}{\overline{\mathbb{F}}}_{\ell }\stackrel{\sim }{\to }{H}_{\mathrm{c}}^i(X_{n,{\overline{\mathbb{F}}}_q},{\overline{\mathbb{F}}}_{\ell })[\overline{\psi }]$$

Full text

††footnotetext: 2020 Mathematics Subject Classification. Primary: 20C33; Secondary: 11F27.††footnotetext: Keywords: Weil representation, Howe correspondence, Lusztig induction, finite unitary group

1 Introduction

For a reductive dual pair $(G,G^{\prime})$ over a field (cf. [How79, §5]), by regarding the Weil representation as a representation of $G\times G^{\prime}$ and decomposing it into irreducible components, we have a correspondence from irreducible representations of $G$ to representations of $G^{\prime}$, which is called the Howe correspondence for $(G,G^{\prime})$. Let $q$ be a power of a prime number $p$. If $q$ is odd, Weil representations of symplectic groups over $\mathbb{F}_{q}$ are studied in [Sai72] and [How73] after [Wei64]. Weil representations of general linear groups over $\mathbb{F}_{q}$ and unitary groups for the extension $\mathbb{F}_{q^{2}}/\mathbb{F}_{q}$ are constructed in [Gér77] for any $q$ (cf. [Leh74] in the unitary case).

In this paper, we give a geometric construction of Weil representations of unitary groups. A finite unitary group $\operatorname{U}_{n}(\mathbb{F}_{q})$ of degree $n$ is regarded as a subgroup of an automorphism group of a certain Heisenberg group $\operatorname{H}(h_{n})$ determined by a hermitian form $h_{n}$ on $\mathbb{F}_{q^{2}}^{n}$, which we call a unitary Heisenberg group (cf. [Gér77, Lemma 3.3] and §2.1). For a non-trivial character $\psi$ of the center of $\operatorname{H}(h_{n})$, there exists a unique irreducible representation $\rho_{\operatorname{H}(h_{n}),\psi}$ of $\operatorname{H}(h_{n})$ which contains $\psi$. In [Gér77, Theorem 3.3], an irreducible representation $\rho_{\operatorname{HU}(h_{n}),\psi}$ of the semidirect product $\operatorname{H}(h_{n})\rtimes\operatorname{U}_{n}(\mathbb{F}_{q})$ extending $\rho_{\operatorname{H}(h_{n}),\psi}$ is constructed, which we call the Heisenberg–Weil representation. The Weil representation is defined to be the restriction of $\rho_{\operatorname{HU}(h_{n}),\psi}$ to the unitary group.

Our geometric construction is very simple. We regard the unitary Heisenberg group as the set of rational points of an algebraic variety, which has a natural action of $\operatorname{H}(h_{n})\rtimes\operatorname{U}_{n}(\mathbb{F}_{q})$. This algebraic variety is isomorphic to the affine smooth variety defined by $a^{q}+a=\sum_{i=1}^{n}x_{i}^{q+1}$ in $\mathbb{A}_{\mathbb{F}_{q}}^{n+1}$, which we denote by $X_{n}$. Let $\ell\neq p$ be a prime number. We show the following:

Theorem**.**

We have an isomorphism

[TABLE]

as representations of $\operatorname{H}(h_{n})\rtimes\operatorname{U}_{n}(\mathbb{F}_{q})$.

We give also a branching formula for the Weil representation of a unitary group in Proposition 6.5. Thanks to the geometric construction, the branching formula is a simple consequence of the Künneth formula.

By taking the mod $\ell$ cohomology, we can obtain a modular Heisenberg–Weil representation of a unitary group without ambiguity of semi-simplifications. Another geometric approach for the Weil representations of symplectic groups is given in [GH07], which is quite different from ours.

For a unitary group of even degree, we can give a geometric construction of the Weil representation using a rational form of the variety $X_{2n}$, which is denoted by $X_{2n}^{\prime}$. Using the Frobenius action coming from the rationality of $X_{2n}^{\prime}$, we can construct a representation of $\operatorname{U}_{2n}(\mathbb{F}_{q})\rtimes\operatorname{Gal}(\mathbb{F}_{q^{2}}/\mathbb{F}_{q})$ as the middle ${\ell}$-adic cohomology of the variety. We write $W_{n}$ for the obtained $\operatorname{U}_{2n}(\mathbb{F}_{q})\rtimes\operatorname{Gal}(\mathbb{F}_{q^{2}}/\mathbb{F}_{q})$-representation. This Frobenius action is important for an application to the Howe correspondence as we explain below.

In characteristic two, a general formulation of the Howe correspondence for symplectic-orthogonal case is not known (cf. [Tie10, §1]). However, there is a trial to construct such a correspondence in [GT04, §7], where they consider a dual pair $(\operatorname{Sp}_{2n},\operatorname{O}_{2}^{\pm})$ and construct a representation of the product group $\operatorname{Sp}_{2n}(\mathbb{F}_{q})\times\operatorname{O}_{2}^{\pm}(\mathbb{F}_{q})$. In this paper, we give a geometric construction of the Howe correspondence for the reductive dual pair $(\operatorname{Sp}_{2n},\operatorname{O}_{2}^{-})$ as follows. The group $\operatorname{O}_{2}^{-}(\mathbb{F}_{q})$ is isomorphic to the dihedral group $D_{2(q+1)}$. Hence, we can identify $\operatorname{O}_{2}^{-}(\mathbb{F}_{q})$ with $\operatorname{U}_{1}(\mathbb{F}_{q})\rtimes\operatorname{Gal}(\mathbb{F}_{q^{2}}/\mathbb{F}_{q})$. We have natural homomorphisms $\operatorname{Sp}_{2n}(\mathbb{F}_{q})\hookrightarrow\operatorname{U}_{2n}(\mathbb{F}_{q})$ and $\operatorname{U}_{2n}(\mathbb{F}_{q})\times\operatorname{U}_{1}(\mathbb{F}_{q})\to\operatorname{U}_{2n}(\mathbb{F}_{q})$ (cf. [Sri79, §1]). Hence, we have a homomorphism

[TABLE]

Inflating $W_{n}$ under this homomorphism, we obtain a representation of $\operatorname{Sp}_{2n}(\mathbb{F}_{q})\times\operatorname{O}_{2}^{-}(\mathbb{F}_{q})$. Using this representation, we define the Howe correspondence for $(\operatorname{Sp}_{2n},\operatorname{O}_{2}^{-})$. We note that the Frobenius action coming from the rationality of $X_{2n}^{\prime}$ plays an important role in the construction of the Howe correspondence. Since the construction is geometric, we can relate the representation of $\operatorname{Sp}_{2n}(\mathbb{F}_{q})\times\operatorname{O}_{2}^{-}(\mathbb{F}_{q})$ with a Lusztig induction in a geometric way. Using the relation, we can show that the Howe correspondence preserves unipotency for any $q$. The preservation of the unipotency is proved in [AM93, Theorem 3.5] for symmetric and even-orthogonal pairs if $p$ is odd and $q$ is large enough.

In Section 2, we recall the Heisenberg–Weil representation for a unitary group and give a geometric realization of the representation in the $\ell$-adic cohomology of the variety $X_{n}$. In Section 3, we give a geometric realization the Heisenberg–Weil representation for a unitary group of even degree using another coordinate. We study also a Frobenius action, which we need later. In Section 4, we relate the cohomology of the variety $X_{n}$ with the cohomology of a Fermat variety or its complement in a projective space. In Section 5, we recall some facts on the Lusztig induction. In Section 6, we relate the Weil representation of a unitary group with a Lusztig induction. We give also a branching formula for the Weil representation of a unitary group. In Section 7, we construct a representation of $\operatorname{Sp}_{2n}(\mathbb{F}_{q})\times\operatorname{O}_{2}^{-}(\mathbb{F}_{q})$ and define the Howe correspondence for $(\operatorname{Sp}_{2n},\operatorname{O}_{2}^{-})$. We show that this Howe correspondence is compatible with the usual one, which is defined if $p\neq 2$, up to an explicit sign. We relate the representation with a Lusztig induction and show the preservation of unipotency under the Howe correspondence.

In [IT19], we construct Shintani lifts for Weil representations of unitary groups as an application of the geometric construction in this paper. In a subsequent paper [IT20], we study mod $\ell$ cohomology and a mod $\ell$ Howe correspondence using results in this paper.

Acknowledgements

The authors would like to thank a referee for helpful comments and suggestions. This work was supported by JSPS KAKENHI Grant Numbers 17K18722, 20K03529, 21H00973.

Notation

For a scheme $X$ over a field $F$ and a field extension $F^{\prime}/F$, let $X_{F^{\prime}}$ denote the base change of $X$ to $F^{\prime}$.

For a finite group $G$, let $1_{G}$ denote the trivial representation of $G$ over a field. We simply write $1$ for $1_{G}$ when $G$ is clear from the context. For a finite group $G$, an irreducible $G$-representation $\lambda$ and a finite-dimensional $G$-representation $\rho$, let $\rho[\lambda]$ denote the $\lambda$-isotypic part of $\rho$.

2 Heisenberg–Weil representation for unitary group

2.1 Unitary Heisenberg group

We recall the unitary Heisenberg group. Let $q$ be a power of a prime number $p$. Let $V$ be a finite-dimensional $\mathbb{F}_{q^{2}}$-vector space with a nondegenerate $\varepsilon$-hermitian form $h$, where $\varepsilon\in\{\pm 1\}$. We put

[TABLE]

We regard $\operatorname{H}(V,h)$ as a group with multiplication

[TABLE]

We put $\mathbb{F}_{q,\varepsilon}=\{a\in\mathbb{F}_{q^{2}}\mid a+\varepsilon a^{q}=0\}$. We sometimes abbreviate $\pm 1$ as $\pm$. The center $Z$ of $\operatorname{H}(V,h)$ equals $\{0\}\times\mathbb{F}_{q,\varepsilon}$. The quotient $\operatorname{H}(V,h)/Z$ is identified with $V$. Hence, $\operatorname{H}(V,h)$ is a Heisenberg group.

Let $\ell\neq p$ be a prime number. For each non-trivial character $\psi$ of $Z$ over $\overline{\mathbb{Q}}_{\ell}$, there exists a unique irreducible representation of $\operatorname{H}(V,h)$ whose restriction to $Z$ contains $\psi$ by the Stone–von Neumann theorem. We denote by $\rho_{\operatorname{H}(V,h),\psi}$ the unique irreducible representation of $\operatorname{H}(V,h)$ containing $\psi$. The dimension of $\rho_{\operatorname{H}(V,h),\psi}$ equals the square root of the index $[\operatorname{H}(V,h):Z]$.

Let $\operatorname{U}(V,h)$ denote the isometry group of $(V,h)$. Then, $\operatorname{U}(V,h)$ acts on $\operatorname{H}(V,h)$ by $(v,a)\mapsto(gv,a)$ for $g\in\operatorname{U}(V,h)$ and $(v,a)\in\operatorname{H}(V,h)$. We put

[TABLE]

Remark 2.1**.**

Assume that $h$ is hermitian. We take an element $\xi\in\mathbb{F}_{q^{2}}$ such that $\xi^{q-1}=-1$. Then

[TABLE]

is a skew-hermitian form. We have an isomorphism $\operatorname{H}(V,h)\xrightarrow{\sim}\operatorname{H}(V,\xi h);\ (v,a)\mapsto(v,\xi a)$ and an identity $\operatorname{U}(V,h)=\operatorname{U}(V,\xi h)$.

Lemma 2.2**.**

There is the unique irreducible representation $\rho_{\operatorname{HU}(h),\psi}$ of $\operatorname{HU}(h)$ which is characterized by

• •

$(\rho_{\operatorname{HU}(h),\psi})|_{\operatorname{H}(V,h)}\simeq\rho_{\operatorname{H}(V,h),\psi}$* as $\operatorname{H}(V,h)$-representations and*

• •

$\operatorname{Tr}\rho_{\operatorname{HU}(h),\psi}(g)=(-1)^{n}(-q)^{N(g)}$* for $g\in\operatorname{U}(V,h)$, where we put $n=\dim_{\mathbb{F}_{q^{2}}}V$ and $N(g)=\dim_{\mathbb{F}_{q^{2}}}\operatorname{Ker}(g-1)$.*

Proof.

