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

We give a geometric construction of the Heisenberg–Weil representation of a ﬁnite unitary group by the middle ´etale 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 (Sp 2 n , O − 2 ) over any ﬁnite ﬁeld including characteristic two. As an application, we show that unipotency is preserved under the Howe correspondence.


Introduction
For a reductive dual pair (G, G ′ ) over a field (cf. [How79,§5]), by regarding the Weil representation as a representation of G × G ′ and decomposing it into irreducible components, we have a correspondence from irreducible representations of G to representations of G ′ , which is called the Howe correspondence for (G, G ′ ). Let q be a power of a prime number p. If q is odd, Weil representations of symplectic groups over F q are studied in [Sai72] and [How73] after [Wei64]. Weil representations of general linear groups over F q and unitary groups for the extension F q 2 /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 U n (F q ) of degree n is regarded as a subgroup of an automorphism group of a certain Heisenberg group H(h n ) determined by a hermitian form h n on F n q 2 , which we call a unitary Heisenberg group (cf. [Gér77,Lemma 3.3] and §2.1). For a non-trivial character ψ of the center of H(h n ), there exists a unique irreducible representation ρ H(hn),ψ of H(h n ) which contains ψ. In [Gér77, Theorem 3.3], an irreducible representation ρ HU(hn),ψ of the semidirect product H(h n ) ⋊ U n (F q ) extending ρ H(hn),ψ is constructed, which we call the Heisenberg-Weil representation. The Weil representation is defined to be the restriction of ρ HU(hn),ψ 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 H(h n ) ⋊ U n (F q ). This algebraic variety is isomorphic to the affine smooth variety defined by a q + a = n i=1 x q+1 i in A n+1 Fq , which we denote by X n . Let ℓ = p be a prime number. We show the following: Theorem. We have an isomorphism H n c (X n,Fq , Q ℓ )[ψ] ≃ ρ HU(hn),ψ as representations of H(h n ) ⋊ U n (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 ℓ 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 . Using the Frobenius action coming from the rationality of X ′ 2n , we can construct a representation of U 2n (F q ) ⋊ Gal(F q 2 /F q ) as the middle ℓ-adic cohomology of the variety. We write W n for the obtained U 2n (F q ) ⋊ Gal(F q 2 /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 symplecticorthogonal 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 (Sp 2n , O ± 2 ) and construct a representation of the product group Sp 2n (F q ) × O ± 2 (F q ). In this paper, we give a geometric construction of the Howe correspondence for the reductive dual pair (Sp 2n , O − 2 ) as follows. The group O − 2 (F q ) is isomorphic to the dihedral group D 2(q+1) . Hence, we can identify O − 2 (F q ) with U 1 (F q ) ⋊ Gal(F q 2 /F q ). We have natural homomorphisms Sp 2n (F q ) ֒→ U 2n (F q ) and U 2n (F q ) × U 1 (F q ) → U 2n (F q ) (cf. [Sri79,§1]). Hence, we have a homomorphism Inflating W n under this homomorphism, we obtain a representation of Sp 2n (F q )×O − 2 (F q ). Using this representation, we define the Howe correspondence for (Sp 2n , O − 2 ). We note that the Frobenius action coming from the rationality of X ′ 2n plays an important role in the construction of the Howe correspondence. Since the construction is geometric, we can relate the representation of Sp 2n (F q ) × O − 2 (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 ℓ-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 Sp 2n (F q ) × O − 2 (F q ) and define the Howe correspondence for (Sp 2n , O − 2 ).
We show that this Howe correspondence is compatible with the usual one, which is defined if p = 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 ℓ cohomology and a mod ℓ Howe correspondence using results in this paper.

Notation
For a scheme X over a field F and a field extension 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 λ and a finite-dimensional G-representation ρ, let ρ[λ] denote the λ-isotypic part of ρ.
2 Heisenberg-Weil representation for unitary group

