2 Definitions and notation

In this article, p will always denote an odd prime number and q will be a power of p. If M is a matrix with coefficients in a field of characteristic p, then $M^{(q)}$ denotes the matrix M with entries raised to the power q. The trivial representation of a given group will be denoted $\mathbf 1$ . Given a reductive group G with Levi complement L, the associated Harish-Chandra induction and restriction functors are denoted $\mathrm R_L^G$ and ${}^*\mathrm R_L^G$ , respectively.

3 Deligne–Lusztig varieties and the variety $S_{\theta }$

Let $\mathbf G$ be a connected reductive group over $\mathbb F$ , together with a split $\mathbb F_q$ -structure given by a geometric Frobenius morphism F. For $\mathbf H$ any F-stable subgroup of $\mathbf G$ , we write $H := \mathbf H^F$ for its group of $\mathbb F_q$ -rational points. Let $(\mathbf T,\mathbf B)$ be a pair consisting of a maximal F-stable torus $\mathbf T$ contained in an F-stable Borel subgroup $\mathbf B$ . Let $(\mathbf W,\mathbf S)$ be the associated Coxeter system, where $\mathbf W = \mathrm {N}_{\mathbf G}(\mathbf T)/\mathbf T$ . Since the $\mathbb F_q$ -structure on $\mathbf G$ is split, the Frobenius F acts trivially on $\mathbf W$ . For $I\subset \mathbf S$ , let $\mathbf P_I, \mathbf U_I, \mathbf L_I$ be, respectively, the standard parabolic subgroup of type I, its unipotent radical and its unique Levi complement containing $\mathbf T$ . Let $\mathbf W_I$ be the subgroup of $\mathbf W$ generated by I. For $\mathbf P$ any parabolic subgroup of $\mathbf G$ , the associated coarse Deligne–Lusztig variety is

$$ \begin{align*}X_{\mathbf P} := \{g\mathbf P\in \mathbf G/\mathbf P\,|\,g^{-1}F(g)\in \mathbf P F(\mathbf P)\}.\end{align*} $$

We say that $X_{\mathbf P}$ is a (parabolic) Deligne–Lusztig variety when in addition the parabolic subgroup $\mathbf P$ contains an F-stable Levi complement. Note that $\mathbf P$ itself need not be F-stable. We may give an equivalent definition using the Coxeter system $(\mathbf W,\mathbf S)$ . For $I\subset \mathbf S$ , let ${}^I\mathbf W^{I}$ be the set of elements $w\in \mathbf W$ which are I-reduced-I. For $w\in \, ^I\mathbf W^{I}$ , the associated coarse Deligne–Lusztig variety is

$$ \begin{align*}X_I(w) := \{g\mathbf P_I\in \mathbf G/\mathbf P_I \,|\, g^{-1}F(g)\in \mathbf P_Iw\mathbf P_I\}.\end{align*} $$

The variety $X_I(w)$ is a parabolic Deligne–Lusztig variety when $w^{-1}Iw = I$ , and it is defined over $\mathbb F_q$ . The dimension is given by $\dim X_I(w) = l(w)+ \dim \mathbf G/\mathbf P_{I\cap wIw^{-1}} - \dim \mathbf G/\mathbf P_I$ where $l(w)$ denotes the length of w with respect to $\mathbf S$ . Let $\theta \geq 0$ , and let V be a $2\theta $ -dimensional $\mathbb F_{q}$ -vector space equipped with a non-degenerate symplectic form $(\cdot ,\cdot ):V\times V \to \mathbb F_{q}$ . Fix a basis $(e_1,\dots ,e_{2\theta })$ in which $(\cdot ,\cdot )$ is described by the matrix

$$ \begin{align*}\Omega := \begin{pmatrix} 0 & A_{\theta} \\ -A_{\theta} & 0 \end{pmatrix},\end{align*} $$

where $A_{\theta }$ denotes the matrix having $1$ on the anti-diagonal and $0$ everywhere else. If k is a field extension of $\mathbb F_q$ , let $V_k := V \otimes _{\mathbb F_q} k$ denote the scalar extension to k equipped with its induced k-symplectic form $(\cdot ,\cdot )$ . Let $\tau :V_k \xrightarrow {\sim } V_k$ denote the map $\mathrm {id}\otimes \sigma $ , where $\sigma (x) := x^q$ . If $U \subset V_k$ , let $U^{\perp }$ denote its orthogonal. We consider the finite symplectic group $\mathrm {Sp}(V,(\cdot ,\cdot )) \simeq \mathrm {Sp}(2\theta ,\mathbb F_q)$ , where the RHS is defined with respect to $\Omega $ . It can be identified with $G = \mathbf G^F$ where $\mathbf G$ is the symplectic group $\mathrm {Sp}(V_{\mathbb F},(\cdot ,\cdot )) \simeq \mathrm {Sp}(2\theta ,\mathbb F)$ and $F(M) := M^{(q)}$ . Let $\mathbf T \subset \mathbf G$ be the maximal torus of diagonal symplectic matrices, and let $\mathbf B \subset \mathbf G$ be the Borel subgroup of upper-triangular symplectic matrices. The Weyl system of $(\mathbf T,\mathbf B)$ is identified with $(W_{\theta },\mathbf S)$ where $W_{\theta }$ is the finite Coxeter group of type $B_{\theta }$ and $\mathbf S = \{s_1,\dots ,s_{\theta }\}$ is the set of simple reflexions. They satisfy the following relations:

$$ \begin{align*} (s_{\theta}s_{\theta-1})^4 & = 1, & (s_is_{i-1})^3 & = 1, & & \forall \; 2\leq i \leq \theta-1, \\ (s_is_j)^2 & = 1, & & & & \forall\; |i-j| \geq 2. \end{align*} $$

Concretely, the simple reflexion $s_i$ acts on V by exchanging $e_i$ and $e_{i+1}$ as well as $e_{2\theta -i}$ and $e_{2\theta -i+1}$ for $1\leq i \leq \theta -1$ , whereas $s_{\theta }$ exchanges $e_{\theta }$ and $e_{\theta +1}$ . We define

$$ \begin{align*}I:=\{s_1,\dots ,s_{\theta-1}\} = \mathbf S\setminus \{s_{\theta}\}.\end{align*} $$

We consider the coarse Deligne–Lusztig variety $X_{I}(s_{\theta })$ . Since $s_{\theta }s_{\theta -1}s_{\theta } \not \in I$ , it is not a parabolic Deligne–Lusztig variety. Let $S_{\theta } := \overline {X_{I}(s_{\theta })}$ be its closure in $\mathbf G/\mathbf P_I$ . This variety has been introduced in [Reference Rapoport, Terstiege and Wilson14], as it is isomorphic to the closed Bruhat–Tits strata of type $\theta $ in the supersingular locus of the $\mathrm {GU}(1,n-1)$ Shimura variety over a ramified prime. We recall some geometric facts on $S_{\theta }$ which are proved in [Reference Rapoport, Terstiege and Wilson14, Section 5].

Proposition 3.1 The variety $S_{\theta }$ is normal and projective. The variety $S_1$ is isomorphic to the projective line $\mathbb P^1$ , and for $\theta \geq 2$ the variety $S_{\theta }$ has isolated singularities. If k is a field extension of $\mathbb F_q$ , we have

$$ \begin{align*}S_{\theta}(k) \simeq \{U \subset V_k \,|\, U^{\perp} = U \text{ and }U\cap \tau(U) \overset{\leq 1}{\subset} U\},\end{align*} $$

where $\overset {\leq 1}{\subset }$ denotes an inclusion of subspaces with index at most $1$ . There is a decomposition

$$ \begin{align*}S_{\theta} = X_{I}(\mathrm{id}) \sqcup X_{I}(s_{\theta}),\end{align*} $$

where $X_{I}(\mathrm {id})$ is closed and of dimension $0$ , and $X_{I}(s_{\theta })$ is open, dense of dimension $\theta $ . They correspond respectively to k-points U having $U = \tau (U)$ or $U\cap \tau (U) \subsetneq U$ . If $\theta \geq 2,$ then $S_{\theta }$ is singular at the points of $X_I(\mathrm {id})$ .