This follows from [Gér77, Theorem 3.3 and Theorem 4.9.2] and Remark 2.1. ∎

We call the representation $\rho_{\operatorname{HU}(h),\psi}$ in Lemma 2.2 the Heisenberg–Weil representation of $\operatorname{HU}(h)$ for $\psi$ (cf. [Leh74, Proposition 3.1]). We put

[TABLE]

which is independent of $\psi$ by Lemma 2.2. The representation $\omega_{\operatorname{U}(V,h)}$ is called the Weil representation of $\operatorname{U}(V,h)$.

2.2 Cohomology of a curve

For a finite abelian group $A$, we simply write $A^{\vee}$ for the character group $\mathrm{Hom}(A,\overline{\mathbb{Q}}_{\ell}^{\times})$.

Let $\Lambda\in\{\overline{\mathbb{Q}}_{\ell},\overline{\mathbb{F}}_{\ell}\}$. For a separated and of finite type scheme $Y$ over $\mathbb{F}_{q}$ which admits a left action of a finite group $G$, let $G$ act on $H_{\rm c}^{i}(Y_{\overline{\mathbb{F}}_{q}},\Lambda)$ as $(g^{\ast})^{-1}$ for $g\in G$. We write $\mathbb{A}^{i}$ for an $i$-dimensional affine space over $\overline{\mathbb{F}}_{q}$. We write $\mathbb{G}_{\mathrm{m}}$ for $\operatorname{Spec}\overline{\mathbb{F}}_{q}[t^{\pm 1}]$.

For $\psi\in\operatorname{Hom}(\mathbb{F}_{q,\varepsilon},\Lambda^{\times})$, let $\mathscr{L}_{\psi}$ denote the rank one sheaf on $\mathbb{A}_{\mathbb{F}_{q}}^{1}$ obtained as the pushforward via $\psi^{-1}\in\operatorname{Hom}(\mathbb{F}_{q,\varepsilon},\Lambda^{\times})$ of the $\mathbb{F}_{q,\varepsilon}$-torsor over $\mathbb{A}_{\mathbb{F}_{q}}^{1}=\operatorname{Spec}\mathbb{F}_{q}[t]$ defined by $z^{q}+\varepsilon z=t$ (cf. [Del77, Sommes trig., Définition 1.7]). For a variety $X$ over $\mathbb{F}_{q}$ and a regular function $f\colon X\to\mathbb{A}_{\mathbb{F}_{q}}^{1}$, let $\mathscr{L}_{\psi}(f)$ denote the pull-back of $\mathscr{L}_{\psi}$ by $f$. We put

[TABLE]

For $\chi\in\operatorname{Hom}(\mu_{q+1},\Lambda^{\times})$, let $\mathscr{K}_{\mathbb{G}_{\mathrm{m}},\chi}$ denote the Kummer sheaf on $\mathbb{G}_{\mathrm{m},\mathbb{F}_{q^{2}}}$ obtained as the pushforward via $\chi^{-1}$ of the $\mu_{q+1}$-torsor over $\mathbb{G}_{\mathrm{m},\mathbb{F}_{q^{2}}}=\operatorname{Spec}\mathbb{F}_{q^{2}}[t^{\pm 1}]$ defined by $y^{q+1}=t$.

Let $C$ be the affine smooth curve defined by $z^{q}+z=x^{q+1}$ over $\mathbb{F}_{q^{2}}$. The group $\mathbb{F}_{q,+}$ acts on $C$ by $z\mapsto z+a$ for $a\in\mathbb{F}_{q,+}$.

Let $\psi\in\operatorname{Hom}(\mathbb{F}_{q,+},\Lambda^{\times})\setminus\{1\}$ in the rest of this section. The first claim of the following lemma is a variant of [IT17, Lemma 7.1]. The cohomology of a variant is studied also in [BR06, §3.3.1].

Lemma 2.3**.**

Assume that $\Lambda=\overline{\mathbb{Q}}_{\ell}$.

1. (1)

We have $H_{\rm c}^{i}(C_{\overline{\mathbb{F}}_{q}},\overline{\mathbb{Q}}_{\ell})[\psi]=0$ for $i\neq 1$ and an isomorphism

[TABLE]

as $\mu_{q+1}$-representations, where $\overline{\mathbb{Q}}_{\ell}(\chi)$ denotes $\overline{\mathbb{Q}}_{\ell}$ with action of $\mu_{q+1}$ by $\chi$. In particular, we have $\dim H_{\rm c}^{1}(C_{\overline{\mathbb{F}}_{q}},\overline{\mathbb{Q}}_{\ell})[\psi]=q$. 2. (2)

We regard $\psi$ as an element of $\mathrm{Hom}(\mathbb{F}_{q,+},\overline{\mathbb{Z}}_{\ell}^{\times})$ via the factorization through $\overline{\mathbb{Z}}_{\ell}^{\times}\subset\overline{\mathbb{Q}}_{\ell}^{\times}$. The $\overline{\mathbb{Z}}_{\ell}$-module $H_{\rm c}^{1}(C_{\overline{\mathbb{F}}_{q}},\overline{\mathbb{Z}}_{\ell})[\psi]$ is free and $H^{i}_{\rm c}(C_{\overline{\mathbb{F}}_{q}},\overline{\mathbb{Z}}_{\ell})[\psi]=0$ for $i\neq 1$. 3. (3)

Let $\overline{\psi}$ denote the composite of $\psi$ and the reduction map $\overline{\mathbb{Z}}_{\ell}^{\times}\to\overline{\mathbb{F}}_{\ell}^{\times}$. Then we have an isomorphism $(H_{\rm c}^{i}(C_{\overline{\mathbb{F}}_{q}},\overline{\mathbb{Z}}_{\ell})[\psi])\otimes_{\overline{\mathbb{Z}}_{\ell}}\overline{\mathbb{F}}_{\ell}\xrightarrow{\sim}H_{\rm c}^{i}(C_{\overline{\mathbb{F}}_{q}},\overline{\mathbb{F}}_{\ell})[\overline{\psi}]$ for $i\geq 0$.

Proof.

We have isomorphisms

[TABLE]

as in [IT15, 2.2]. We have

[TABLE]

by [Del77, Sommes trig., Proposition 4.2]. The first assertion follows from (2.3) and (2.4). The second assertion follows from a well-known fact that the compactly supported $\mathcal{O}$-cohomology of a smooth affine curve over $\overline{\mathbb{F}}_{q}$ is torsion free. The third assertion follows from the second assertion. ∎

2.3 Geometric construction

Let $n$ be a positive integer. We write an element $v\in\mathbb{F}_{q}^{n}$ as $v=(v_{k})$. We consider the nondegenerate hermitian form on $\mathbb{F}_{q^{2}}^{n}$ defined by

[TABLE]

We put

[TABLE]

Let $X_{n}$ be the affine smooth variety over $\mathbb{F}_{q^{2}}$ defined by

[TABLE]

in $\mathbb{A}_{\mathbb{F}_{q^{2}}}^{n+1}=\operatorname{Spec}\mathbb{F}_{q^{2}}[x_{1},\ldots,x_{n},z]$. Let $\operatorname{H}(h_{n})$ act on $X_{n}$ by

[TABLE]

and $\operatorname{U}(h_{n})$ act on $X_{n}$ by

[TABLE]

where we regard $x=(x_{k})$ as a column vector. The variety $X_{n}$ admits an action of $\operatorname{HU}(h_{n})$. Let $\mathbb{F}_{q,+}$ act on $X_{n}$ by $z\mapsto z+a$ for $a\in\mathbb{F}_{q,+}$. The cohomology of a variant of $X_{n}$ is studied in [Dud09, Lemma 3.6].

We consider the morphism

[TABLE]

Since we have a cartesian diagram

[TABLE]

we obtain an isomorphism

[TABLE]

for $i\geq 0$ by using the proper base change theorem and taking $\psi$-parts.

We prepare a lemma to treat the case where $(n,q)=(2,2)$. We put $\iota=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}\in\operatorname{U}(h_{2})$.

Lemma 2.4**.**

Assume that $\Lambda=\overline{\mathbb{Q}}_{\ell}$ and $q=2$. We have

[TABLE]

Proof.

Let $e_{1},e_{2}$ be a basis of $H_{\rm c}^{1}(C_{\overline{\mathbb{F}}_{q}},\overline{\mathbb{Q}}_{\ell})[\psi]$. Using the morphism

[TABLE]

for $1\leq i\leq 2$, we have an isomorphism

[TABLE]

of sheaves on $\mathbb{A}_{\mathbb{F}_{q}}^{2}$. Hence we have an isomorphism

[TABLE]

given by (2.7), the Künneth formula in [Del77, Sommes trig., (2.4.1)*] and the first isomorphism in (2.3). Under the isomorphism (2.8), the action of $\iota$ is $e_{i}\otimes e_{j}\mapsto-e_{j}\otimes e_{i}$ for $1\leq i,j\leq 2$, where the minus sign appears because of the anti-commutativity of cup products. Hence we have the claim. ∎

Theorem 2.5**.**

Assume that $\Lambda=\overline{\mathbb{Q}}_{\ell}$. We have $H_{\rm c}^{i}(X_{n,\overline{\mathbb{F}}_{q}},\overline{\mathbb{Q}}_{\ell})[\psi]=0$ for $i\neq n$ and $\dim H_{\rm c}^{n}(X_{n,\overline{\mathbb{F}}_{q}},\overline{\mathbb{Q}}_{\ell})[\psi]=q^{n}$. Further, we have an isomorphism

[TABLE]

as $\operatorname{HU}(h_{n})$-representations.

Proof.

The first assertion follows from (2.7), the Künneth formula, Lemma 2.3 (1) and the first isomorphism in (2.3) in the same way as the proof of Lemma 2.4. By the first assertion, $H_{\rm c}^{n}(X_{n,\overline{\mathbb{F}}_{q}},\overline{\mathbb{Q}}_{\ell})[\psi]$ is isomorphic to $\rho_{\operatorname{H}(h_{n}),\psi}$ as $\operatorname{H}(h_{n})$-representations by the Stone–von Neumann theorem. We write $\det$ for the composite $\operatorname{HU}(h_{n})\xrightarrow{\mathrm{pr}}\operatorname{U}(h_{n})\xrightarrow{\det}\mu_{q+1}$.

Assume that $q$ is odd. By [Gér77, (1) in the proof of Theorem 3.3], the finite special unitary group $\operatorname{SU}_{n}(\mathbb{F}_{q})$ is perfect except for $(n,q)=(2,3)$. Hence, if $(n,q)\neq(2,3)$, any character of $\operatorname{U}(h_{n})$ factors through $\det$. Even if $(n,q)=(2,3)$, any character of $\operatorname{U}(h_{n})$ does by [Enn63, the table in p.28].

Assume that $q$ is even. By [Gér77, (8) in the proof of Theorem 3.3], any character of $\operatorname{U}(h_{n})$ factors through $\det$ except for $(n,q)=(2,2)$. Further, assume that $(n,q)=(2,2)$. By [Gér77, (5) in the proof of Theorem 3.3], the unitary group $\operatorname{U}(\mathbb{F}_{4}^{2},h_{2})$ is isomorphic to $\mathbb{Z}/3\mathbb{Z}\times\mathfrak{S}_{3}$. Let $\mathrm{sgn}\colon\mathfrak{S}_{3}\to\overline{\mathbb{Q}}_{\ell}^{\times}$ be the sign character. Then the character group of $\operatorname{U}(\mathbb{F}_{4}^{2},h_{2})$ is generated by $\det$ and

[TABLE]

(cf. [Enn63, the table in p.28]).

By Schur’s lemma, there exists a character $\chi\in\mu_{q+1}^{\vee}$ such that

[TABLE]

Let $g=\mathrm{diag}(\zeta_{1},\ldots,\zeta_{n})\in\operatorname{U}(h_{n})$ with $\zeta_{i}\in\mu_{q+1}\setminus\{1\}$. By the Künneth formula and Lemma 2.3 (1), we have

[TABLE]

By Lemma 2.2 and (2.9), we have

[TABLE]

Hence, we have $\chi(\zeta_{1}\cdots\zeta_{n})=1$ for any $\zeta_{i}\in\mu_{q+1}\setminus\{1\}$. This implies $\chi=1$. If $(n,q)=(2,2)$, by Lemma 2.2 and Lemma 2.4, we have

[TABLE]

which implies $j=0$. ∎

We give a relation with mod $\ell$-cohomology.