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 F q 2 -vector space with a nondegenerate ε-hermitian form h, where ε ∈ {±1}. We put We regard H(V, h) as a group with multiplication (2.1) We put F q,ε = {a ∈ F q 2 | a + εa q = 0}. We sometimes abbreviate ±1 as ±. The center Let ℓ = p be a prime number. For each non-trivial character ψ of Z over Q ℓ , there exists a unique irreducible representation of H(V, h) whose restriction to Z contains ψ by the Stone-von Neumann theorem. We denote by ρ H(V,h),ψ the unique irreducible representation of H(V, h) containing ψ. The dimension of ρ H(V,h),ψ equals the square root of the index [H(V, h) : Z].
Let U(V, h) denote the isometry group of (V, h). Then, U(V, h) acts on H(V, h) by (v, a) → (gv, a) for g ∈ U(V, h) and (v, a) ∈ H(V, h). We put Remark 2.1. Assume that h is hermitian. We take an element ξ ∈ F q 2 such that ξ q−1 = −1. Then is a skew-hermitian form. We have an isomorphism H(V, h) Lemma 2.2. There is the unique irreducible representation ρ HU(h),ψ of HU(h) which is characterized by Proof. This follows from [Gér77, Theorem 3.3 and Theorem 4.9.2] and Remark 2.1.
We call the representation ρ HU(h),ψ in Lemma 2.2 the Heisenberg-Weil representation of HU(h) for ψ (cf. [Leh74, Proposition 3.1]). We put which is independent of ψ by Lemma 2.2. The representation ω U(V,h) is called the Weil representation of U(V, h).

Cohomology of a curve
For a finite abelian group A, we simply write A ∨ for the character group Hom(A, Q × ℓ ). Let Λ ∈ {Q ℓ , F ℓ }. For a separated and of finite type scheme Y over F q which admits a left action of a finite group G, let G act on H i c (Y Fq , Λ) as (g * ) −1 for g ∈ G. We write A i for an i-dimensional affine space over F q . We write G m for Spec F q [t ±1 ].
For ψ ∈ Hom(F q,ε , Λ × ), let L ψ denote the rank one sheaf on A 1 Fq obtained as the pushforward via ψ −1 ∈ Hom(F q,ε , Λ × ) of the F q,ε -torsor over A 1 Fq = Spec F q [t] defined by z q + εz = t (cf. [Del77, Sommes trig., Définition 1.7]). For a variety X over F q and a regular function f : X → A 1 Fq , let L ψ (f ) denote the pull-back of L ψ by f . We put For χ ∈ Hom(µ q+1 , Λ × ), let K Gm,χ denote the Kummer sheaf on G m,F q 2 obtained as the pushforward via χ −1 of the µ q+1 -torsor over G m,F q 2 = Spec F q 2 [t ±1 ] defined by y q+1 = t. Let C be the affine smooth curve defined by z q + z = x q+1 over F q 2 . The group F q,+ acts on C by z → z + a for a ∈ F q,+ .
Let ψ ∈ Hom(F q,+ , Λ × )\{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].
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 Ocohomology of a smooth affine curve over F q is torsion free. The third assertion follows from the second assertion.

Geometric construction
Let n be a positive integer. We write an element v ∈ F n q as v = (v k ). We consider the nondegenerate hermitian form on F n q 2 defined by We put H(h n ) = H(F n q 2 , h n ), U(h n ) = U(F n q 2 , h n ). Let X n be the affine smooth variety over F q 2 defined by and U(h n ) act on X n by where we regard x = (x k ) as a column vector. The variety X n admits an action of HU(h n ). Let F q,+ act on X n by z → z + a for a ∈ F q,+ . The cohomology of a variant of X n is studied in [Dud09, Lemma 3.6]. We consider the morphism (2.6) Since we have a cartesian diagram for i ≥ 0 by using the proper base change theorem and taking ψ-parts.
Lemma 2.4. Assume that Λ = Q ℓ and q = 2. We have Proof. Let e 1 , e 2 be a basis of H 1 c (C F q , Q ℓ )[ψ]. Using the morphism of sheaves on A 2 Fq . Hence we have an isomorphism as 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 n c (X n,Fq , Q ℓ )[ψ] is isomorphic to ρ H(hn),ψ as H(h n )-representations by the Stone-von Neumann theorem. We write det for the composite HU(h n ) Assume that q is odd. By [Gér77, (1) in the proof of Theorem 3.3], the finite special unitary group SU n (F q ) is perfect except for (n, q) = (2, 3). Hence, if (n, q) = (2, 3), any character of U(h n ) factors through det. Even if (n, q) = (2, 3), any character of U(h n ) does by [Enn63, the table in p.28].
Proposition 2.6. Let the notations be as in Lemma 2.3.
is regarded as a mod ℓ version of a Heisenberg-Weil representation of a unitary group.

Another coordinate
We give a construction of the Heisenberg-Weil representation of a finite unitary group of even degree using another coordinate. Let ψ ∈ F ∨ q \ {1} in this section.

Cohomology of a surface
Let X be the affine smooth surface over F q defined by Let Fr q ∈ Gal(F q /F q ) be the geometric Frobenius automorphism defined by x → x q −1 for x ∈ F q . When we consider a closed subscheme of a variety, we suppose that it is equipped with the reduced scheme structure.
Let F × q 2 ⋊ Fr Z q be the semidirect product under the natural action. We sometimes Lemma 3.1. We have an isomorphism 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 → S; (w, x, y) → (w, x q , y). Hence we have an where we use the projection formula at the fourth isomorphism.
. Hence, we obtain the claim.