For $0 \leq \theta ' \leq \theta $ , define

$$ \begin{align*}I_{\theta'} := \{s_1,\dots ,s_{\theta-\theta'-1}\}\end{align*} $$

and $w_{\theta '} := s_{\theta +1-\theta '}\dots s_{\theta }$ . In particular, $I_0 = I$ , $I_{\theta -1} = I_{\theta } = \emptyset $ , $w_0 = \mathrm {id}$ and $w_1 = s_{\theta }$ .

Proposition 3.2 There is a stratification into locally closed subvarieties

$$ \begin{align*}S_{\theta} = \bigsqcup_{\theta'=0}^{\theta} X_{I_{\theta'}}(w_{\theta'}).\end{align*} $$

The stratum $X_{I_{\theta '}}(w_{\theta '})$ corresponds to points U such that $\dim (U\cap \tau (U) \cap \dots \cap \tau ^{\theta '+1}(U)) = \theta - \theta '$ . The closure in $S_{\theta }$ of a stratum $X_{I_{\theta '}}(w_{\theta '})$ is the union of all the strata $X_{I_{t}}(w_{t})$ for $t\leq \theta '$ . The stratum $X_{I_{\theta '}}(w_{\theta '})$ is of dimension $\theta '$ , and $X_{I_{\theta }}(w_{\theta })$ is open, dense, and irreducible.

It follows, in particular, that $S_{\theta }$ is irreducible as well. The strata $X_{I_{\theta '}}(w_{\theta '})$ are parabolic Deligne–Lusztig varieties, and it turns out that they are related to Coxeter varieties for symplectic groups of smaller sizes. For $0 \leq \theta ' \leq \theta $ , define

$$ \begin{align*}K_{\theta'} := \{s_1,\dots s_{\theta-\theta'-1}, s_{\theta-\theta'+1}, \dots , s_\theta\} = \mathbf S \setminus \{s_{\theta-\theta'}\}.\end{align*} $$

Note that $K_0 = I_0 = I$ and $K_{\theta } = \mathbf S$ . We have $I_{\theta '} \subset K_{\theta '}$ with equality if and only if $\theta '=0$ .

Proposition 3.3 There is an $\mathrm {Sp}(2\theta ,\mathbb F_p)$ -equivariant isomorphism

$$ \begin{align*}X_{I_{\theta'}}(w_{\theta'}) \simeq \mathrm{Sp}(2\theta,\mathbb F_q)/U_{K_{\theta'}} \times_{L_{K_{\theta'}}} X_{I_{\theta'}}^{\mathbf L_{K_{\theta'}}}(w_{\theta'}),\end{align*} $$

where $X_{I_{\theta '}}^{\mathbf L_{K_{\theta '}}}(w_{\theta '})$ is a Deligne–Lusztig variety for $\mathbf L_{K_{\theta '}}$ . The zero-dimensional variety $\mathrm {Sp}(2\theta ,\mathbb F_q)/U_{K_{\theta '}}$ has a left action of $\mathrm {Sp}(2\theta ,\mathbb F_q)$ and a right action of $L_{K_{\theta '}}$ .

The Levi complement $\mathbf L_{K_{\theta '}}$ is isomorphic to $\mathrm {GL}(\theta -\theta ') \times \mathrm {Sp}(2\theta ')$ , and its Weyl group is isomorphic to $\mathfrak S_{\theta -\theta '}\times W_{\theta '}$ . Via this decomposition, the permutation $w_{\theta '}$ corresponds to $\mathrm {id}\times w_{\theta '}$ . The Deligne–Lusztig variety $X_{I_{\theta '}}^{\mathbf L_{K_{\theta '}}}(w_{\theta '})$ decomposes as a product

$$ \begin{align*}X_{I_{\theta'}}^{\mathbf L_{K_{\theta'}}}(w_{\theta'}) = X_{\mathbf I_{\theta'}}^{\mathrm{GL(\theta-\theta')}}(\mathrm{id}) \times X_{\emptyset}^{\mathrm{Sp}(2\theta')}(w_{\theta'}).\end{align*} $$

The variety $X_{\mathbf I_{\theta '}}^{\mathrm {GL(\theta -\theta ')}}(\mathrm {id})$ is just a single point, but $X_{\emptyset }^{\mathrm {Sp}(2\theta ')}(w_{\theta '})$ is the Coxeter variety for the symplectic group of size $2\theta '$ . Indeed, $w_{\theta '}$ is a Coxeter element, that is, the product of all the simple reflexions of the Weyl group of $\mathrm {Sp}(2\theta ')$ .

4 Unipotent representations of the finite symplectic group

Recall that a (complex) irreducible representation of a finite group of Lie type $G = \mathbf G^{F}$ is said to be unipotent, if it occurs in the Deligne–Lusztig induction of the trivial representation of some maximal rational torus. Equivalently, it is unipotent if it occurs in the cohomology (with coefficient in $\overline {\mathbb Q_{\ell }}$ and $\ell \not = p$ ) of some Deligne–Lusztig variety of the form $X_{\mathbf B}$ , with $\mathbf B$ a Borel subgroup of $\mathbf G$ containing a maximal rational torus. In this section, we recall the classification of the unipotent representations of the finite symplectic groups. The underlying combinatorics is described by Lusztig’s notion of symbols. Our main reference is [Reference Geck and Malle5, Section 4.4].

Definition 4.1 Let $\theta \geq 1,$ and let d be an odd positive integer. The set of symbols of rank $\theta $ and defect d is

$$ \begin{align*}\hspace{-10pt}\mathcal Y^1_{d,\theta} := \left\{ S = (X,Y) \,\bigg |\, \begin{array}{l} X = (x_1,\dots ,x_{r+d})\\ Y = (y_1,\dots ,y_r) \end{array}, x_i,y_j \in \mathbb Z_{\geq 0}, \begin{array}{l} x_{i+1}-x_i \geq 1,\\ y_{j+1}-y_j \geq 1, \end{array} \mathrm{rk}(S) = \theta\right\} \bigg / (\text{shift}),\end{align*} $$

where the shift operation is defined by $\text {shift}(X,Y) := (\{0\}\sqcup (X+1),\{0\}\sqcup (Y+1))$ , and where the rank of S is given by

$$ \begin{align*}\mathrm{rk}(S) := \sum_{s\in S} s - \left\lfloor\frac{(\#S-1)^2}{4}\right\rfloor.\end{align*} $$

Note that the formula defining the rank is invariant under the shift operation, therefore, it is well defined. By [Reference Lusztig12], we have $\mathrm {rk}(S) \geq \left \lfloor \frac {d^2}{4}\right \rfloor $ so, in particular, $\mathcal Y^1_{d,\theta }$ is empty for d big enough. We write $\mathcal Y^1_{\theta }$ for the union of the $\mathcal Y^1_{d,\theta }$ with d odd, this is a finite set.