Proposition 2.6**.**

Let the notations be as in Lemma 2.3.

1. (1)

The $\overline{\mathbb{Z}}_{\ell}$-module $H_{\rm c}^{n}(X_{n,\overline{\mathbb{F}}_{q}},\overline{\mathbb{Z}}_{\ell})[\psi]$ is free and $H_{\rm c}^{i}(X_{n,\overline{\mathbb{F}}_{q}},\overline{\mathbb{Z}}_{\ell})[\psi]=0$ for $i\neq n$. 2. (2)

We have an isomorphism

[TABLE]

as $\overline{\mathbb{F}}_{\ell}[\operatorname{HU}(h_{n})]$-modules for $i\geq 0$.

Proof.

These assertions follow from the Künneth formula, and Lemma 2.3 (2) and (3), respectively. ∎

The $\overline{\mathbb{Z}}_{\ell}[\operatorname{HU}(h_{n})]$-module $H_{\rm c}^{n}(X_{n,\overline{\mathbb{F}}_{q}},\overline{\mathbb{Z}}_{\ell})[\psi]$ is a $\overline{\mathbb{Z}}_{\ell}[\operatorname{HU}(h_{n})]$-lattice in $H_{\rm c}^{n}(X_{n,\overline{\mathbb{F}}_{q}},\overline{\mathbb{Q}}_{\ell})[\psi]$ by Proposition 2.6 (1). By Proposition 2.6 (2), the $\overline{\mathbb{F}}_{\ell}[\operatorname{HU}(h_{n})]$-module $H_{\rm c}^{n}(X_{n,\overline{\mathbb{F}}_{q}},\overline{\mathbb{F}}_{\ell})[\overline{\psi}]$ is regarded as a mod $\ell$ version of a Heisenberg–Weil representation of a unitary group.

3 Another coordinate

We give a construction of the Heisenberg–Weil representation of a finite unitary group of even degree using another coordinate. Let $\psi\in\mathbb{F}_{q}^{\vee}\setminus\{1\}$ in this section.

3.1 Cohomology of a surface

Let $X$ be the affine smooth surface over $\mathbb{F}_{q}$ defined by

[TABLE]

in $\mathbb{A}_{\mathbb{F}_{q}}^{3}=\operatorname{Spec}\mathbb{F}_{q}[x,y,z]$. Let $\mathbb{F}_{q^{2}}^{\times}$ act on $X_{\overline{\mathbb{F}}_{q}}$ by $(x,y,z)\mapsto(\zeta^{-1}x,\zeta^{q}y,z)$ for $\zeta\in\mathbb{F}_{q^{2}}^{\times}$. Let $\mathrm{Fr}_{q}\in\operatorname{Gal}(\overline{\mathbb{F}}_{q}/\mathbb{F}_{q})$ be the geometric Frobenius automorphism defined by $x\mapsto x^{q^{-1}}$ for $x\in\overline{\mathbb{F}}_{q}$. When we consider a closed subscheme of a variety, we suppose that it is equipped with the reduced scheme structure.

Let $\mathbb{F}_{q^{2}}^{\times}\rtimes\mathrm{Fr}_{q}^{\mathbb{Z}}$ be the semidirect product under the natural action. We sometimes regard $(\mathbb{F}_{q}^{\times})^{\vee}$ as a subset of $(\mathbb{F}_{q^{2}}^{\times})^{\vee}$ via the norm map. For $\chi\in(\mathbb{F}_{q}^{\times})^{\vee}\subset(\mathbb{F}_{q^{2}}^{\times})^{\vee}$, we define $\widetilde{\chi}\in(\mathbb{F}_{q^{2}}^{\times}\rtimes\mathrm{Fr}_{q}^{\mathbb{Z}})^{\vee}$ by $\widetilde{\chi}(\zeta,\mathrm{Fr}_{q}^{m})=\chi(\zeta)$ for $\zeta\in\mathbb{F}_{q^{2}}^{\times}$ and $m\in\mathbb{Z}$. For $\chi\in(\mathbb{F}_{q^{2}}^{\times})^{\vee}\setminus(\mathbb{F}_{q}^{\times})^{\vee}$, we define $\widetilde{\chi}\in(\mathbb{F}_{q^{2}}^{\times}\rtimes\mathrm{Fr}_{q}^{2\mathbb{Z}})^{\vee}$ by $\widetilde{\chi}(\zeta,\mathrm{Fr}_{q}^{2m})=\chi(\zeta)$ for $\zeta\in\mathbb{F}_{q^{2}}^{\times}$ and $m\in\mathbb{Z}$, and put

[TABLE]

For $\chi,\chi^{\prime}\in(\mathbb{F}_{q^{2}}^{\times})^{\vee}\setminus(\mathbb{F}_{q}^{\times})^{\vee}$, we have $\pi_{\chi}\simeq\pi_{\chi^{\prime}}$ if and only if $\chi$ and $\chi^{\prime}$ are equal in $((\mathbb{F}_{q^{2}}^{\times})^{\vee}\setminus(\mathbb{F}_{q}^{\times})^{\vee})/\mathrm{Fr}_{q}^{\mathbb{Z}}$.

Lemma 3.1**.**

We have an isomorphism

[TABLE]

as $\mathbb{F}_{q^{2}}^{\times}\rtimes\mathrm{Fr}_{q}^{\mathbb{Z}}$-representations.

Proof.

By changing a variable as $w=z+xy^{q}$ in (3.1), the surface $X$ is defined by $w^{q}-w=x^{q}(y^{q^{2}}-y)$. Let $S$ be the surface defined by $w^{q}-w=x(y^{q^{2}}-y)$. We have the finite purely inseparable map $X\to S;\ (w,x,y)\mapsto(w,x^{q},y)$. Hence we have an isomorphism $H_{\rm c}^{2}(X_{\overline{\mathbb{F}}_{q}},\overline{\mathbb{Q}}_{\ell})\simeq H_{\rm c}^{2}(S_{\overline{\mathbb{F}}_{q}},\overline{\mathbb{Q}}_{\ell})$.

Let $\pi\colon\mathbb{A}_{\mathbb{F}_{q^{2}}}^{2}\to\mathbb{A}_{\mathbb{F}_{q^{2}}}^{2};\ (x,y)\mapsto(x,y^{q^{2}}-y)$. For any $\psi^{\prime}\in\mathbb{F}_{q^{2}}^{\vee}$, there exists a unique element $\alpha\in\mathbb{F}_{q^{2}}$ such that $\psi^{\prime}(x)=\psi(\operatorname{Tr}_{\mathbb{F}_{q^{2}}/\mathbb{F}_{q}}(\alpha x))$ for any $x\in\mathbb{F}_{q^{2}}$. We have isomorphisms

[TABLE]

where we use the projection formula at the fourth isomorphism. Let $\mathscr{F}_{\psi}$ denote the $\ell$-adic Fourier transformation on $\mathbb{A}^{1}$ associated to $\psi$ in [Lau87, Définition (1.2.1.1)]. By [Lau87, Proposition (1.2.2.2)], we have

[TABLE]

Hence, we obtain the claim. ∎

3.2 Construction using another coordinate

Let $n$ be a positive integer. Let $V=\mathbb{F}_{q}^{2n}$. We consider the skew-hermitian form on $V_{\mathbb{F}_{q^{2}}}=\mathbb{F}_{q^{2}}^{2n}$ defined by

[TABLE]

Let $X_{2n}^{\prime}$ be the affine smooth variety over $\mathbb{F}_{q}$ defined by

[TABLE]

in $\mathbb{A}_{\mathbb{F}_{q}}^{2n+1}=\operatorname{Spec}\mathbb{F}_{q}[x_{1},\ldots,x_{2n},z]$. The group $\operatorname{H}(V_{\mathbb{F}_{q^{2}}},h_{2n}^{\prime})$ acts on $X_{2n}^{\prime}$ by

[TABLE]

for $(v,a)\in\operatorname{H}(V_{\mathbb{F}_{q^{2}}},h_{2n}^{\prime})$. The group $\operatorname{U}(V_{\mathbb{F}_{q^{2}}},h_{2n}^{\prime})$ acts on $X_{2n}^{\prime}$ by $(x,z)\mapsto(gx,z)$ for $g\in\operatorname{U}(V_{\mathbb{F}_{q^{2}}},h_{2n}^{\prime})$, where we regard $x=(x_{k})$ as a column vector. Hence, $\operatorname{HU}(h_{2n}^{\prime})$ acts on $X_{2n}^{\prime}$. Let $\mathbb{F}_{q}$ act on $X_{2n}^{\prime}$ by $z\mapsto z+a$ for $a\in\mathbb{F}_{q}$.

Remark 3.2**.**

Let $h_{2n}$ be the hermitian form on $V_{\mathbb{F}_{q^{2}}}=\mathbb{F}_{q^{2}}^{2n}$ defined in (2.5). Take $\xi\in\mathbb{F}_{q^{2}}$ such that $\xi^{q-1}=-1$. Then we have an isomorphism $f\colon(V_{\mathbb{F}_{q^{2}}},\xi h_{2n})\to(V_{\mathbb{F}_{q^{2}}},h_{2n}^{\prime})$ as skew-hermitian forms. This induces isomorphisms $\operatorname{H}(V_{\mathbb{F}_{q^{2}}},\xi h_{2n})\simeq\operatorname{H}(V_{\mathbb{F}_{q^{2}}},h_{2n}^{\prime})$ and $\operatorname{U}(V_{\mathbb{F}_{q^{2}}},\xi h_{2n})\simeq\operatorname{U}(V_{\mathbb{F}_{q^{2}}},h_{2n}^{\prime})$ by Remark 2.1. Further, $f$ and $z\mapsto\xi z$ gives an isomorphism $X_{2n}\simeq X_{2n,\mathbb{F}_{q^{2}}}^{\prime}$ over $\mathbb{F}_{q^{2}}$, which is compatible with group actions under the above isomorphisms.

Lemma 3.3**.**

We have $H_{\rm c}^{i}(X_{2n,\overline{\mathbb{F}}_{q}}^{\prime},\overline{\mathbb{Q}}_{\ell})[\psi]=0$ for $i\neq 2n$ and $\dim H_{\rm c}^{2n}(X_{2n,\overline{\mathbb{F}}_{q}}^{\prime},\overline{\mathbb{Q}}_{\ell})[\psi]=q^{2n}$. Further, we have an isomorphism

[TABLE]

as $\operatorname{HU}(h_{2n}^{\prime})$-representations.

Proof.

These follow from Theorem 2.5 and Remark 3.2. ∎

Lemma 3.4**.**

The action of $\mathrm{Fr}_{q}^{2}$ on $H_{\rm c}^{2n}(X_{2n,\overline{\mathbb{F}}_{q}}^{\prime},\overline{\mathbb{Q}}_{\ell}(n))[\psi]$ is trivial.

Proof.

This follows from Lemma 3.1 and the Künneth formula. ∎

4 Relation with Fermat variety

4.1 Unipotent representations of unitary group

Let $n$ be a positive integer. We follow [HM78, §1]. Let $G_{n}$ denote a general linear group $\operatorname{GL}_{n}$ over $\overline{\mathbb{F}}_{q}$. We consider the Frobenius endomorphism $F\colon G_{n}\to G_{n};\ (x_{i,j})\mapsto(x_{j,i}^{q})^{-1}$. Let $\operatorname{U}_{n}$ be the unitary algebraic group over $\mathbb{F}_{q}$ defined by $h_{n}$. Then we have $G_{n}^{F}=\operatorname{U}_{n}(\mathbb{F}_{q})=\operatorname{U}(h_{n})$.