Construction using another coordinate
Let n be a positive integer. Let V = F 2n q . We consider the skew-hermitian form on Let X ′ 2n be the affine smooth variety over F q defined by 1. Further, f and z → ξz gives an isomorphism X 2n ≃ X ′ 2n,F q 2 over F q 2 , which is compatible with group actions under the above isomorphisms.
is trivial. Proof. This follows from Lemma 3.1 and the Künneth formula.

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 GL n over F q . We consider the Frobenius endomorphism F : The Weyl group W n is isomorphic to the symmetric group S n . The set of conjugacy classes of S n is identified with the set of partitions of n, which we denote by Λ n . We have the natural bijection Let T ∈ T n and θ ∈ (T F ) ∨ . Let R Gn T (θ) denote the Deligne-Lusztig character in the notation of [Lus76,p.204] and [Lus78, Corollary 2.4]. Let χ λ denote the irreducible character of S n corresponding to λ ∈ Λ n normalized such that χ λ is the sign representation if λ = (1 n ). For λ, ρ ∈ Λ n , let χ λ ρ denote the value of χ λ at the class corresponding to ρ. Let z ρ be the cardinality of the centralizer of the class in S n corresponding to ρ. For λ ∈ Λ n , we define a class function ψ λ on G F n by We follow the definition of unipotency in [DL76, Definition 7.8]. By [LS77,§2], the set {ψ λ } λ∈Λn equals the set of all unipotent characters of G F n up to sign.

Geometric relation
Let Λ ∈ {Q ℓ , F ℓ } and ψ ∈ Hom(F q,+ , Λ × ) \ {1}. Let π be as in (2.6). We put U = π −1 (G m,Fq ) and Z = π −1 (0). Then U F q 2 and Z F q 2 admit left actions induced by the natural action of U n (F q ) on A n F q 2 . Let L ψ be as in §2.2. In this section, all maps between U n (F q )-representations are U n (F q )-equivariant.
Lemma 4.1. We have a long exact sequence Proof. We have H i c (A n , π * L ψ ) = 0 for i = n by Theorem 2.5 and the Künneth formula. We have (π * L ψ )| Z = Λ. Hence the assertion follows.
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 n i=1 x q+1 Let Y n be the affine smooth variety defined by n i=1 y q+1 Let µ q+1 act on Y n,F q 2 by (y i ) 1≤i≤n → (ζy i ) 1≤i≤n for ζ ∈ µ q+1 . Then Y n,F q 2 is a µ q+1 -torsor over Y n,F q 2 . For χ ∈ µ ∨ q+1 , let K Yn,χ denote the smooth Λ-sheaf on Y n,F q 2 associated to f Yn and χ −1 . We note that K Yn,χ is defined over Y n if χ 2 = 1.
In the sequel, we assume that q + 1 is invertible in Λ.
Lemma 4.3. We have an isomorphism H i c (U Fq , π * L ψ )[χ] ≃ H i−1 c (Y n,Fq , K Yn,χ ) as U n (F q )representations for χ ∈ Hom(µ q+1 , Λ × ) and any integer i. If Λ = Q ℓ and χ 2 = 1, then we have an isomorphism as representations of U n (F q ) and Gal(F q /F q ), where χ ′ is the character of F × q of the same order as χ and δ χ ′ ,ψ denotes the unramified character of Gal(F q /F q ) such that the q-th power geometric Frobenius element acts by the Gauss sum τ (χ ′ , ψ) defined in [Del77, Sommes trig., (4.1.1)].
Proof. We consider the affine smooth variety over F q . The space U F q 2 admits a left action induced by the natural action of U n (F q ) on A n F q 2 . Let µ 2 q+1 act on U F q 2 by for (ζ 1 , ζ 2 ) ∈ µ 2 q+1 . Clearly, the action of µ 2 q+1 on U F q 2 is free. We have a natural identification We have the morphism q+1 . The isomorphism ϕ induces an isomorphism By taking the quotients of the both sides of ϕ by µ 2 q+1 /H and µ 2 q+1 /φ(H), we obtain an isomorphism ϕ : be the first projection and the second projection respectively. We have the commutative diagram For smooth Λ-sheaves F on Y n,F q 2 and G on G m,F q 2 , let F ⊠ G denote the sheaf pr * 1 F ⊗ pr * 2 G on Y n,F q 2 × G m,F q 2 . Let K Gm,χ be as in §2.2. We identify as µ q+1 ∼ − → µ 2 q+1 /φ(H); ζ → (ζ, 1). We have by the Künneth formula. Let K ′ χ denote the smooth Λ-sheaf on U F q 2 /µ q+1 associated to χ −1 and the µ q+1 -torsor U F q 2 → U F q 2 /µ q+1 . We identify as µ q+1 ≃ µ 2 q+1 /H; ζ → (ζ, 1). We have K ′ χ ≃ ϕ * (K Yn,χ ⊠ K Gm,χ −1 ). By the Künneth formula and (2.4), we have isomorphisms The last claim follows from the above arguments and [Del77, Sommes trig., Proposition 4.2 (ii)].
We study the cohomology of Z Fq . Let Z 0 ⊂ Z be the open subscheme defined by (x 1 , . . . , x n ) = 0 = (0, . . . , 0). Now, we regard Z 0 as a closed subscheme of A n x n ] is a G m,Fq -bundle. By restricting this to Z 0 , we have a G m,Fq -bundle The morphism π 0 factors as where Z 0 /µ q+1 denotes the quotient of Z 0 under the natural action of the group scheme of (q + 1)-st roots of unity over F q . (2) We have (3) The action of µ q+1 on H i c (Z 0 Fq , Λ) is trivial for any i. Proof. Since the morphism π 0 is a G m -bundle and any G 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)].
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.
(1) We have a Gal(F q /F q )-equivariant isomorphism (2) Assume that n ≥ 2. We have a Gal(F q /F q )-equivariant isomorphism
We show the second assertion. By Lemma 4.3, we have isomorphisms (4.9) Consider the composite 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 U n (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.