Example 4.1 In general, a symbol $S = (X,Y)$ will be written

$$ \begin{align*}S = \begin{pmatrix} x_1 & \dots & x_{r} & \dots & x_{r+d}\\ y_1 & \dots & y_{r} & & \end{pmatrix}.\end{align*} $$

We refer to X and Y as the first and second rows of S. The six elements of $\mathcal Y^1_{2}$ are given by

$$ \begin{align*} \begin{pmatrix} 2 \\ {} \end{pmatrix}, && \begin{pmatrix} 0 & 1 \\ 2 & \end{pmatrix}, && \begin{pmatrix} 0 & 2 \\ 1 & \end{pmatrix}, && \begin{pmatrix} 1 & 2 \\ 0 & \end{pmatrix}, && \begin{pmatrix} 0 & 1 & 2\\ 1 & 2 & \end{pmatrix}, && \begin{pmatrix} 0 & 1 & 2 \\ & & \end{pmatrix}. \end{align*} $$

The last symbol has defect $3,$ whereas all the other symbols have defect $1$ .

The symbols can be used to classify the unipotent representations of the finite symplectic group (cf. [Reference Lusztig12, Theorem 8.2]).

If $S \in \mathcal Y^1_{\theta }$ , we write $\rho _S$ for the associated unipotent representation of $\mathrm {Sp}(2\theta ,\mathbb F_q)$ . The classification is done so that the symbols

$$ \begin{align*} \begin{pmatrix} \theta \\ {} \end{pmatrix}, && \begin{pmatrix} 0 & \dots & \theta - 1 & \theta \\ 1 & \dots & \theta & \end{pmatrix}, \end{align*} $$

correspond respectively to the trivial and to the Steinberg representations. Let $S = (X,Y)$ be a symbol, and let $k\geq 1$ . A k-hook h in S is an integer $z\geq k$ such that $z\in X,z-k \not \in X$ or $z\in Y,z-k \not \in Y$ . A k-cohook c in S is an integer $z\geq k$ such that $z \in X, z-k \not \in Y$ or $z\in Y, z-k \not \in X$ . The integer k is referred to as the length of the hook h or the cohook c, and it is denoted $\ell (h)$ or $\ell (c)$ . The hook formula gives an expression of $\dim (\rho _S)$ in terms of hooks and cohooks.

Proposition 4.3 We have

$$ \begin{align*}\dim(\rho_S) = q^{a(S)} \frac{\prod_{i=1}^{\theta} \left(q^{2i}-1\right)}{2^{b'(S)}\prod_{h}\left(q^{\ell(h)}-1\right)\prod_{c}\left(q^{\ell(c)}+1\right)},\end{align*} $$

where the products in the denominator run over all the hooks h and all the cohooks c in S, and the numbers $a(S)$ and $b'(S)$ are given by

$$ \begin{align*} a(S) = \sum_{\{s,t\}\subset S} \min(s,t) - \sum_{i\geq 1} \binom{\#S-2i}{2}, & & b'(S) = \left\lfloor\frac{\#S-1}{2}\right\rfloor - \#\left(X\cap Y\right). \end{align*} $$

For $\delta \geq 0$ , we define the symbol

$$ \begin{align*}S_{\delta} := \begin{pmatrix} 0 & \dots & 2\delta \\ & & \end{pmatrix} \in \mathcal Y^1_{2\delta+1,\delta(\delta+1)}.\end{align*} $$

The next theorem states that cuspidal unipotent representations correspond to cuspidal symbols.

The determination of the cuspidal unipotent representations leads to a description of the unipotent Harish-Chandra series.

Definition 4.3 Let $\delta \geq 0$ such that $\theta = \delta (\delta +1) + a$ for some $a\geq 0$ . We write

$$ \begin{align*}L_{\delta} \simeq \mathrm{GL}(1,\mathbb F_q)^a \times \mathrm{Sp}(2\delta(\delta+1),\mathbb F_q)\end{align*} $$

for the block-diagonal Levi complement in $\mathrm {Sp}(2\theta ,\mathbb F_q)$ , with one middle block of size $2\delta (\delta +1)$ and other blocks of size $1$ . We write $\rho _{\delta } := (\mathbf {1})^a \boxtimes \rho _{S_{\delta }}$ , which is a cuspidal representation of $L_{\delta }$ .

In particular, the defect of the symbol S of rank $\theta $ classifies the unipotent Harish-Chandra series of $\mathrm {Sp}(2\theta ,\mathbb F_p)$ . If $\delta \geq 0$ is such that $\delta (\delta +1) \leq \theta $ , we write $\mathcal E_{\delta }$ for the Harish-Chandra series determined by $(L_{\delta },\rho _{\delta })$ . The previous proposition says that $\mathcal E_{\delta }$ is in bijection with $\mathcal Y^1_{2\delta +1,\theta }$ . Representations in a given series $\mathcal E_{\delta }$ can also be labeled in an alternative way.

Let $S \in \mathcal Y^1_{2\delta +1,\theta }$ where $\delta (\delta +1) \leq \theta $ . Up to taking suitable shifts, we may consider representatives of S and of $S_{\delta }$ whose rows have the same length. We obtain a bipartition $(\alpha ,\beta )$ of $\theta - \delta (\delta +1)$ by subtracting component-wise, the rows of $S_{\delta }$ from the rows of S, and reordering them decreasingly while ignoring the potential $0$ components.

Example 4.7 Recall the six symbols of $\mathcal Y^1_2$ described in Example 4.1. The associated bipartitions are, respectively,

$$ \begin{align*} ((2),\emptyset), & & (\emptyset,(2)), & & ((1),(1)), & & ((1^2),\emptyset), & & (\emptyset,(1^2)) & & (\emptyset,\emptyset). \end{align*} $$

The five symbols of defect $1$ correspond to bipartitions of $2$ , and the last symbol of defect $3$ corresponds to the empty bipartition of $0$ .

It is well-known that the set $\mathrm {Irr}(W_n)$ of equivalence classes of irreducible representations of the Coxeter group $W_n$ are in bijection with the set of bipartitions of n (cf. [Reference Geck and Pfeiffer6, Section 5.5]). Through this bijection, the bipartitions $((n),\emptyset )$ and $(\emptyset , (1^n))$ correspond, respectively, to the trivial and to the signature characters of $W_n$ .

One may check that the trivial (resp. the sign) character of $W_{\theta }$ corresponds to the trivial (resp. the Steinberg) representation of $\mathrm {Sp}(2\theta ,\mathbb F_q)$ in $\mathcal E_0$ . Given a symbol $S \in \mathcal Y^1_{2\delta +1,\theta }$ and the corresponding bipartition $(\alpha ,\beta )$ of $\theta - \delta (\delta +1)$ , we will sometimes also write $\rho _{\delta ,\alpha ,\beta }$ instead of $\rho _S$ . It turns out that the labeling of the unipotent representations of $\mathrm {Sp}(2\theta ,\mathbb F_q)$ in terms of bipartitions is particularly well suited in order to compute Harish-Chandra inductions and restrictions. To this end, it is also convenient to identify partitions with their Young diagrams.

Example 4.10 Let us consider the following Young diagram of size $4$

The Young diagrams which can be obtained by adding a box are given by

Clearly, the set of partitions of n is in bijection with the set of Young diagrams of size n. Likewise, bipartitions of n correspond to pairs of Young diagrams of combined sizes n. In the following, we will have to compute inductions of the following form:

$$ \begin{align*}\mathrm R_{\mathrm{GL}(a,\mathbb F_q) \times \mathrm{Sp}(2\theta',\mathbb F_q)}^{\mathrm{Sp}(2\theta,\mathbb F_q)} \, \mathbf 1 \boxtimes \rho_{S'},\end{align*} $$

where $\theta = a + \theta '$ and $S' \in \mathcal Y^{1}_{d,\theta '}$ is a symbol. In particular, we assume that $\theta ' \geq \delta (\delta +1)$ where $d = 2\delta +1$ .