Let $T_{0}$ denote the $F$-stable maximal torus of $G_{n}$ consisting of diagonal matrices. We set $\mathcal{W}_{n}=N_{G_{n}}(T_{0})/T_{0}$. Let $\mathscr{T}_{n}$ denote the set of $G_{n}^{F}$-conjugacy classes of $F$-stable maximal tori. The conjugacy classes of $\mathcal{W}_{n}$ is identified with $\mathscr{T}_{n}$ by $x^{-1}F(x)\mapsto xT_{0}x^{-1}$ for $x^{-1}F(x)\in\mathcal{W}_{n}$. The Weyl group $\mathcal{W}_{n}$ is isomorphic to the symmetric group $\mathfrak{S}_{n}$. The set of conjugacy classes of $\mathfrak{S}_{n}$ is identified with the set of partitions of $n$, which we denote by $\Lambda_{n}$. We have the natural bijection

[TABLE]

such that $T((1^{n}))=T_{0}$.

Let $T\in\mathscr{T}_{n}$ and $\theta\in(T^{F})^{\vee}$. Let $R_{T}^{G_{n}}(\theta)$ denote the Deligne–Lusztig character in the notation of [Lus76, p.204] and [Lus78, Corollary 2.4]. Let $\chi^{\lambda}$ denote the irreducible character of $\mathfrak{S}_{n}$ corresponding to $\lambda\in\Lambda_{n}$ normalized such that $\chi^{\lambda}$ is the sign representation if $\lambda=(1^{n})$. For $\lambda,\rho\in\Lambda_{n}$, let $\chi_{\rho}^{\lambda}$ denote the value of $\chi^{\lambda}$ at the class corresponding to $\rho$. Let $z_{\rho}$ be the cardinality of the centralizer of the class in $\mathfrak{S}_{n}$ corresponding to $\rho$. For $\lambda\in\Lambda_{n}$, we define a class function $\psi^{\lambda}$ on $G_{n}^{F}$ by

[TABLE]

We follow the definition of unipotency in [DL76, Definition 7.8]. By [LS77, §2], the set $\{\psi^{\lambda}\}_{\lambda\in\Lambda_{n}}$ equals the set of all unipotent characters of $G_{n}^{F}$ up to sign.

4.2 Geometric relation

Let $\Lambda\in\{\overline{\mathbb{Q}}_{\ell},\overline{\mathbb{F}}_{\ell}\}$ and $\psi\in\operatorname{Hom}(\mathbb{F}_{q,+},\Lambda^{\times})\setminus\{1\}$. Let $\pi$ be as in (2.6). We put $U=\pi^{-1}(\mathbb{G}_{\mathrm{m},\mathbb{F}_{q}})$ and $Z=\pi^{-1}(0)$. Then $U_{\mathbb{F}_{q^{2}}}$ and $Z_{\mathbb{F}_{q^{2}}}$ admit left actions induced by the natural action of $\operatorname{U}_{n}(\mathbb{F}_{q})$ on $\mathbb{A}_{\mathbb{F}_{q^{2}}}^{n}$. Let $\mathscr{L}_{\psi}$ be as in §2.2. In this section, all maps between $\operatorname{U}_{n}(\mathbb{F}_{q})$-representations are $\operatorname{U}_{n}(\mathbb{F}_{q})$-equivariant.

Lemma 4.1**.**

We have a long exact sequence

[TABLE]

Proof.

We have $H_{\rm c}^{i}(\mathbb{A}^{n},\pi^{\ast}\mathscr{L}_{\psi})=0$ for $i\neq n$ by Theorem 2.5 and the Künneth formula. We have $(\pi^{\ast}\mathscr{L}_{\psi})|_{Z}=\Lambda$. Hence the assertion follows. ∎

Lemma 4.2**.**

Assume that $\Lambda=\overline{\mathbb{Q}}_{\ell}$. Let $\chi\in\mu_{q+1}^{\vee}$. Then $H_{\rm c}^{n}(\mathbb{A}^{n},\pi^{\ast}\mathscr{L}_{\psi})[\chi]$ is an irreducible $\operatorname{U}_{n}(\mathbb{F}_{q})$-representation, and we have

[TABLE]

Proof.

The assertion follows from Theorem 2.5 and [Gér77, Corollary 4.5(a) and its proof]. ∎

Let $S_{n}$ be the Fermat variety defined by $\sum_{i=1}^{n}x_{i}^{q+1}=0$ in $\mathbb{P}_{\mathbb{F}_{q}}^{n-1}$. Let $Y_{n}=\mathbb{P}_{\mathbb{F}_{q}}^{n-1}\setminus S_{n}$. Let $\operatorname{U}_{n}(\mathbb{F}_{q})$ act on $\mathbb{P}_{\mathbb{F}_{q^{2}}}^{n-1}$ by usual left multiplication. Then $S_{n,\mathbb{F}_{q^{2}}}$ is stable under the action.

Let $\widetilde{Y}_{n}$ be the affine smooth variety defined by $\sum_{i=1}^{n}y_{i}^{q+1}=1$ in $\mathbb{A}_{\mathbb{F}_{q}}^{n}$. Let

[TABLE]

Let $\mu_{q+1}$ act on $Y_{n,\mathbb{F}_{q^{2}}}$ by $(y_{i})_{1\leq i\leq n}\mapsto(\zeta y_{i})_{1\leq i\leq n}$ for $\zeta\in\mu_{q+1}$. Then $Y_{n,\mathbb{F}_{q^{2}}}$ is a $\mu_{q+1}$-torsor over $Y_{n,\mathbb{F}_{q^{2}}}$. For $\chi\in\mu_{q+1}^{\vee}$, let $\mathscr{K}_{Y_{n},\chi}$ denote the smooth $\Lambda$-sheaf on $Y_{n,\mathbb{F}_{q^{2}}}$ associated to $f_{Y_{n}}$ and $\chi^{-1}$. We note that $\mathscr{K}_{Y_{n},\chi}$ is defined over $Y_{n}$ if $\chi^{2}=1$.

In the sequel, we assume that $q+1$ is invertible in $\Lambda$.

Lemma 4.3**.**

We have an isomorphism $H_{\rm c}^{i}(U_{\overline{\mathbb{F}}_{q}},\pi^{\ast}\mathscr{L}_{\psi})[\chi]\simeq H_{\rm c}^{i-1}(Y_{n,\overline{\mathbb{F}}_{q}},\mathscr{K}_{Y_{n},\chi})$ as $\operatorname{U}_{n}(\mathbb{F}_{q})$-representations for $\chi\in\operatorname{Hom}(\mu_{q+1},\Lambda^{\times})$ and any integer $i$. If $\Lambda=\overline{\mathbb{Q}}_{\ell}$ and $\chi^{2}=1$, then we have an isomorphism

[TABLE]

as representations of $\operatorname{U}_{n}(\mathbb{F}_{q})$ and $\operatorname{Gal}(\overline{\mathbb{F}}_{q}/\mathbb{F}_{q})$, where $\chi^{\prime}$ is the character of $\mathbb{F}_{q}^{\times}$ of the same order as $\chi$ and $\delta_{\chi^{\prime},\psi}$ denotes the unramified character of $\operatorname{Gal}(\overline{\mathbb{F}}_{q}/\mathbb{F}_{q})$ such that the $q$-th power geometric Frobenius element acts by the Gauss sum $\tau(\chi^{\prime},\psi)$ defined in [Del77, Sommes trig., (4.1.1)].

Proof.

We consider the affine smooth variety

[TABLE]

over $\mathbb{F}_{q}$. The space $\widetilde{U}_{\mathbb{F}_{q^{2}}}$ admits a left action induced by the natural action of $\operatorname{U}_{n}(\mathbb{F}_{q})$ on $\mathbb{A}_{\mathbb{F}_{q^{2}}}^{n}$. Let $\mu_{q+1}^{2}$ act on $\widetilde{U}_{\mathbb{F}_{q^{2}}}$ by

[TABLE]

for $(\zeta_{1},\zeta_{2})\in\mu_{q+1}^{2}$. Clearly, the action of $\mu_{q+1}^{2}$ on $\widetilde{U}_{\mathbb{F}_{q^{2}}}$ is free. We have a natural identification

[TABLE]

We have the morphism

[TABLE]

We put $H=\{(1,\zeta)\in\mu_{q+1}^{2}\mid\zeta\in\mu_{q+1}\}$. Then $f_{U}$ is an $H$-torsor over $\mathbb{F}_{q^{2}}$.

Let $f_{\mathbb{G}_{\mathrm{m}}}\colon\mathbb{G}_{\mathrm{m},\mathbb{F}_{q}}\to\mathbb{G}_{\mathrm{m},\mathbb{F}_{q}};\ t\mapsto t^{q+1}$. We have the isomorphism

[TABLE]

Let $\mu_{q+1}^{2}$ act on $\widetilde{Y}_{n,\mathbb{F}_{q^{2}}}\times\mathbb{G}_{\mathrm{m},\mathbb{F}_{q^{2}}}$ by

[TABLE]

for $(\zeta_{1},\zeta_{2})\in\mu_{q+1}^{2}$. The morphism $\widetilde{\varphi}$ is compatible with $\phi\colon\mu_{q+1}^{2}\xrightarrow{\sim}\mu_{q+1}^{2};\ (\zeta_{1},\zeta_{2})\mapsto(\zeta_{1}/\zeta_{2},\zeta_{2})$ in the sense that $\widetilde{\varphi}(gx)=\phi(g)\widetilde{\varphi}(x)$ for $x\in\widetilde{U}_{\mathbb{F}_{q^{2}}}$ and $g\in\mu_{q+1}^{2}$. The isomorphism $\widetilde{\varphi}$ induces an isomorphism

[TABLE]

compatible with the isomorphism $\mu_{q+1}^{2}/H\xrightarrow{\sim}\mu_{q+1}^{2}/\phi(H)$ induced by $\phi$. By taking the quotients of the both sides of $\varphi$ by $\mu_{q+1}^{2}/H$ and $\mu_{q+1}^{2}/\phi(H)$, we obtain an isomorphism $\overline{\varphi}\colon U_{\mathbb{F}_{q^{2}}}/\mu_{q+1}\xrightarrow{\sim}Y_{n,\mathbb{F}_{q^{2}}}\times\mathbb{G}_{\mathrm{m},\mathbb{F}_{q^{2}}}$. Let

[TABLE]

be the first projection and the second projection respectively. We have the commutative diagram

[TABLE]

where $f^{\prime}$ is a morphism induced by $f_{Y_{n}}\times f_{\mathbb{G}_{\mathrm{m}}}$.

For smooth $\Lambda$-sheaves $\mathscr{F}$ on $Y_{n,\mathbb{F}_{q^{2}}}$ and $\mathscr{G}$ on $\mathbb{G}_{\mathrm{m},\mathbb{F}_{q^{2}}}$, let $\mathscr{F}\boxtimes\mathscr{G}$ denote the sheaf $\mathrm{pr}_{1}^{\ast}\mathscr{F}\otimes\mathrm{pr}_{2}^{\ast}\mathscr{G}$ on $Y_{n,\mathbb{F}_{q^{2}}}\times\mathbb{G}_{\mathrm{m},\mathbb{F}_{q^{2}}}$. Let $\mathscr{K}_{\mathbb{G}_{\mathrm{m}},\chi}$ be as in §2.2. We identify as $\mu_{q+1}\xrightarrow{\sim}\mu_{q+1}^{2}/\phi(H);\ \zeta\mapsto(\zeta,1)$. We have

[TABLE]

by the Künneth formula. Let $\mathscr{K}^{\prime}_{\chi}$ denote the smooth $\Lambda$-sheaf on $U_{\mathbb{F}_{q^{2}}}/\mu_{q+1}$ associated to $\chi^{-1}$ and the $\mu_{q+1}$-torsor $U_{\mathbb{F}_{q^{2}}}\to U_{\mathbb{F}_{q^{2}}}/\mu_{q+1}$. We identify as $\mu_{q+1}\simeq\mu_{q+1}^{2}/H;\ \zeta\mapsto(\zeta,1)$. We have $\mathscr{K}^{\prime}_{\chi}\simeq\overline{\varphi}^{\ast}\left(\mathscr{K}_{Y_{n},\chi}\boxtimes\mathscr{K}_{\mathbb{G}_{\mathrm{m}},\chi^{-1}}\right)$. By the Künneth formula and (2.4), we have isomorphisms

[TABLE]

as $\operatorname{U}_{n}(\mathbb{F}_{q})$-representations. The last claim follows from the above arguments and [Del77, Sommes trig., Proposition 4.2 (ii)]. ∎

We study the cohomology of $Z_{\overline{\mathbb{F}}_{q}}$. Let $Z^{0}\subset Z$ be the open subscheme defined by $(x_{1},\ldots,x_{n})\neq\mathbf{0}=(0,\ldots,0)$. Now, we regard $Z^{0}$ as a closed subscheme of $\mathbb{A}_{\mathbb{F}_{q}}^{n}\setminus\{\mathbf{0}\}$. The morphism $\mathbb{A}_{\mathbb{F}_{q}}^{n}\setminus\{\mathbf{0}\}\to\mathbb{P}^{n-1}_{\mathbb{F}_{q}};\ (x_{i})_{1\leq i\leq n}\mapsto[x_{1}:\cdots:x_{n}]$ is a $\mathbb{G}_{\mathrm{m},\mathbb{F}_{q}}$-bundle. By restricting this to $Z^{0}$, we have a $\mathbb{G}_{\mathrm{m},\mathbb{F}_{q}}$-bundle

[TABLE]

The morphism $\pi_{0}$ factors as

[TABLE]

where $Z^{0}/\mu_{q+1}$ denotes the quotient of $Z^{0}$ under the natural action of the group scheme of $(q+1)$-st roots of unity over $\mathbb{F}_{q}$.