Lusztig induction
We recall Lusztig induction in [Lus76]. Let G be a connected reductive group defined over F q with an 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

The morphism
π P : Y P → Y P ; g → gM is known to be an M F -torsor (cf. [DM14, §7.3]). We give a proof of it.
Lemma 5.1. The morphism π P is an M F -torsor.
For a representation ρ of M F , we consider a virtual G F -module Assume that M = P ∩ F (P ). Then we have Assume that there is a decomposition M = T × M ′ as algebraic groups over F q where T is a torus and M ′ is a reductive group. Let χ ∈ (T F ) ∨ . Let K Y P ,χ be the smooth Q ℓ -sheaf on Y P associated to χ −1 under the natural projection M F → T F and π P . We have (5.1) By [DL76, Corollary 7.14], we have as characters of M ′F , where the summation runs over all M ′F -conjugacy classes of Fstable maximal tori T ′ of M ′ . We have as characters of M F . By applying R G M ⊂P to this, we have as characters of G F by [Lus76, Corollary 5]. Therefore we have in the rest of this paper.

Unipotency
We use notation in Subsection 4.1. Let P ⊂ G n be the parabolic subgroup consisting of matrices (x i,j ) such that x i,1 = 0 for 2 ≤ i ≤ n. Let M be the F -stable Levi subgroup of P . Note that P is not stable by F . We have M ≃ G 1 × G n−1 diagonally embedded in G n . We have G F n = U n (F q ) and . Hence, we have the morphism π n : Y P → Y n,Fq ; x → (x i,1 ) 1≤i≤n . This induces a morphism ϕ n : Y P / U n−1 (F q ) → Y n,Fq . We have an identification Y P /M F ≃ Y P by Lemma 5.1. As shown in [HM78,p.259], π n induces an isomorphism ϕ n : where the vertical morphisms are natural ones. Since f Y P and f Yn are µ q+1 -torsors and ϕ n is an isomorphism, ϕ n is also an isomorphism.
Proposition 6.1. We have as characters of G F n . Proof. Since ϕ n is an isomorphism, the claim follows from Proposition 4.8 and (5.1).
In the sequel, we write ω Un for ω U(F n q 2 ,hn) 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.
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 ǫ G be the F q -rank of a linear algebraic group G over F q . For any integer m ≥ 1, let m p be the largest power of p dividing m and m p ′ = m/m p . We go back to our situation. We have ǫ Gn = [n/2] (cf. [DM91, §15.1]). Hence, we have ǫ Gn ǫ M = (−1) n−1 . By [DM91, Proposition 12.17], we have for any χ ∈ µ ∨ q+1 . This is compatible with Lemma 4.2 and Proposition 6.1.
Proof. By (2.7), we have isomorphisms , where we use the Künneth formula at the second isomorphism. By taking the χ-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 ω be a symplectic form on V obtained by the restriction of h ′ 2n to V . By regarding V F q 2 as a vector space over F q , we consider the symplectic form associated to h ′ 2n : Let Sp(V, ω) and Sp(V F q 2 , ω ′ ) be the isometry groups of (V, ω) and (V F q 2 , ω ′ ), respectively. We have Sp(V, ω) ⊂ U(V F q 2 , h ′ 2n ) ⊂ Sp(V F q 2 , ω ′ ). We put W = F q 2 . Let h 1 : W × W → F q 2 ; (x, y) → x q y. We have U(W, h 1 ) = µ q+1 . Under the natural isomorphism V ⊗ Fq W ≃ V F q 2 , the skew-hermitian form ω ⊗ h 1 corresponds to h ′ 2n . This induces a morphism We have natural actions of Gal(F q 2 /F q ) ≃ Z/2Z on U(W, h 1 ) and U(V F q 2 , h ′ 2n ). Then the homomorphism (7.1) extends naturally to We regard this as an element of Sp(V F q 2 , ω ′ ). Then we have the injective homomorphism for k ∈ Z/2Z, which is well-defined by Lemma 3.4. By the natural action of Gal(F q 2 /F q ) on V F q 2 and F q 2 , we have a natural action of Gal(F q 2 /F q ) ≃ Z/2Z on HU(h ′ 2n ). We write π h ′ 2n ,ψ for the HU(h ′ 2n ) ⋊ (Z/2Z)-representation H 2n c (X ′ 2n,Fq , Q ℓ )(n) [ψ]. Then the action of 1 ∈ Z/2Z induces an isomorphism between π h ′ 2n ,ψ [χ] and π h ′ 2n ,ψ [χ −1 ] for χ ∈ µ ∨ q+1 \{1}. We consider the quadratic form Q : W → F q ; x → x q+1 . We set Since Q(x) = h 1 (x, x), we have a natural inclusion U(W, h 1 ) ֒→ O(W, Q). We regard F W : W → W ; x → x q as an element of O(W, Q). Then we have the isomorphism since O(W, Q) is isomorphic to the dihedral group D 2(q+1) by [KL90, Proposition 2.9.1]. We identify U(W, h 1 ) ⋊ (Z/2Z) with O(W, Q) by the above isomorphism. We consider the symmetric form s 1 : Hence, we have the natural map O(W, Q) ֒→ O(W, s 1 ), which is an isomorphism if q is odd. We have a natural homomorphism Sp(V, ω) × O(W, s 1 ) → Sp(V F q 2 , ω ′ ), since ω ⊗ s 1 corresponds to ω ′ under the natural isomorphism V ⊗ Fq W ≃ V F q 2 . We consider the composite By the construction, we obtain the commutative diagram where the going up arrows are natural homomorphisms. We put ω SpO,n,ψ = π h ′ 2n ,ψ | Sp(V,ω)×O(W,Q) . For a character χ 0 ∈ µ ∨ q+1 such that χ 2 0 = 1, the χ 0 -isotypic part ω SpO,n,ψ [χ 0 ] is stable under the action of Z/2Z. For κ ∈ {±1}, let ω SpO,n,ψ [χ 0 ] κ denote the κ-eigenspace of 1 ∈ Z/2Z on ω SpO,n,ψ [χ 0 ]. Let ν be the quadratic character of µ q+1 if p = 2.
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.
Proof. The first isormophism follows from Frobenius reciprocity and the second one is clear.
Lemma 7.7. We have Proof. By [Gér77,(2) in the proof of Corollary 4.5], the number of Sp(V, ω)-orbits in V F q 2 equals ρ HU(h ′ 2n ),ψ , ρ HU(h ′ 2n ),ψ Sp(V,ω) (cf. [Gér77, Theorem 4.5(a)]). We have the orbit {0}. We say that an element of V F q 2 is decomposable if it is written as v ⊗ a for some v ∈ V and a ∈ F q 2 . If an element of V F q 2 is not decomposable, we say that it is indecomposable. Let {e 1 , e 2 } be the basis of F q 2 . An element of V F q 2 is written as v 1 ⊗ e 1 + v 2 ⊗ e 2 with v i ∈ V . The set of the orbits of non-zero decomposable elements is identified with P 1 (F q ). The set of the orbits of indecomposable elements is identified with . Hence, the required assertion follows.
Let π ′ P 0 : Y ′ P 0 → Y ′ P 0 be the natural morphism. Then we have the isomorphisms Y P → Y ′ P 0 ; g → gg w , Y P → Y ′ P 0 ; gP → gg w P 0 which give the commutative diagram Therefore π ′ P 0 is an M w 0 -torsor by Lemma 5.1.
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 ϕ ′ is an isomorphism.