Theorem 4.11 Let $S' \in \mathcal Y^1_{d,\theta '}$ and $(\alpha ',\beta ')$ the associated bipartition of $\theta ' - \delta (\delta +1)$ . We have

$$ \begin{align*}\mathrm R_{\mathrm{GL}(a,\mathbb F_q) \times \mathrm{Sp}(2\theta',\mathbb F_q)}^{\mathrm{Sp}(2\theta,\mathbb F_q)} \, \mathbf 1 \boxtimes \rho_{S'} = \sum_{\alpha,\beta} \rho_{\delta,\alpha,\beta},\end{align*} $$

where $(\alpha ,\beta )$ runs over all the bipartitions of $\theta - \delta (\delta +1)$ such that for some $0 \leq d \leq a$ , the Young diagram of $\alpha $ (resp. of $\beta $ ) can be obtained from the Young diagram of $\alpha '$ (resp. of $\beta '$ ) by adding a succession of d boxes (resp. $a - d$ boxes), no two of them lying in the same column.

This computation is a consequence of the Howlett–Lehrer comparison theorem [Reference Howlett and Lehrer9], stating that induction in a finite group of Lie type can be computed inside a corresponding Weyl group. We then apply Pieri’s rule for Coxeter groups of type $B_n$ (see [Reference Geck and Pfeiffer6, Paragraph 6.1.9]). We will see concrete examples of such computations in the following sections. There is a similar rule in order to compute certain Harish-Chandra restrictions. We write ${}^*\mathrm R_{\mathrm {Sp}(2\theta ',\mathbb F_q)}^{\mathrm {Sp}(2\theta ,\mathbb F_q)}$ for the restriction to the symplectic part of the Harish-Chandra restriction functor from $\mathrm {Sp}(2\theta ,\mathbb F_q)$ to the Levi complement $\mathrm {GL}(a,\mathbb F_q) \times \mathrm {Sp}(2\theta ',\mathbb F_q)$ .

Theorem 4.12 Let $S \in \mathcal Y^1_{d,\theta }$ and $(\alpha ,\beta )$ the associated bipartition of $\theta - \delta (\delta +1)$ . We have

$$ \begin{align*}{}^*\mathrm R_{\mathrm{Sp}(2\theta',\mathbb F_q)}^{\mathrm{Sp}(2\theta,\mathbb F_q)} \, \rho_{S} = \sum_{\alpha',\beta'} \rho_{\delta,\alpha',\beta'},\end{align*} $$

where $(\alpha ',\beta ')$ runs over all the bipartitions of $\theta ' - \delta (\delta +1)$ such that for some $0 \leq d \leq a$ , the Young diagram of $\alpha '$ (resp. of $\beta '$ ) can be obtained from the Young diagram of $\alpha $ (resp. of $\beta $ ) by removing a succession of d boxes (resp. $a - d$ boxes), no two of them lying in the same column.

5 The cohomology of the Coxeter variety for the symplectic group

In this section, we compute the cohomology of Coxeter varieties of finite symplectic groups, in terms of the classification of the unipotent characters that we recalled in Theorem 4.2. We write $X^{\theta } := X_{\emptyset }(\mathrm {cox})$ for the Coxeter variety attached to the symplectic group $\mathrm {Sp}(2\theta ,\mathbb F_q)$ , and $\mathrm {H}_c^{\bullet }(X^{\theta })$ instead of $\mathrm {H}_c^{\bullet }(X^{\theta }\otimes \mathbb F,\overline {\mathbb Q_{\ell }})$ where $\ell \not = p$ . While $X^{\theta }$ might actually depend on the choice of the Coxeter element $\mathrm {cox}$ , its cohomology $\mathrm H^{\bullet }_c(X^{\theta })$ does not. We first recall known facts on the cohomology of $X^{\theta }$ from Lusztig’s work in [Reference Lusztig11].

In other words, there exists a uniquely determined family of pairwise distinct symbols $S^\theta _0,\dots ,S^{\theta }_{\theta }$ and $T^{\theta }_0,\dots ,T^{\theta }_{\theta -2}$ in $\mathcal Y^{1}_{\theta }$ such that

$$ \begin{align*} {\forall} 0{\leq} i {\leq} {\theta-2}, & \qquad {\mathrm{H}}{}^{\theta+i}{}_{c}(X{}^{\theta}) \qquad {\simeq} {\rho{}_{S}{}^{\theta}{}_{i}} \oplus {\rho{}_{T}{}^{\theta}{}_{i}}, \\ {}\text{for } i = \theta-1,{\theta}, & \qquad {\mathrm{H}}{}^{\theta+i}{}_{c}(X{}^{\theta}) \qquad \simeq {\rho{}_{S}{}^{\theta}{}_{i}}. \end{align*} $$

The representation $\rho _{S^{\theta }_i}$ (resp. $\rho _{T^{\theta }_i}$ ) corresponds to the eigenspace of the Frobenius F on $\bigoplus _{i\geq 0} \mathrm H_c^{\theta +i}(X^{\theta })$ attached to $q^i$ (resp. to $-q^{i+1}$ ). Moreover, we know that $\rho _{S^{\theta }_{\theta }}$ is the trivial representation, therefore

$$ \begin{align*} S^{\theta}_{\theta} = \begin{pmatrix} \theta \\ {} \end{pmatrix}.\end{align*} $$

Lusztig also gives a formula computing the dimension of the eigenspaces. Specializing to the case of the symplectic group, it reduces to the following statement.

Proposition 5.2 For $0\leq i \leq \theta $ , we have

$$ \begin{align*}\deg(\rho_{S^{\theta}_i}) = q^{(\theta-i)^2} \prod_{s=1}^{\theta-i} \frac{q^{s+i}-1}{q^s-1}\prod_{s=0}^{\theta-i-1} \frac{q^{s+i}+1}{q^s+1}.\end{align*} $$

For $0 \leq j \leq \theta -2$ , we have

$$ \begin{align*}\deg(\rho_{T^{\theta}_j}) = q^{(\theta-j-1)^2}\frac{(q^{\theta-1}-1)(q^\theta-1)}{2(q+1)}\prod_{s=1}^{\theta-j-2}\frac{q^{s+j}-1}{q^s-1}\prod_{s=2}^{\theta-j-1}\frac{q^{s+j}+1}{q^s+1}.\end{align*} $$

Our goal in this section is to determine the symbols $S^{\theta }_i$ and $T^{\theta }_j$ explicitly. This is done in the following proposition.

Proposition 5.3 For $0\leq i \leq \theta $ and $0\leq j \leq \theta -2$ , we have

$$ \begin{align*} S^{\theta}_i = \begin{pmatrix} 0 & \dots & \theta-i-1 & \,\,\,\theta \\ 1 & \dots & \theta-i & \end{pmatrix}, && T^{\theta}_j = \begin{pmatrix} 0 & \dots & \theta-j-3 & \theta-j-2 & \theta-j-1 & \theta \\ 1 & \dots & \theta-j-2 & & & \end{pmatrix}. \end{align*} $$

We note that the statement is coherent with the two dimension formulae that we provided earlier. That is, the degree of $\rho _{S^{\theta }_i}$ (resp. of $\rho _{T^{\theta }_j}$ ) computed with the hook formula of Proposition 4.3, agrees with the dimension of the eigenspace of $q^i$ (resp. of $-q^{j+1}$ ) in the cohomology of $X^{\theta }$ as given in the previous paragraph.