Lemma 4.4**.**

1. (1)

We have the spectral sequence

[TABLE] 2. (2)

We have

[TABLE] 3. (3)

The action of $\mu_{q+1}$ on $H_{\rm c}^{i}(Z^{0}_{\overline{\mathbb{F}}_{q}},\Lambda)$ is trivial for any $i$.

Proof.

Since the morphism $\overline{\pi}_{0}$ is a $\mathbb{G}_{\mathrm{m}}$-bundle and any $\mathbb{G}_{\mathrm{m}}$-bundle is the complement of the zero section in a line bundle, we have the second assertion by [SGA77, VII, Proposition 1.3(ii)].

We show the first assertion. We have the Leray spectral sequence

[TABLE]

For $\chi\in\operatorname{Hom}(\mu_{q+1},\Lambda^{\times})$, let $\mathscr{K}^{0}_{\chi}$ denote the smooth $\Lambda$-sheaf on $Z^{0}_{\mathbb{F}_{q^{2}}}/\mu_{q+1}$ associated to the finite Galois étale covering $f_{Z^{0}}$ and $\chi^{-1}$. We have $f_{Z^{0},!}\Lambda\simeq\bigoplus_{\chi\in\mu_{q+1}^{\vee}}\mathscr{K}^{0}_{\chi}$. We have $R\overline{\pi}_{0,!}\mathscr{K}^{0}_{\chi}=0$ for $\chi\in\operatorname{Hom}(\mu_{q+1},\Lambda^{\times})\setminus\{1\}$ by [Del77, Sommes trig., Théorème 2.7]. Hence we have $R^{b}\pi_{0,!}\Lambda\simeq R^{b}\overline{\pi}_{0,!}(f_{Z^{0},!}\Lambda)\simeq R^{b}\overline{\pi}_{0,!}\Lambda$ for any $b$. Hence the first claim follows. The third assertion follows from the proof of the first one. ∎

Lemma 4.5**.**

The action of $\mu_{q+1}$ on $H_{\rm c}^{i}(Z_{\overline{\mathbb{F}}_{q}},\Lambda)$ is trivial for any $i$.

Proof.

We have an exact sequence

[TABLE]

and an isomorphism

[TABLE]

for $i\geq 2$. Hence the assertion follows from Lemma 4.4 (3). ∎

Corollary 4.6**.**

Let $\chi\in\operatorname{Hom}(\mu_{q+1},\Lambda^{\times})\setminus\{1\}$. We have an isomorphism

[TABLE]

for any $i$. Furthermore, if $i\neq n$, this is isomorphic to zero.

Proof.

The former claim follows from Lemma 4.1 and Lemma 4.5. The latter claim follows from (2.7), Theorem 2.5 and Proposition 2.6. ∎

Lemma 4.7**.**

Assume that $\Lambda=\overline{\mathbb{Q}}_{\ell}$.

1. (1)

We have a $\operatorname{Gal}(\overline{\mathbb{F}}_{q}/\mathbb{F}_{q})$-equivariant isomorphism

[TABLE] 2. (2)

Assume that $n\geq 2$. We have a $\operatorname{Gal}(\overline{\mathbb{F}}_{q}/\mathbb{F}_{q})$-equivariant isomorphism

[TABLE]

Proof.

By Lemma 4.3, we have an isomorphism

[TABLE]

Let $\phi_{n}$ be the unipotent representation of $\operatorname{U}_{n}(\mathbb{F}_{q})$ corresponding to the partition $(n-1,1)$ of $n$ (*cf. *§4.1). By [HM78, the proof of Theorem 1], we have

[TABLE]

Note that differentials between $E_{2}$-terms of (4.3) are zero except

[TABLE]

for $m\in\mathbb{Z}$.

We show the first assertion. We have $H_{\rm c}^{n-1}(Z_{\overline{\mathbb{F}}_{q}},\overline{\mathbb{Q}}_{\ell})\neq 0$ by Lemma 4.4, (4.4), (4.5) and [HM78, Theorem 1]. By (4.6) and (4.7), the $\operatorname{U}_{n}(\mathbb{F}_{q})$-representation $H_{\rm c}^{n}(U_{\overline{\mathbb{F}}_{q}},\pi^{\ast}\mathscr{L}_{\psi})[1_{\mu_{q+1}}]$ is irreducible. Hence the claim follows from Lemma 4.1 and Lemma 4.5.

We show the second assertion. By Lemma 4.3, we have isomorphisms

[TABLE]

where we use [HM78, the last line in p.257] if $n\geq 3$. By the first assertion and Lemma 4.1, we have an isomorphism

[TABLE]

Consider the composite

[TABLE]

where the third morphism is induced by the spectral sequence (4.3) and the last morphism is as in [HM78, (i),(ii) in p.258]. Then the kernels of the morphisms in (4.10) are direct sums of trivial $\operatorname{U}_{n}(\mathbb{F}_{q})$-representations by (4.9) and [HM78, Theorem 1 and (i),(ii) in p.258]. On the other hand, the source and the target of (4.10) are irreducible and non-trivial by Lemma 4.2 and (4.7). Therefore (4.10) is an isomorphism. ∎

Proposition 4.8**.**

Assume that $\Lambda=\overline{\mathbb{Q}}_{\ell}$. For $\chi\in\mu_{q+1}^{\vee}$, we have

[TABLE]

as representations of $\operatorname{U}_{n}(\mathbb{F}_{q})$.

Proof.

If $\chi\neq 1$, the claim follows from (2.7), Lemma 4.3 and Corollary 4.6. If $\chi=1$ and $n=1$, the claim is trivial. If $\chi=1$ and $n\geq 2$, the claim follows from (2.7), Lemma 4.7 (2) and the fact that $H_{\rm c}^{i}(Y_{n,\overline{\mathbb{F}}_{q}},\overline{\mathbb{Q}}_{\ell})=0$ if $i\neq n-1,2(n-1)$ (cf. [HM78, Proof of Theorem 1]). ∎

5 Lusztig induction

We recall Lusztig induction in [Lus76]. Let $G$ be a connected reductive group defined over $\overline{\mathbb{F}}_{q}$ with an $\mathbb{F}_{q}$-rational structure. Let $F$ be the corresponding Frobenius endomorphism on $G$.

Let $P$ be a parabolic subgroup of $G$, and $M$ be a Levi subgroup of $P$. Assume that $M$ is $F$-stable. We put

[TABLE]

Let $G^{F}\times M^{F}$ act on $\widetilde{Y}_{P}$ by $\widetilde{Y}_{P}\to\widetilde{Y}_{P};\ x\mapsto gxg^{\prime-1}$ for $(g,g^{\prime})\in G^{F}\times M^{F}$. We set

[TABLE]

The morphism

[TABLE]

is known to be an $M^{F}$-torsor (cf. [DM14, §7.3]). We give a proof of it.

Lemma 5.1**.**

The morphism $\pi_{P}$ is an $M^{F}$-torsor.

Proof.

Let $X=\left\{g\in G\mid g^{-1}F(g)\in F(P)\right\}$. We regard $\widetilde{Y}_{P}$ as a closed subscheme of $X$. We consider the morphisms $\phi^{\prime}\colon F(U_{P})\times M\to F(P);\ (u,m)\mapsto m^{-1}uF(m)$ and $\phi\colon\widetilde{Y}\times M\to X;\ (g,m)\mapsto gm$. Let $L\colon G\to G;\ g\mapsto g^{-1}F(g)$. We have the cartesian diagram

[TABLE]

We consider the morphisms

[TABLE]

We can easily check $\phi^{\prime}=\phi_{3}\circ\phi_{2}\circ\phi_{1}$. The morphisms $\phi_{i}$ for $i=1,3$ are isomorphisms. The morphism $\phi_{2}$ is an $M^{F}$-torsor. Hence, $\phi$ is an $M^{F}$-torsor. By taking the quotient of $\phi$ by $M$, we obtain the claim. ∎

For a representation $\rho$ of $M^{F}$, we consider a virtual $G^{F}$-module

[TABLE]

which is introduced in [Lus76, p.203].

Assume that $M=P\cap F(P)$. Then we have

[TABLE]

Assume that there is a decomposition $M=T\times M^{\prime}$ as algebraic groups over $\mathbb{F}_{q}$ where $T$ is a torus and $M^{\prime}$ is a reductive group. Let $\chi\in(T^{F})^{\vee}$. Let $\mathscr{K}_{Y_{P},\chi}$ be the smooth $\overline{\mathbb{Q}}_{\ell}$-sheaf on $Y_{P}$ associated to $\chi^{-1}$ under the natural projection $M^{F}\to T^{F}$ and $\pi_{P}$. We have

[TABLE]

By [DL76, Corollary 7.14], we have

[TABLE]

as characters of $M^{\prime F}$, where the summation runs over all $M^{\prime F}$-conjugacy classes of $F$-stable maximal tori $T^{\prime}$ of $M^{\prime}$. We have

[TABLE]

as characters of $M^{F}$. By applying $R^{G}_{M\subset P}$ to this, we have

[TABLE]

as characters of $G^{F}$ by [Lus76, Corollary 5]. Therefore we have

[TABLE]

6 Unitary group

Let $\psi\in\mathbb{F}_{q}^{\vee}\setminus\{1\}$ in the rest of this paper.

6.1 Unipotency

We use notation in Subsection 4.1. Let $P\subset G_{n}$ be the parabolic subgroup consisting of matrices $(x_{i,j})$ such that $x_{i,1}=0$ for $2\leq i\leq n$. Let $M$ be the $F$-stable Levi subgroup of $P$. Note that $P$ is not stable by $F$. We have $M\simeq G_{1}\times G_{n-1}$ diagonally embedded in $G_{n}$. We have $G_{n}^{F}=\operatorname{U}_{n}(\mathbb{F}_{q})$ and $M^{F}=\operatorname{U}_{1}(\mathbb{F}_{q})\times\operatorname{U}_{n-1}(\mathbb{F}_{q})$.

For $x=(x_{i,j})\in\widetilde{Y}_{P}$, we have $\sum_{i=1}^{n}x_{i,1}^{q+1}=1$ by $F^{-1}(x)x\in F(U_{P})$. Hence, we have the morphism $\widetilde{\pi}_{n}\colon\widetilde{Y}_{P}\to\widetilde{Y}_{n,\overline{\mathbb{F}}_{q}};\ x\mapsto(x_{i,1})_{1\leq i\leq n}$. This induces a morphism $\widetilde{\varphi}_{n}\colon\widetilde{Y}_{P}/\operatorname{U}_{n-1}(\mathbb{F}_{q})\to\widetilde{Y}_{n,\overline{\mathbb{F}}_{q}}$. We have an identification $\widetilde{Y}_{P}/M^{F}\simeq Y_{P}$ by Lemma 5.1. As shown in [HM78, p.259], $\widetilde{\pi}_{n}$ induces an isomorphism $\varphi_{n}\colon Y_{P}\simeq\widetilde{Y}_{P}/M^{F}\xrightarrow{\sim}Y_{n,\overline{\mathbb{F}}_{q}}$. We have the commutative diagram

[TABLE]

where the vertical morphisms are natural ones. Since $f_{Y_{P}}$ and $f_{Y_{n}}$ are $\mu_{q+1}$-torsors and $\varphi_{n}$ is an isomorphism, $\widetilde{\varphi}_{n}$ is also an isomorphism.