Proof We use induction on $\theta \geq 0$ . Since we already know that $S^{\theta }_{\theta }$ is the symbol corresponding to the trivial representation, the proposition is proved for $\theta =0$ . Thus we may assume $\theta \geq 1$ . We consider the block diagonal Levi complement $L \simeq \mathrm {GL}(1,\mathbb F_q) \times \mathrm {Sp}(2(\theta -1),\mathbb F_q)$ , and we write ${}^*\mathrm R_{\theta -1}^{\theta }$ for the restriction to $\mathrm {Sp}(2(\theta -1),\mathbb F_q)$ of the Harish-Chandra restriction from $\mathrm {Sp}(2\theta ,\mathbb F_q)$ to L. According to [Reference Lusztig11, Corollary 2.10], for all $0\leq i \leq \theta $ , we have an $\mathrm {Sp}(2(\theta -1),\mathbb F_q)\times \langle F \rangle $ -equivariant isomorphism

(*) $$\begin{align} {}^*\mathrm R_{\theta-1}^{\theta} \left(\mathrm H^{\theta+i}_c(X^{\theta})\right) \simeq \mathrm H^{\theta-1+i}_c(X^{\theta-1}) \oplus \mathrm H^{\theta-1+(i-1)}_c(X^{\theta-1})(-1). \end{align} $$

The right-hand side can be computed by induction hypothesis whereas the left-hand side can be computed using Theorem 4.12. We fix $0 \leq i \leq \theta - 1$ and $0 \leq j \leq \theta - 2$ . We denote by $(\delta ,\alpha ,\beta )$ and $(\nu ,\gamma ,\delta )$ the alternate labeling by bipartitions of the representations $\rho _{S^{\theta }_i}$ and $\rho _{T^{\theta }_j}$ , respectively. Recall that the restriction ${}^*\mathrm R_{\theta -1}^{\theta }\,\rho _{\delta ,\alpha ,\beta }$ is the sum of all the representations $\rho _{\delta ,\alpha ',\beta '}$ where $(\alpha ',\beta ')$ is obtained from $(\alpha ,\beta )$ by removing a single box in one of their Young diagrams. The similar description also holds for ${}^*\mathrm R_{\theta -1}^{\theta }\,\rho _{\nu ,\gamma ,\delta }$ . First, we determine $S^{\theta }_i$ by identifying the $q^i$ -eigenspace of the Frobenius in (*). We distinguish different cases depending on the values of $\theta $ and i.

• Case $\boldsymbol {\theta =1}$ . In this case, $i = 0$ . The right-hand side of (*) is $\rho _{S^0_0} \simeq \rho _{0,\emptyset ,\emptyset }$ with eigenvalue $1$ . Thus, $\delta = 0$ and the bipartition $(\alpha ,\beta )$ consists of a single box. Therefore, $(\alpha ,\beta ) = ((1),\emptyset )$ or $(\emptyset ,(1))$ . By Theorem 5, we know that $\rho _{0,\alpha ,\beta }$ has degree q. This forces $(\alpha ,\beta ) = (\emptyset ,(1))$ .

• Case $\boldsymbol {\theta =2}$ and $\mathbf {i=0}$ . The eigenspace attached to $1$ on the right-hand side of (*) is $\rho _{S^{1}_0} \simeq \rho _{0,\emptyset ,(1)}$ . Thus, $\delta = 0$ and there is a single removable box in the bipartition $(\alpha ,\beta )$ . When we remove it, we obtain $(\emptyset ,(1))$ . Therefore, $(\alpha ,\beta ) = (\emptyset ,(2))$ or $(\emptyset ,(1^2))$ . By Theorem 5, we know that $\rho _{0,\alpha ,\beta }$ has degree $q^4$ , thus $(\alpha ,\beta ) = (\emptyset ,(1^2))$ .

• Case $\boldsymbol {\theta>2}$ and $\mathbf {i=0}$ . The eigenspace attached to $1$ on the right-hand side of (*) is $\rho _{S^{\theta -1}_0} \simeq \rho _{0,\emptyset ,(1^{\theta -1})}$ . Thus, $\delta = 0$ and there is a single removable box in the bipartition $(\alpha ,\beta )$ . When we remove it, we obtain $(\emptyset ,(1^{\theta -1}))$ . The only such bipartition is $(\alpha ,\beta ) = (\emptyset ,(1^{\theta }))$ .

• Case $\boldsymbol {\theta>2}$ and $\mathbf {1\leq i \leq k-1}$ . The eigenspace attached to $p^i$ on the right-hand side of (*) is $\rho _{S^{\theta -1}_i} \oplus \rho _{S^{\theta -1}_{i-1}} \simeq \rho _{0,(i),(1^{\theta -1-i})} \oplus \rho _{0,(i-1),(1^{\theta -i})}$ . Thus, $\delta = 0$ and there are exactly two removable boxes in the bipartition $(\alpha ,\beta )$ . When we remove one of them, we obtain either $((i),(1^{\theta -1-i}))$ or $((i-1),(1^{\theta -i}))$ . The only such bipartition is $(\alpha ,\beta ) = ((i),(1^{\theta -i}))$ .

It remains to determine $T^{\theta }_j$ for $0\leq j \leq \theta -2$ .

• Case $\boldsymbol {\theta =2}$ . The eigenspace attached to $-p$ on the right-hand side of (*) is $0$ . Thus, the symbol $T^2_0 \in \mathcal Y^1_2$ has no hook at all, implying that it is cuspidal. Since $\mathrm {Sp}(4,\mathbb F_q)$ admits only one unipotent cuspidal representation, we deduce that $\nu = 1$ and $(\gamma ,\delta ) = (\emptyset ,\emptyset )$ .

• Case $\mathbf {k=3}$ . First, when $j=0$ , the eigenspace attached to $-p$ on the right-hand side of (*) is $\rho _{T^{2}_0} \simeq \rho _{1,\emptyset ,\emptyset }$ . Thus, $\nu = 1$ and there is a single box in the bipartition $(\gamma ,\delta )$ . Therefore, $(\gamma ,\delta ) = ((1),\emptyset )$ or $(\emptyset ,(1))$ . By Theorem 5, we know that $\rho _{1,\gamma ,\delta }$ has degree $q^4\frac {(q^2-1)(q^3-1)}{2(q+1)}$ , thus $(\gamma ,\delta ) = (\emptyset ,(1))$ .Then when $j=1$ , the eigenspace attached to $-p^2$ on the right-hand side of (*) is $\rho _{T^{2}_0} \simeq \rho _{1,\emptyset ,\emptyset }$ . Thus, $\nu = 1$ and as in the case $j=0$ we have $(\gamma ,\delta ) = ((1),\emptyset )$ or $(\emptyset ,(1))$ . We can deduce that it is equal to the former by comparing the dimensions or by using the fact that the symbols $T_j^{\theta }$ are pairwise distinct.

• Case $\boldsymbol {\theta =4}$ and $\mathbf {j=0}$ . The eigenspace attached to $-p$ on the right-hand side of (*) is $\rho _{T^{3}_0} \simeq \rho _{1,\emptyset ,(1)}$ . Thus, $\nu = 1$ and there is a single removable box in the bipartition $(\gamma ,\delta )$ . When we remove it, we obtain $(\emptyset ,(1))$ . Therefore, $(\gamma ,\delta ) = (\emptyset ,(2))$ or $(\emptyset ,(1^2))$ . By Theorem 5, we know that $\rho _{1,\gamma ,\delta }$ has degree $q^9\frac {(q^3-1)(q^4-1)}{2(q+1)}$ , thus $(\gamma ,\delta ) = (\emptyset ,(1^2))$ .

• Case $\boldsymbol {\theta>4}$ and $\mathbf {j=0}$ . The eigenspace attached to $-p$ on the right-hand side of (*) is $\rho _{T^{\theta -1}_0} \simeq \rho _{1,\emptyset ,(1^{\theta -3})}$ . Thus, $\nu = 1$ and there is a single removable box in the bipartition $(\gamma ,\delta )$ . When we remove it, we obtain $(\emptyset ,(1^{\theta -3}))$ . The only such bipartition is $(\gamma ,\delta ) = (\emptyset ,(1^{\theta -2}))$ .

• Case $\boldsymbol {\theta =4}$ and $\mathbf {j=}\boldsymbol {\theta -2}$ . The eigenspace attached to $-p^{3}$ on the right-hand side of (*) is $\rho _{T^{3}_{1}} \simeq \rho _{1,(1),\emptyset }$ . Thus, $\nu = 1$ and there is a single removable box in the bipartition $(\gamma ,\delta )$ . When we remove it, we obtain $((1),\emptyset )$ . Therefore, $(\gamma ,\delta ) = ((2),\emptyset )$ or $((1^2),\emptyset )$ . By Theorem 5, we know that $\rho _{1,\gamma ,\delta }$ has degree $q\frac {(q^3-1)(q^4-1)}{2(q+1)}$ , thus $(\gamma ,\delta ) = ((2),\emptyset )$ .

• Case $\boldsymbol {\theta>4}$ and $\mathbf {j =} \boldsymbol {\theta -2}$ . The eigenspace attached to $-p^{\theta -1}$ on the right-hand side of (*) is $\rho _{T^{\theta -1}_{\theta -3}} \simeq \rho _{1,(\theta -3),\emptyset }$ . Thus, $\nu = 1$ and there is a single removable box in the bipartition $(\gamma ,\delta )$ . When we remove it, we obtain $((\theta -3),\emptyset )$ . The only such bipartition is $(\gamma ,\delta ) = ((\theta -2),\emptyset )$ .

• Case $\mathbf {1\leq j \leq } \boldsymbol {\theta -3}$ . The eigenspace attached to $-p^{j+1}$ on the right-hand side of (*) is $\rho _{T^{\theta -1}_{j}} \oplus \rho _{T^{\theta -1}_{j-1}} \simeq \rho _{1,(j),(1^{\theta -3-j})}\oplus \rho _{1,(j-1),(1^{\theta -2-j})}$ . Thus, $\nu = 1$ and there are exactly two removable boxes in the bipartition $(\gamma ,\delta )$ . When we remove one of them, we obtain either $((j),(1^{\theta -3-j}))$ or $((j-1),(1^{\theta -2-j}))$ . The only such bipartition is $(\gamma ,\delta ) = ((j),(1^{\theta -2-j}))$ .

6 The cohomology of $S_{\theta }$

The last three sections of the article are devoted to proving the main theorem below, which describes the cohomology of the variety $S_{\theta }$ . Since $S_0$ is a point and $S_1 \simeq \mathbb P^1$ , the cases $\theta = 0$ or $1$ are trivial. From now and up to the end of the article, we assume that $\theta \geq 2$ .

In order to compute the cohomology of $S_{\theta }$ , we use the stratification by parabolic Deligne–Lusztig varieties which we recalled in Proposition 3.2, and we analyze the associated spectral sequence. It is given in its first page by

(E) $$\begin{align} E^{\theta',i}_1 = \mathrm H^{\theta'+i}_c(X_{I_{\theta'}}(w_{\theta'})) \implies \mathrm H^{\theta'+i}(S_{\theta}). \end{align} $$

Let us first determine each term explicitly. By Proposition 3.3 and using the notations introduced there, we have an isomorphism

$$ \begin{align*}X_{I_{\theta'}}(w_{\theta'}) \simeq \mathrm{Sp}(2\theta,\mathbb F_q)/U_{K_{\theta'}} \times_{L_{K_{\theta'}}} X_{I_{\theta'}}^{\mathbf L_{K_{\theta'}}}(w_{\theta'}).\end{align*} $$

Taking cohomology, this identity translates into some Harish-Chandra induction

$$ \begin{align*}\mathrm H^{\bullet}_c(X_{I_{\theta'}}(w_{\theta'})) \simeq \mathrm R_{L_{K_{\theta'}}}^{\mathrm{Sp}(2\theta,\mathbb F_q)} \, \mathrm H^{\bullet}_c(X_{I_{\theta'}}^{\mathbf L_{K_{\theta'}}}(w_{\theta'})).\end{align*} $$

The Deligne–Lusztig variety $X_{I_{\theta '}}^{\mathbf L_{K_{\theta '}}}(w_{\theta '})$ for the Levi complement $L_{K_{\theta '}} \simeq \mathrm {GL}(\theta -\theta ',\mathbb F_q) \times \mathrm {Sp}(2\theta ',\mathbb F_q)$ is isomorphic to the Coxeter variety $X^{\theta '}$ with the $\mathrm {GL}$ -part acting trivially. Thus, the terms $E_1^{\theta ',i}$ are the Harish-Chandra inductions of the cohomology groups of the Coxeter varieties $X^{\theta '}$ , which we have determined in Proposition 5.3. Let us compute these inductions explicitly.

Lemma 6.3 Let $0 \leq i \leq \theta ' \leq \theta $ . We have $E_1^{\theta ',i} = A^{\theta ',i} \oplus B^{\theta ',i}$ with

$$ \begin{align*} A^{\theta',i} = \bigoplus_{\alpha,\beta} \rho_{0,\alpha,\beta}, & \qquad B^{\theta',i} = \bigoplus_{\gamma,\delta} \rho_{1,\gamma,\delta}, \end{align*} $$

where $(\alpha ,\beta )$ runs over all the bipartitions of $\theta $ such that, for some $0 \leq d \leq \theta - \theta '$ , we have

$$ \begin{align*}\begin{cases} \alpha = (i + d - s, s) \text{ for some } 0 \leq s \leq \min(d,i),\\ \beta = (\theta-\theta'-d, 1^{\theta'-i})\text{ or } (\theta-\theta'-d+1,1^{\theta'-1-i}), \end{cases}\end{align*} $$

and if $i \leq \theta '-2$ , $(\gamma ,\delta )$ runs over all the bipartitions of $\theta - 2$ such that, for some $0 \leq d \leq \theta - \theta '$ , we have

$$ \begin{align*}\begin{cases} \gamma = (i + d - s, s) \text{ for some } 0 \leq s \leq \min(d,i),\\ \delta = (\theta-\theta'-d, 1^{\theta'-2-i}) \text{ or } (\theta-\theta'-d+1,1^{\theta'-3-i}). \end{cases}\end{align*} $$

The summand $A^{\theta ',i}$ (resp. $B^{\theta ',i}$ ) is the eigenspace of the Frobenius for the eigenvalue $q^i$ (resp. the eigenvalue $-q^{i+1}$ ).

Proof Using Pieri’s rule for Coxeter groups of type $B_n$ as we recalled in Proposition 4.11, we must decompose the Harish-Chandra inductions

$$ \begin{align*} R_{\mathrm{GL}(\theta-\theta',\mathbb F_q) \times \mathrm{Sp}(2\theta',\mathbb F_q)}^{\mathrm{Sp}(2\theta,\mathbb F_q)} \, \mathbf 1 \boxtimes \rho_{S_i^{\theta'}}, & \qquad R_{\mathrm{GL}(\theta-\theta',\mathbb F_q) \times \mathrm{Sp}(2\theta',\mathbb F_q)}^{\mathrm{Sp}(2\theta,\mathbb F_q)} \, \mathbf 1 \boxtimes \rho_{T_j^{\theta'}}, \end{align*} $$

where $S_i^{\theta '}$ and $T_j^{\theta '}$ are the symbols determined in Proposition 5.3. The Young diagrams of the bipartitions associated with these symbols have the form

The problem is to determine all the pairs of Young diagrams one may obtain after adding a succession of $\theta -\theta '$ boxes to the pair of diagrams above, with no two boxes in the same column. This computation has already been done in [Reference Muller13, Section 5], and leads to the claimed formula.