Proposition 6.1**.**

We have

[TABLE]

as characters of $G_{n}^{F}$.

Proof.

Since $\widetilde{\varphi}_{n}$ is an isomorphism, the claim follows from Proposition 4.8 and (5.1). ∎

In the sequel, we write $\omega_{\operatorname{U}_{n}}$ for $\omega_{\operatorname{U}(\mathbb{F}_{q^{2}}^{n},h_{n})}$ defined in (2.2). We give a geometric proof of the following fact using a relation between $X_{n}$ and a certain Deligne–Lusztig variety.

Corollary 6.2**.**

The $\operatorname{U}_{n}(\mathbb{F}_{q})$-representation $\omega_{\operatorname{U}_{n}}[1_{\mu_{q+1}}]$ is unipotent. For each $\chi\in\mu_{q+1}^{\vee}\setminus\{1\}$, the $\operatorname{U}_{n}(\mathbb{F}_{q})$-representation $\omega_{\operatorname{U}_{n}}[\chi]$ is not unipotent.

Proof.

The claim follows from (5.2) and Proposition 6.1 (cf. [DL76, the sentence after Definition 7.8]). ∎

Remark 6.3**.**

This is a special case of the preservation of unipotency under the Howe correspondence for a unitary pair, which is proved in [AM93] if $q$ is odd and large enough. The assumption on $q$ is necessary in [AM93], since the proof depends on [Sri79].

Remark 6.4**.**

Let $\epsilon_{G}$ be the $\mathbb{F}_{q}$-rank of a linear algebraic group $G$ over $\overline{\mathbb{F}}_{q}$. For any integer $m\geq 1$, let $m_{p}$ be the largest power of $p$ dividing $m$ and $m_{p^{\prime}}=m/m_{p}$. We go back to our situation. We have $\epsilon_{G_{n}}=[n/2]$ (cf. [DM91, §15.1]). Hence, we have $\epsilon_{G_{n}}\epsilon_{M}=(-1)^{n-1}$. By [DM91, Proposition 12.17], we have

[TABLE]

for any $\chi\in\mu_{q+1}^{\vee}$. This is compatible with Lemma 4.2 and Proposition 6.1.

6.2 Branching formula

We consider the embedding $\iota\colon\operatorname{U}_{n}\hookrightarrow\operatorname{U}_{n+1};\ g\mapsto\mathrm{diag}(g,1)$. Let $\chi\in\mu_{q+1}^{\vee}$. We regard the $\operatorname{U}_{n+1}(\mathbb{F}_{q})$-representation $\omega_{\operatorname{U}_{n+1}}[\chi]$ as a $\operatorname{U}_{n}(\mathbb{F}_{q})$-representation by $\iota$. We give a geometric proof of the following branching formula (cf. [Tie97, Lemma 4.4] in $\operatorname{SU}_{n}$ case).

Proposition 6.5**.**

Let $\chi\in\mu_{q+1}^{\vee}$. We have a decomposition

[TABLE]

as $\operatorname{U}_{n}(\mathbb{F}_{q})$-representations.

Proof.

By (2.7), we have isomorphisms

[TABLE]

where we use the Künneth formula at the second isomorphism. By taking the $\chi$-isotypic part of the above isomorphism, we obtain the assertion by Lemma 2.3 (1) and Theorem 2.5. ∎

7 Symplectic orthogonal pair

7.1 Representation of a dual pair

7.1.1 Construction

We use the same notation as §3.2. Let $\omega$ be a symplectic form on $V$ obtained by the restriction of $h_{2n}^{\prime}$ to $V$. By regarding $V_{\mathbb{F}_{q^{2}}}$ as a vector space over $\mathbb{F}_{q}$, we consider the symplectic form associated to $h_{2n}^{\prime}$:

[TABLE]

Let $\operatorname{Sp}(V,\omega)$ and $\operatorname{Sp}(V_{\mathbb{F}_{q^{2}}},\omega^{\prime})$ be the isometry groups of $(V,\omega)$ and $(V_{\mathbb{F}_{q^{2}}},\omega^{\prime})$, respectively. We have $\operatorname{Sp}(V,\omega)\subset\operatorname{U}(V_{\mathbb{F}_{q^{2}}},h_{2n}^{\prime})\subset\operatorname{Sp}(V_{\mathbb{F}_{q^{2}}},\omega^{\prime})$.

We put $W=\mathbb{F}_{q^{2}}$. Let $h_{1}\colon W\times W\to\mathbb{F}_{q^{2}};\ (x,y)\mapsto x^{q}y$. We have $\operatorname{U}(W,h_{1})=\mu_{q+1}$. Under the natural isomorphism $V\otimes_{\mathbb{F}_{q}}W\simeq V_{\mathbb{F}_{q^{2}}}$, the skew-hermitian form $\omega\otimes h_{1}$ corresponds to $h_{2n}^{\prime}$. This induces a morphism

[TABLE]

We have natural actions of $\operatorname{Gal}(\mathbb{F}_{q^{2}}/\mathbb{F}_{q})\simeq\mathbb{Z}/2\mathbb{Z}$ on $\operatorname{U}(W,h_{1})$ and $\operatorname{U}(V_{\mathbb{F}_{q^{2}}},h_{2n}^{\prime})$. Then the homomorphism (7.1) extends naturally to

[TABLE]

The $\mathbb{F}_{q}$-automorphism $F\colon V_{\mathbb{F}_{q^{2}}}\to V_{\mathbb{F}_{q^{2}}};\ (v_{i})\mapsto(v_{i}^{q})$ preserves $\omega^{\prime}$. We regard this as an element of $\operatorname{Sp}(V_{\mathbb{F}_{q^{2}}},\omega^{\prime})$. Then we have the injective homomorphism

[TABLE]

Let $\mathbb{Z}/2\mathbb{Z}$ act on $H_{\rm c}^{2n}(X_{2n,\overline{\mathbb{F}}_{q}}^{\prime},\overline{\mathbb{Q}}_{\ell})(n)[\psi]$ by

[TABLE]

for $k\in\mathbb{Z}/2\mathbb{Z}$, which is well-defined by Lemma 3.4. By the natural action of $\operatorname{Gal}(\mathbb{F}_{q^{2}}/\mathbb{F}_{q})$ on $V_{\mathbb{F}_{q^{2}}}$ and $\mathbb{F}_{q^{2}}$, we have a natural action of $\operatorname{Gal}(\mathbb{F}_{q^{2}}/\mathbb{F}_{q})\simeq\mathbb{Z}/2\mathbb{Z}$ on $\operatorname{HU}(h_{2n}^{\prime})$. We write $\pi_{h_{2n}^{\prime},\psi}$ for the $\operatorname{HU}(h_{2n}^{\prime})\rtimes(\mathbb{Z}/2\mathbb{Z})$-representation $H_{\rm c}^{2n}(X_{2n,\overline{\mathbb{F}}_{q}}^{\prime},\overline{\mathbb{Q}}_{\ell})(n)[\psi]$. Then the action of $1\in\mathbb{Z}/2\mathbb{Z}$ induces an isomorphism between $\pi_{h_{2n}^{\prime},\psi}[\chi]$ and $\pi_{h_{2n}^{\prime},\psi}[\chi^{-1}]$ for $\chi\in\mu_{q+1}^{\vee}\setminus\{1\}$.

We consider the quadratic form $Q\colon W\to\mathbb{F}_{q};\ x\mapsto x^{q+1}$. We set

[TABLE]

Since $Q(x)=h_{1}(x,x)$, we have a natural inclusion $\operatorname{U}(W,h_{1})\hookrightarrow\operatorname{O}(W,Q)$. We regard $F_{W}\colon W\to W;\ x\mapsto x^{q}$ as an element of $\operatorname{O}(W,Q)$. Then we have the isomorphism

[TABLE]

since $\operatorname{O}(W,Q)$ is isomorphic to the dihedral group $D_{2(q+1)}$ by [KL90, Proposition 2.9.1]. We identify $\operatorname{U}(W,h_{1})\rtimes(\mathbb{Z}/2\mathbb{Z})$ with $\operatorname{O}(W,Q)$ by the above isomorphism.

We consider the symmetric form $s_{1}\colon W\times W\to\mathbb{F}_{q};\ (x,y)\mapsto\operatorname{Tr}_{\mathbb{F}_{q^{2}}/\mathbb{F}_{q}}(h_{1}(x,y))$. Note that

[TABLE]

Hence, we have the natural map $\operatorname{O}(W,Q)\hookrightarrow\operatorname{O}(W,s_{1})$, which is an isomorphism if $q$ is odd. We have a natural homomorphism $\operatorname{Sp}(V,\omega)\times\operatorname{O}(W,s_{1})\to\operatorname{Sp}(V_{\mathbb{F}_{q^{2}}},\omega^{\prime})$, since $\omega\otimes s_{1}$ corresponds to $\omega^{\prime}$ under the natural isomorphism $V\otimes_{\mathbb{F}_{q}}W\simeq V_{\mathbb{F}_{q^{2}}}$. We consider the composite

[TABLE]

By the construction, we obtain the commutative diagram

[TABLE]

where the going up arrows are natural homomorphisms. We put

[TABLE]

For a character $\chi_{0}\in\mu_{q+1}^{\vee}$ such that $\chi_{0}^{2}=1$, the $\chi_{0}$-isotypic part $\omega_{\mathrm{SpO},n,\psi}[\chi_{0}]$ is stable under the action of $\mathbb{Z}/2\mathbb{Z}$. For $\kappa\in\{\pm 1\}$, let $\omega_{\mathrm{SpO},n,\psi}[\chi_{0}]^{\kappa}$ denote the $\kappa$-eigenspace of $1\in\mathbb{Z}/2\mathbb{Z}$ on $\omega_{\mathrm{SpO},n,\psi}[\chi_{0}]$. Let $\nu$ be the quadratic character of $\mu_{q+1}$ if $p\neq 2$.

Lemma 7.1**.**

1. (1)

Assume that $p=2$. We have

[TABLE]

for $\kappa\in\{\pm 1\}$ and $\chi\in\mu_{q+1}^{\vee}\setminus\{1\}$. 2. (2)

Assume that $p\neq 2$. We have

[TABLE]

for $\kappa\in\{\pm 1\}$ and $\chi\in\mu_{q+1}^{\vee}\setminus\{1,\nu\}$.

Proof.

We set $a_{n,\chi}=\dim\omega_{\mathrm{SpO},n,\psi}[\chi]$ for $\chi\in\mu_{q+1}^{\vee}$. We have a $\mu_{q+1}$-equivariant isomorphism $X_{2n,\mathbb{F}_{q^{2}}}\simeq X_{2n,\mathbb{F}_{q^{2}}}^{\prime}$. Hence, we have

[TABLE]

by Lemma 4.2.