Recall that an eigenvalue $\alpha \in \overline {\mathbb Q_{\ell }}$ of the Frobenius on the cohomology of a variety defined over $\mathbb F_q$ is said to be of weight $w\in \mathbb Z$ if, for any isomorphism $\iota :\overline {\mathbb Q_{\ell }} \simeq \mathbb C$ , we have $|\iota (\alpha )| = q^{\frac {w}{2}}$ .

Analyzing the $\mathrm {Sp}(2\theta ,\mathbb F_q)$ -action, we may decompose each term $E_1^{\theta ',i}$ in the following way:

$$ \begin{align*} A^{\theta',i} = A^{\theta',i}_0 \oplus A^{\theta',i}_1, & \qquad B^{\theta',i} = B^{\theta',i}_0 \oplus B^{\theta',i}_1, \end{align*} $$

where for $\epsilon = 0,1$ the term $A^{\theta ',i}_{\epsilon }$ is the sum of all the irreducible components $\rho _{0,\alpha ,\beta }$ with the partition $\beta $ , written as $\beta = (\beta _1 \geq \dots \geq \beta _r> 0)$ , satisfies $r = \theta ' + \epsilon - i$ , and if $i \leq \theta '-2$ the term $B^{\theta ',i}_{\epsilon }$ is the sum of all the irreducible components $\rho _{1,\gamma ,\delta }$ with the partition $\delta $ , written as $\delta = (\delta _1 \geq \dots \geq \delta _s> 0)$ , satisfies $s = \theta ' - 2 + \epsilon - i$ . We observe that $A^{\theta ,i}_1 = B^{\theta ,i}_1 = 0$ , and for $0 \leq i \leq \theta ' < \theta $ (resp. $0 \leq j + 2 \leq \theta ' < \theta $ ), we have isomorphisms $A^{\theta ',i}_1 \simeq A^{\theta '+1,i}_0 $ and $B^{\theta ',j}_1 \simeq B^{\theta '+1,j}_0$ . Consider a differential

$$ \begin{align*}d^{\theta\prime,i}: E_1^{\theta\prime,i} \to E_1^{\theta\prime+1,i}.\end{align*} $$

Since the differentials are Frobenius equivariant, they decompose as a sum $d^{\theta \prime ,i} = d_{A}^{\theta \prime ,i} \oplus d_{B}^{\theta ',i}$ where

$$ \begin{align*} d_{A}^{\theta\prime,i}: A^{\theta\prime,i} \to A^{\theta\prime+1,i}, & d_{B}^{\theta\prime,i}: B^{\theta\prime,i} \to B^{\theta\prime+1,i}. \end{align*} $$

The $\mathrm {Sp}(2\theta ,\mathbb F_q)$ -equivariance then forces

$$ \begin{align*} \mathrm{Im}(d_{A}^{\theta\prime-1,i}) \subset A_0^{\theta\prime,i} \subset \mathrm{Ker}(d_{A}^{\theta\prime,i}), & \qquad \mathrm{Im}(d_{B}^{\theta\prime-1,i}) \subset B_0^{\theta\prime,i} \subset \mathrm{Ker}(d_{B}^{\theta\prime,i}). \end{align*} $$

In order to help visualize the situation, the page $E_1$ is drawn in Figure 1. Since the sequence degenerates in $E_2$ and the resulting filtration splits, it is clear that for all $0 \leq i \leq \theta $ , the cohomology group $\mathrm H^{2i}(S_{\theta })$ contains $A_0^{i,i} \oplus B^{i+1,i-1}_0$ (the term $B^{i+1,i-1}_0$ is nonzero if and only if $0 < i < \theta $ ). We point out that in Theorem 6.1, (2) can be rephrased as $\mathrm H^{2i}(S_{\theta }) \simeq A_0^{i,i} \oplus B^{i+1,i-1}_0$ for all $0 \leq i \leq \theta $ .

Figure 1:

The first page of the spectral sequence (E).

Proof Let us fix $0\leq i \leq \theta ' \leq \theta $ . The equality $\mathrm {Im}(d_{A}^{\theta '-1,i}) = A_0^{\theta ',i}$ (resp. $\mathrm {Im}(d_{B}^{\theta \prime -1,i}) = B_0^{\theta \prime ,i}$ ) is equivalent to $\mathrm {Ker}(d_{A}^{\theta '-1,i}) = A_0^{\theta '-1,1}$ (resp. $\mathrm {Ker}(d_{B}^{\theta '-1,i}) = B_0^{\theta '-1,i}$ ). Thus, the vanishing of a term $E_2^{\theta ',i} = \mathrm {Ker}(d_{A}^{\theta ',i})/\mathrm {Im}(d_{A}^{\theta '-1,i}) \oplus \mathrm {Ker}(d_{B}^{\theta ',i})/\mathrm {Im}(d_{B}^{\theta '-1,i})$ is equivalent to the equalities

$$ \begin{align*} \mathrm{Ker}(d_{A}^{\theta\prime-1,i}) = A_0^{\theta\prime-1,i}, & \qquad \mathrm{Im}(d_A^{\theta\prime,i}) = A_0^{\theta\prime+1,i}, \\ \mathrm{Ker}(d_{B}^{\theta\prime-1,i}) = B_0^{\theta\prime-1,i}, & \qquad \mathrm{Im}(d_B^{\theta\prime,i}) = B_0^{\theta\prime+1,i}. \end{align*} $$

The lemma follows easily.

7 Desingularization of $S_{\theta }$ and purity of the Frobenius action on cohomology

In order to make out how the spectral sequence simplifies in the second page, it is necessary to get more information on the expected weights of the Frobenius on the abutment. To this end, we introduce the blow-up $\pi :S_{\theta }'\to S_{\theta }$ at its singular points. We denote by $E := \pi ^{-1}(Z)$ the exceptional divisor, where in the notations of Proposition 3.1, $Z = X_I(\mathrm {id})$ is the singular locus of $S_{\theta }$ . Recall that $\dim Z = 0$ . In this section, we prove the following proposition.

Since $S_{\theta }'\setminus E$ is isomorphic to the smooth locus of $S_{\theta }$ , it is enough to prove that the blow-up is smooth at points of the exceptional divisor. To this end, we exhibit a certain affine neighborhood of any singular point of $S_{\theta }$ . Recall the symplectic space V introduced in the first section. Let $L(\theta )$ denote the Lagrangian Grassmannian of V, that is, the smooth projective variety defined over $\mathbb F_q$ whose k-rational points, for any field extension $k/\mathbb F_q$ , correspond to subspaces of $U\subset V_k$ such that $U^{\perp } = U$ . Recall from Proposition 3.1 that $S_{\theta }$ is the closed subvariety of $L(\theta )$ consisting of those U such that $U\cap \tau (U) \overset {\leq 1}{\subset } U$ .

Let $\mathrm {Sym}_{\theta } \simeq \mathbb A^{\frac {\theta (\theta +1)}{2}}$ denote the variety of $\theta \times \theta $ symmetric matrices over $\mathbb F_q$ . Let $V_{\theta }$ denote the closed subvariety consisting of all $M\in \mathrm {Sym}_{\theta }$ such that $M^{(q)} - M$ has rank at most one.