Let

[TABLE]

Let $\mathbf{1}\in A$ denote the element $(\chi_{k})_{1\leq k\leq n}$ with $\chi_{k}=1$ for any $1\leq k\leq n$. For each $(\chi_{k})_{1\leq k\leq n}\in A$, let $V_{(\chi_{k})_{1\leq k\leq n}}$ denote the subspace $\bigotimes_{k=1}^{n}\omega_{\mathrm{SpO},1,\psi}[\chi_{k}]\subset\omega_{\mathrm{SpO},n,\psi}[1_{\mu_{q+1}}]$. By Lemma 3.1, we have $\dim\omega_{\mathrm{SpO},1,\psi}[1_{\mu_{q+1}}]=q$ and the action of $\mathbb{Z}/2\mathbb{Z}$ on $\omega_{\mathrm{SpO},1,\psi}[1_{\mu_{q+1}}]$ is trivial. Further, if $p\neq 2$, the dimensions of the $1$-eigenspace and the $(-1)$-eigenspace in $\omega_{\mathrm{SpO},1,\psi}[\nu]$ are same by Lemma 3.1. Therefore, we have $\dim V_{\mathbf{1}}=q^{n}$ and the action of $\mathbb{Z}/2\mathbb{Z}$ on $V_{\mathbf{1}}$ is trivial. Further, the dimensions of the $1$-eigenspace and the $(-1)$-eigenspace in $\bigoplus_{(\chi_{k})_{1\leq i\leq n}\in A\setminus\{\mathbf{1}\}}V_{(\chi_{k})_{1\leq k\leq n}}$ are same. Hence, the dimension of $\omega_{\mathrm{SpO},n,\psi}[1_{\mu_{q+1}}]^{\kappa}$ equals

[TABLE]

The second equality in the claim (2) is proved similarly. ∎

Remark 7.2**.**

It is possible to calculate values of characters of representations in Lemma 7.1 using the geometric constructions in this paper and the Grothendieck–Lefschetz trace formula. See [IT20, Proposition 4.9] for an example of such a calculation.

7.1.2 Compatibility

In §7.1.2, we always assume $p\neq 2$. Then we can construct a representation of the dual pair $\operatorname{Sp}(V,\omega)\times\operatorname{O}(W,Q)$ as the restriction of the Weil representation of $\operatorname{Sp}(V_{\mathbb{F}_{q^{2}}},\omega^{\prime})$. We show that the two constructions are compatible. We put

[TABLE]

with multiplication

[TABLE]

Then, $\operatorname{Sp}(V_{\mathbb{F}_{q^{2}}},\omega^{\prime})$ acts on $\operatorname{H}(V_{\mathbb{F}_{q^{2}}},\omega^{\prime})$ by $(v,a)\mapsto(gv,a)$ for $g\in\operatorname{Sp}(V_{\mathbb{F}_{q^{2}}},\omega^{\prime})$ and $(v,a)\in\operatorname{H}(V_{\mathbb{F}_{q^{2}}},\omega^{\prime})$. We put

[TABLE]

Let $\rho_{\operatorname{HSp}(\omega^{\prime}),\psi}$ be the Heisenberg–Weil representation of $\operatorname{HSp}(\omega^{\prime})$ associated to $\psi$ (cf. [Gér77, Theorem 2.4(a’)] and [GH07, 2.2]). We write $\rho_{\operatorname{Sp}(\omega^{\prime}),\psi}$ for the restriction of $\rho_{\operatorname{HSp}(\omega^{\prime}),\psi}$ to $\operatorname{Sp}(V_{\mathbb{F}_{q^{2}}},\omega^{\prime})$. The representation $\rho_{\operatorname{Sp}(\omega^{\prime}),\psi}$ is called the Weil representation of $\operatorname{Sp}(V_{\mathbb{F}_{q^{2}}},\omega^{\prime})$ associated to $\psi$. We put

[TABLE]

By (7.2) and the isomorphism

[TABLE]

we have a morphism $\operatorname{HU}(h_{2n}^{\prime})\rtimes(\mathbb{Z}/2\mathbb{Z})\to\operatorname{HSp}(\omega^{\prime})$. We have the commutative diagram

[TABLE]

where the arrows in the square are natural homomorphisms. Let $F\in\operatorname{Sp}(V_{\mathbb{F}_{q^{2}}},\omega^{\prime})$ be the $q$-th power Frobenius. Then $F$ takes $\rho_{\operatorname{Sp}(\omega^{\prime}),\psi}[\chi]$ isomorphically to $\rho_{\operatorname{Sp}(\omega^{\prime}),\psi}[\chi^{-1}]$ for $\chi\in\mu_{q+1}^{\vee}$. Let $\chi\in\mu_{q+1}^{\vee}$ such that $\chi^{2}=1$. For $\kappa\in\{\pm 1\}$, let $\rho_{\operatorname{Sp}(\omega^{\prime}),\psi}[\chi]^{\kappa}$ denote the $\kappa$-eigenspace of $F$ on $\rho_{\operatorname{Sp}(\omega^{\prime}),\psi}[\chi]$.

Lemma 7.3**.**

Let $\eta_{0}$ be the quadratic character of $\mathbb{F}_{q}^{\times}$. Then we have $\operatorname{Tr}\rho_{\operatorname{Sp}(\omega^{\prime}),\psi}(F)=(\eta_{0}(-1)q)^{n}$.

Proof.

We imitate arguments in [Des08, Lemma 2.1]. Let $L=\mathbb{F}_{q}^{n}\oplus\{\mathbf{0}\}$ and $L^{\prime}=\{\mathbf{0}\}\oplus\mathbb{F}_{q}^{n}$ in $V$. These are maximal totally isotropic subspaces with respect to $\omega$. Then $\rho_{\operatorname{Sp}(\omega^{\prime}),\psi}$ is realized in the space $\overline{\mathbb{Q}}_{\ell}[L^{\prime}_{\mathbb{F}_{q^{2}}}]$ of $\overline{\mathbb{Q}}_{\ell}$-valued functions on $L^{\prime}_{\mathbb{F}_{q^{2}}}$ as in [Gér77, (3) in the proof of Theorem 2.4]. Let $\sigma\colon\mathbb{F}_{q^{2}}\to\mathbb{F}_{q^{2}};\ x\mapsto x^{q}$. Since $F$ stabilizes $L_{\mathbb{F}_{q^{2}}}$ and $L^{\prime}_{\mathbb{F}_{q^{2}}}$, we have $(\rho_{\operatorname{Sp}(\omega^{\prime}),\psi}(F)f)(v)=\eta_{0}(-1)^{n}f((1\otimes\sigma)v)$ for $v\in L^{\prime}_{\mathbb{F}_{q^{2}}}$ by [Gér77, (2.7)]. Hence $\operatorname{Tr}\rho_{\operatorname{Sp}(\omega^{\prime}),\psi}(F)$ equals $\eta_{0}(-1)^{n}$ times the number of the fixed points of $1\otimes\sigma$ on $L^{\prime}_{\mathbb{F}_{q^{2}}}$. ∎

Lemma 7.4**.**

The restriction $\rho_{\operatorname{HSp}(\omega^{\prime}),\psi}|_{\operatorname{HU}(h_{2n}^{\prime})}$ is isomorphic to $\rho_{\operatorname{HU}(h_{2n}^{\prime}),\psi}\otimes(\nu\circ\det)$.

Proof.

This follows from [Gér77, Theorem 3.3(a”)] and Lemma 2.2. ∎

We regard the quadratic character $\operatorname{O}(W,Q)\to\operatorname{O}(W,Q)/\operatorname{U}(W,h_{1})\cong\mathbb{Z}/2\mathbb{Z}\hookrightarrow\overline{\mathbb{Q}}_{\ell}^{\times}$ as a character of $\operatorname{Sp}(V,\omega)\times\operatorname{O}(W,Q)$ naturally, which we denote by $\kappa_{0}$.

Proposition 7.5**.**

We have an isomorphism $\rho_{\operatorname{Sp}(\omega^{\prime}),\psi}|_{\operatorname{Sp}(V,\omega)\times\operatorname{O}(W,Q)}\simeq\omega_{\mathrm{SpO},n,\psi}\otimes\kappa^{n(q-1)/2}_{0}$.

Proof.

By Lemma 7.4, the restriction $\rho_{\operatorname{HSp}(\omega^{\prime}),\psi}|_{\operatorname{HSpU}(h_{2n}^{\prime})}$ is isomorphic to $\pi_{h_{2n}^{\prime},\psi}|_{\operatorname{HSpU}(h_{2n}^{\prime})}$. Hence there exists a character $\eta\in(\mathbb{Z}/2\mathbb{Z})^{\vee}$ such that

[TABLE]

by Schur’s lemma. By Lemma 7.1 (2) and Lemma 7.3, we have $\eta=\kappa^{n(q-1)/2}_{0}$. ∎

7.2 Howe correspondence

Let $\operatorname{Irr}\operatorname{O}(W,Q)$ be the set of isomorphism classes of irreducible representations of $\operatorname{O}(W,Q)$ over $\overline{\mathbb{Q}}_{\ell}$. For $\sigma\in\operatorname{Irr}\operatorname{O}(W,Q)$, we put

[TABLE]

as an $\operatorname{Sp}(V,\omega)$-representation. We call $\sigma\mapsto\Theta_{n}(\sigma)$ the Howe correspondence for $\operatorname{Sp}(V,\omega)\times\operatorname{O}(W,Q)$. This is compatible with the usual Howe correspondence, which is defined for $p\neq 2$, up to an explicit sign as Proposition 7.5.

We identify $\operatorname{O}(W,Q)$ with $\operatorname{U}(W,h_{1})\rtimes(\mathbb{Z}/2\mathbb{Z})$ as before. We set $\mu^{\vee}=\{\chi_{0}\in\mu_{q+1}^{\vee}\mid\chi_{0}^{2}=1\}$. For a pair $(\chi_{0},\kappa)\in\mu^{\vee}\times\{\pm 1\}$, the map $\sigma_{\chi_{0},\kappa}\colon\operatorname{O}(W,Q)\to\overline{\mathbb{Q}}_{\ell}^{\times};\ (x,m)\mapsto\chi_{0}(x)\kappa^{m}$ for $x\in\mu_{q+1}$ and $m\in\mathbb{Z}/2\mathbb{Z}$ is a character. For a character $\chi\in\mu_{q+1}^{\vee}\setminus\mu^{\vee}$, the two-dimensional representation $\sigma_{\chi}=\mathrm{Ind}_{\operatorname{U}(W,h_{1})}^{\operatorname{O}(W,Q)}\chi$ is irreducible. Note that $\sigma_{\chi}\simeq\sigma_{\chi^{-1}}$ as $\operatorname{O}(W,Q)$-representations. Any irreducible representation of $\operatorname{O}(W,Q)$ is isomorphic to the one of these representations. Note that the orthogonal algebraic group defined by $W$ and $Q$ is not connected. We say that an irreducible representation of $\operatorname{O}(W,Q)$ is unipotent if its restriction to $\operatorname{U}(W,h_{1})$ contains a unipotent representation (cf. [AMR96, 3.A]). The unipotent representations in $\operatorname{Irr}\operatorname{O}(W,Q)$ are $\sigma_{1,+}$ and $\sigma_{1,-}$.

Lemma 7.6**.**

We have isomorphisms

[TABLE]

as $\operatorname{Sp}(V,\omega)$-representations.

Proof.

The first isormophism follows from Frobenius reciprocity and the second one is clear. ∎

Lemma 7.7**.**

We have

[TABLE]

Proof.

By [Gér77, (2) in the proof of Corollary 4.5], the number of $\operatorname{Sp}(V,\omega)$-orbits in $V_{\mathbb{F}_{q^{2}}}$ equals $\langle\rho_{\operatorname{HU}(h_{2n}^{\prime}),\psi},\rho_{\operatorname{HU}(h_{2n}^{\prime}),\psi}\rangle_{\operatorname{Sp}(V,\omega)}$ (cf. [Gér77, Theorem 4.5(a)]). We have the orbit $\{0\}$. We say that an element of $V_{\mathbb{F}_{q^{2}}}$ is decomposable if it is written as $v\otimes a$ for some $v\in V$ and $a\in\mathbb{F}_{q^{2}}$. If an element of $V_{\mathbb{F}_{q^{2}}}$ is not decomposable, we say that it is indecomposable. Let $\{e_{1},e_{2}\}$ be the basis of $\mathbb{F}_{q^{2}}$. An element of $V_{\mathbb{F}_{q^{2}}}$ is written as $v_{1}\otimes e_{1}+v_{2}\otimes e_{2}$ with $v_{i}\in V$. The set of the orbits of non-zero decomposable elements is identified with $\mathbb{P}^{1}(\mathbb{F}_{q})$. The set of the orbits of indecomposable elements is identified with

[TABLE]

by $\operatorname{Sp}(V,\omega)(v_{1}\otimes e_{1}+v_{2}\otimes e_{2})\mapsto\omega(v_{1},v_{2})$. Hence, the required assertion follows. ∎

Proposition 7.8**.**

The representations

[TABLE]

are irreducible and distinct as $\operatorname{Sp}(V,\omega)$-representations, except that $\Theta_{1}(\sigma_{1,-})$.

Proof.

We use the notation in Lemma 7.6. Let $\{\pm 1\}$ act on $\mu_{q+1}^{\vee}\setminus\mu^{\vee}$ by $-1\colon\chi\mapsto\chi^{-1}$. We have an isomorphism

[TABLE]

as $\operatorname{O}(W,Q)\times\operatorname{Sp}(V,\omega)$-representations. Recall $\omega_{\mathrm{SpO},n,\psi}|_{\mathrm{HU}(h^{\prime}_{2n})}\simeq\rho_{\mathrm{HU}(h^{\prime}_{2n}),\psi}$ by Lemma 3.3. By Lemma 7.1, Lemma 7.6, Lemma 7.7 and (7.5), the claim follows. ∎

Corollary 7.9**.**

The $\operatorname{Sp}(V,\omega)$-representation $\omega_{\mathrm{SpO},n,\psi}$ contains a unipotent cuspidal representation if and only if $n=2$. In this case, $\omega_{\mathrm{SpO},2,\psi}[1_{\mu_{q+1}}]^{-}$ is unipotent cuspidal.

Proof.

By [Lus77, 8.11. Remarks (1)], $\operatorname{Sp}(V,\omega)$ admits a unipotent cuspidal representation if and only if $n=s(s+1)$ with some integer $s\geq 1$. Every $\operatorname{Sp}(V,\omega)$-subrepresentation of $\omega_{\mathrm{SpO},n,\psi}$ in Lemma 7.1 is irreducible by Lemma 7.6 and Proposition 7.8. By [Lus77, (8.11.1)] and Lemma 7.1, if $s\geq 2$, the $p$-adic valuation of the dimension of the unipotent cuspidal representation is greater than the dimension of any irreducible representation in $\omega_{\mathrm{SpO},n,\psi}$. Hence $\omega_{\mathrm{SpO},n,\psi}$ does not contain any unipotent cuspidal representation if $s\geq 2$. Therefore, for $\omega_{\mathrm{SpO},n,\psi}$ to contain a unipotent cuspidal representation, we must have $n=2$. Conversely, we show that $\omega_{\mathrm{SpO},2,\psi}[1_{\mu_{q+1}}]^{-}$ is unipotent cuspidal. This is irreducible by Proposition 7.8 and of dimension $q(q-1)^{2}/2$ by Lemma 7.1. This is isomorphic to $\theta_{5}$ in [Eno72, Table IV-2 in Appendix] if $q$ is even and $\theta_{10}$ in [Sri68, §8] if $q$ is odd, which is the unique irreducible representation of dimension $q(q-1)^{2}/2$. These are unipotent cuspidal by [Lus77, Theorem 8.2 and (8.11.1)]. ∎

7.3 Relation with Lusztig induction

We set

[TABLE]

Recall that

[TABLE]

We consider the Frobenius endomorphism $F\colon G\to G;\ g=(x_{i,j})\mapsto(x_{i,j}^{q})$. Let $\mathbf{0}_{n-1,1}$ denote the zero $(n-1)\times 1$-matrix. We consider the parabolic subgroup

[TABLE]

By using (7.6), we have

[TABLE]

We consider the Levi subgroup

[TABLE]

of $P_{0}$ and the unipotent radical

[TABLE]

of $P_{0}$. We have $\dim U_{P_{0}}=2n-1$. Let

[TABLE]

We take $g_{w}\in G(\overline{\mathbb{F}}_{q})$ such that $g_{w}^{-1}F(g_{w})=w$ by Lang’s theorem.

We put $P=g_{w}P_{0}g_{w}^{-1}$, $M=g_{w}M_{0}g_{w}^{-1}$ and $M_{0}^{w}=\{m\in M_{0}\mid F(m)=w^{-1}mw\}$. We put

[TABLE]

Let $\pi_{P_{0}}^{\prime}\colon\widetilde{Y}_{P_{0}}^{\prime}\to Y_{P_{0}}^{\prime}$ be the natural morphism. Then we have the isomorphisms

[TABLE]

which give the commutative diagram

[TABLE]

Therefore $\pi_{P_{0}}^{\prime}$ is an $M_{0}^{w}$-torsor by Lemma 5.1.

Let $L_{0}=\overline{\mathbb{F}}_{q}e_{1}\subset V_{\overline{\mathbb{F}}_{q}}=\overline{\mathbb{F}}_{q}^{2n}$. Let $s\colon V_{\overline{\mathbb{F}}_{q}}\times V_{\overline{\mathbb{F}}_{q}}\to\overline{\mathbb{F}}_{q};\ (v,v^{\prime})\mapsto{}^{\mathrm{t}}\!vJv^{\prime}$. We have the isomorphism $f\colon G/P_{0}\xrightarrow{\sim}\mathbb{P}(V_{\overline{\mathbb{F}}_{q}});\ gP_{0}\mapsto gL_{0}$. For $L\in\mathbb{P}(V_{\overline{\mathbb{F}}_{q}})$, we put

[TABLE]

We have

[TABLE]

Let $Y^{\prime}=\{L\in\mathbb{P}(V_{\overline{\mathbb{F}}_{q}})\mid F(L)\nsubseteq L^{\perp}\}$. For $g\in G$, the condition $F(gL_{0})\nsubseteq(gL_{0})^{\perp}=gL_{0}^{\perp}$ is equivalent to $g^{-1}F(g)L_{0}\nsubseteq L_{0}^{\perp}$. The morphism

[TABLE]

is well-defined since we have $g^{-1}F(g)L_{0}=p_{1}wp_{2}L_{0}=p_{1}wL_{0}\nsubseteq p_{1}L_{0}^{\perp}=L_{0}^{\perp}$ where $g^{-1}F(g)L_{0}=p_{1}wp_{2}$ for $p_{1},p_{2}\in P_{0}$.

Lemma 7.10**.**

The morphism $\varphi_{0}^{\prime}\colon Y_{P_{0}}^{\prime}\to Y^{\prime}$ is an isomorphism.

Proof.

Let $X_{0}=\{g=(g_{i,j})\in G\mid g_{n+1,1}\neq 0\}$. We show that $P_{0}wP_{0}=X_{0}$. We easily check $P_{0}wP_{0}\subset X_{0}$. Let $P_{0}$ act on $X_{0}$ by right multiplication. Then we have a natural map $\phi\colon P_{0}wP_{0}/P_{0}\to X_{0}/P_{0}$. By $P_{0}\cap wP_{0}w^{-1}=M_{0}$, we have the isomorphisms

[TABLE]

We can check that the composite $\phi_{2}\circ\phi\circ\phi_{1}$ is an isomorphism by (7.8). Thus the claim follows.

Let $L\in Y^{\prime}$. We take $g\in G$ such that $L=gL_{0}$. By $g^{-1}F(g)L_{0}\nsubseteq L_{0}^{\perp}$, we have $g^{-1}F(g)\in X_{0}=P_{0}wP_{0}$. Hence, we obtain the required assertion. ∎

Let $S_{2n}^{\prime}$ be the projective smooth variety defined by $\sum_{i=1}^{n}(x_{i}^{q}y_{i}-x_{i}y_{i}^{q})=0$ in $\mathbb{P}_{\mathbb{F}_{q}}^{2n-1}$. Let $Y_{2n}^{\prime}=\mathbb{P}_{\mathbb{F}_{q}}^{2n-1}\setminus S_{2n}^{\prime}$. The canonical isomorphism $\mathbb{P}(V_{\overline{\mathbb{F}}_{q}})\simeq\mathbb{P}^{2n-1}_{\overline{\mathbb{F}}_{q}}$ induces an isomorphism $Y^{\prime}\simeq Y_{2n,\overline{\mathbb{F}}_{q}}^{\prime}$. Let $\varphi^{\prime}\colon Y_{P_{0}}^{\prime}\to Y_{2n,\overline{\mathbb{F}}_{q}}^{\prime}$ be the composite of $\varphi_{0}^{\prime}\colon Y_{P_{0}}^{\prime}\to Y^{\prime}$ and the isomorphism $Y^{\prime}\simeq Y_{2n,\overline{\mathbb{F}}_{q}}^{\prime}$.

Let $\mathbf{x}=(x_{1}\ \cdots\ x_{n})$, $\mathbf{y}=(y_{1}\ \cdots\ y_{n})$, $\mathbf{z}=(z_{1}\ \cdots\ z_{n})$ and $\mathbf{w}=(w_{1}\ \cdots\ w_{n})$. We write as

[TABLE]

We have

[TABLE]

By $g^{-1}F(g)\in wU_{P_{0}}$, we have $F(\mathbf{x})=-\mathbf{z}$ and $F(\mathbf{y})=-\mathbf{w}$. By ${}^{\mathrm{t}}\!AD-{}^{\mathrm{t}}\!CB=E_{n}$, we have $\mathbf{y}{}^{\mathrm{t}}\!F(\mathbf{x})-\mathbf{x}{}^{\mathrm{t}}\!F(\mathbf{y})=1$. Let $\widetilde{Y}_{2n}^{\prime}$ be the affine smooth variety over $\mathbb{F}_{q}$ defined by $\sum_{i=1}^{n}(x_{i}^{q}y_{i}-x_{i}y_{i}^{q})=1$ in $\mathbb{A}_{\mathbb{F}_{q}}^{2n}$. We have the morphisms

[TABLE]

We identify $\operatorname{Sp}_{2n-2}(\mathbb{F}_{q})$ with the subgroup $\{1\}\times\operatorname{Sp}_{2n-2}(\mathbb{F}_{q})\subset M_{0}^{w}$. Then $\widetilde{\pi}^{\prime}$ induces a morphism

[TABLE]

We have the commutative diagram

[TABLE]

where the vertical morphisms are natural ones. Since $f_{Y^{\prime}_{P_{0}}}$ and $f_{Y^{\prime}_{2n}}$ are $\mu_{q+1}$-torsors and $\varphi^{\prime}$ is an isomorphism, $\widetilde{\varphi}^{\prime}$ is also an isomorphism.

Remark 7.11**.**

The results in Subsection 4.2 hold even if we replace $n$, $\pi\colon\mathbb{A}_{\mathbb{F}_{q}}^{n}\to\mathbb{A}_{\mathbb{F}_{q}}^{1}$, $Y_{n}$ etc. by $2n$, $\pi^{\prime}\colon\mathbb{A}_{\mathbb{F}_{q}}^{2n}\to\mathbb{A}_{\mathbb{F}_{q}}^{1}$, $Y^{\prime}_{2n}$ etc.

Proposition 7.12**.**

Let $\chi\in\mu_{q+1}^{\vee}$. We have

[TABLE]

as characters of $G^{F}$.

Proof.

The first equality follows from Lemma 7.6. The equality between the first one and the third one follows from Remark 7.11 (*cf. *Proposition 4.8) and (5.1), since $\widetilde{\varphi}^{\prime}$ is an isomorphism. ∎

Corollary 7.13**.**

Let $\sigma\in\operatorname{Irr}\operatorname{O}(W,Q)$. Assume that $\sigma\neq\sigma_{1,-}$ if $n=1$. Then the $\operatorname{Sp}(V,\omega)$-representation $\Theta_{n}(\sigma)$ is unipotent if and only if $\sigma$ is unipotent.

Proof.

The claim follows from (5.2) and Proposition 7.12. ∎

Remark 7.14**.**

It is also possible to show Corollary 7.13 using results in [GT04] if $p=2$, and results in [TZ96] and [Ngu10] if $p\neq 2$. The proof in this paper is geometric and does not depend on the parity of $p$.

Remark 7.15**.**

We assume that $q$ is odd. If $q$ is large enough as in [AM93, (3.11)(2)], then Corollary 7.13 follows from [AM93, Theorem 3.5(2)].

Remark 7.16**.**

We use the same notation as Remark 6.4. We have $\epsilon_{G}=(-1)^{n}$ and $\epsilon_{M}=(-1)^{n-1}$. As in Remark 6.4, we have

[TABLE]

for any $\chi\in\mu_{q+1}^{\vee}$. This is compatible with Lemma 7.1 and Proposition 7.12.