Proof If $U \in L(\theta )$ is a closed point defined over a finite extension $k/\mathbb F_q$ , one may choose an isotropic supplement $U'$ so that we have a decomposition $V_k = U \oplus U'$ . The symplectic pairing induces an identification between $U'$ and the k-linear dual of U. We may consider the affine variety $\mathrm {Hom}(U,U')$ defined over $\mathrm {Spec}(k)$ , and the subvariety $\mathrm {Hom}(U,U')^{\mathrm {sym}}$ consisting of morphisms $\varphi \in \mathrm {Hom}(U,U')$ such that $\varphi ^* = \varphi $ , where $\varphi ^*:U^{\prime *} \simeq U^{**} \simeq U \to U^*$ is the dual of $\varphi $ . According to [Reference Kolhatkar10, Lemma 2.8], we have $\varphi \in \mathrm {Hom}(U,U')^{\mathrm {sym}}$ if and only $\Gamma _{\varphi }\in L(\theta )$ where $\Gamma _{\varphi }$ denotes the graph of $\varphi $ . The assignment $\varphi \mapsto \Gamma _{\varphi }$ defines an open immersion $\mathrm {Hom}(U,U')^{\mathrm {sym}} \hookrightarrow L(\theta )$ , identifying the former with an open affine neighborhood of U. We have an identification $\mathrm {Hom}(U,U')^{\mathrm {sym}} \simeq \mathrm {Sym}_{\theta } \otimes k$ upon fixing a basis of U and equipping $U'$ with the dual basis.

Now assume that $U \in S_{\theta }$ is a singular point, equivalently an $\mathbb F_q$ -rational point of $S_{\theta }$ . Let us fix a basis $(f_i)_{1\leq i \leq \theta }$ of U and equip $U'$ with the dual basis $(f_i')_{1\leq i \leq \theta }$ . Let $M \in \mathrm {Sym}_{\theta }$ . The graphs $\Gamma _M$ and $\Gamma _{M^{(q)}}$ are generated by the vectors

$$ \begin{align*} g_j := f_j + \sum_{i=1}^{\theta} M_{ij}f_i', & \qquad g_j^{(q)} = f_j + \sum_{i=1}^{\theta} M_{ij}^qf_i', \end{align*} $$

respectively. A direct computation shows that the intersection $\Gamma _M \cap \Gamma _{M^{(q)}}$ is isomorphic to $\mathrm {Ker}(M^{(q)}-M)$ . Since U is defined over $\mathbb F_q$ , the vectors $f_j$ and $f_j'$ are fixed by $\tau $ , thus we have $\tau (\Gamma _M) = \Gamma _{M^{(q)}}$ . It follows that $M \in S_{\theta } \cap \mathrm {Sym}_{\theta }$ if and only if $\dim \mathrm {Ker}(M^{(q)}-M) \geq \theta - 1$ , that is, if and only if $M \in V_{\theta }$ .

Let $V_{\theta }' \subset \mathrm {Sym}_{\theta }$ denote the subvariety of symmetric matrices M of rank at most $1$ . The variety $V_{\theta }'$ is known as a symmetric determinantal variety. It admits a single singular point corresponding to $M=0$ .

Proposition 7.1 has the following consequence regarding the cohomology of $S_{\theta }$ .

Proof By [15, Lemma 0EW3], there is a long exact sequence

$$ \begin{align*}\dots \to \mathrm H^k(S_{\theta}) \to \mathrm H^k(S_{\theta}') \oplus \mathrm H^k(Z) \to \mathrm H^k(E) \to \mathrm H^{k+1}(S_{\theta}) \to \dots .\end{align*} $$

Since $S_{\theta }'$ and E are projective and smooth, the Frobenius action on their kth cohomology group is pure of weight k. For any $0 \leq j \leq \theta $ , the maps $\mathrm H^{2j-1}(E) \to \mathrm H^{2j}(S_{\theta })$ and $\mathrm H^{2j+1}(S_{\theta }) \to \mathrm H^{2j+1}(S_{\theta }')$ vanish since the cohomology of $S_{\theta }$ only contains even weights by Corollary 6.5. Thus, we have in fact exact sequences

$$ \begin{align*}0 \to \mathrm H^{2j}(S_{\theta}) \to \mathrm H^{2j}(S_{\theta}') \oplus \mathrm H^{2j}(Z) \to \mathrm H^{2j}(E) \to \mathrm H^{2j+1}(S_{\theta}) \to 0.\end{align*} $$

It follows that $\mathrm H^{2j}(S_{\theta })$ and $\mathrm H^{2j+1}(S_{\theta })$ are pure of weight $2j$ .

8 Intersection cohomology of $S_{\theta }$

Let $U := S_{\theta } \setminus Z$ denote the smooth locus of $S_{\theta }$ . Let $j:U \hookrightarrow S_{\theta }$ be the open immersion. The intersection complex of $S_{\theta }$ is the intermediate image $j_{!*}\overline {\mathbb Q_{\ell }}$ . We write $\mathrm {IH}^{\bullet }(S_{\theta }) := \mathrm H^{\bullet }(S_{\theta },j_{!*}\overline {\mathbb Q_{\ell }})$ for the intersection cohomology of $S_{\theta }$ . It is a standard fact that $j_{!*}\overline {\mathbb Q_{\ell }}$ is pure of weight $0$ , and therefore, $\mathrm {IH}^k(S_{\theta })$ is pure of weight k for all k. Since Z is $0$ -dimensional, it follows from [Reference Beilinson, Bernstein, Deligne and Gabber1, Proposition 2.1.11] that the intersection complex is given by

$$ \begin{align*}j_{!*}\overline{\mathbb Q_{\ell}} \simeq \tau_{\theta-1}\mathrm Rj_*\overline{\mathbb Q_{\ell}}.\end{align*} $$

The hypercohomology spectral sequence for intersection cohomology reads

(F) $$\begin{align} F_2^{a,b} = \mathrm H^{a}(S_{\theta},\mathrm H^b(j_{!*}\overline{\mathbb Q_{\ell}})) \implies \mathrm{IH}^{a+b}(S_{\theta}). \end{align} $$

For $k\geq 1$ , the sheaf $\mathrm R^kj_{*}\overline {\mathbb Q_{\ell }}$ is skyscraper at the points of Z. Furthermore, we have $j_*\overline {\mathbb Q_{\ell }} = \overline {\mathbb Q_{\ell }}$ since $S_{\theta }$ is irreducible and normal. It follows that

$$ \begin{align*} F_2^{a,b} = \begin{cases} \mathrm H^{a}(S_{\theta}), & \text{if } b=0,\\ \bigoplus_{\overline{z}\in Z} (\mathrm R^bj_{*}\overline{\mathbb Q_{\ell}})_{\overline{z}}, & \text{if } a = 0 \text{ and } 1 \leq b \leq \theta-1,\\ 0, & \text{else.} \end{cases} \end{align*} $$

In particular, the spectral sequence degenerate in $F_{\theta +1}$ , and we have $\mathrm H^k(S_{\theta }) = \mathrm {IH}^k(S_{\theta })$ for all $k> \theta $ . The second page $F_2$ is drawn in Figure 2.

Figure 2:

The second page of the spectral sequence (F) (the differentials in dashed lines correspond to deeper pages).

Proposition 8.1 For $k> \theta $ , we have

$$ \begin{align*} \mathrm H^k(S_{\theta}) \simeq \begin{cases} A_0^{i,i}\oplus B_0^{i+1,i-1}, & \text{if } k = 2i \text{ is even},\\ 0, & \text{if } k \text{ is odd}. \end{cases} \end{align*} $$

The statement also holds when $k = \theta $ is even.

It remains to compute the cohomology of $S_{\theta }$ up to the middle degree. To do so, for $1 \leq k \leq \theta -1,$ we consider the differential $\delta _k:F_{k+1}^{0,k} \to F_{k+1}^{k+1,0}$ in the $(k+1)$ th page of the spectral sequence (F). We note that $F_{k+1}^{0,k} = F_2^{0,k}$ and $F_{k+1}^{k+1,0} = F_2^{k+1,0}$ since, up to the $(k+1)$ th page, both terms have not been touched by any nonzero differential.

Proof of Theorem 6.1

Let $k = 2i < \theta $ be even. Since the differential $\delta _{2i-1}$ vanishes, the term of coordinate $(2i,0)$ in the spectral sequence (F) is unchanged through the deeper pages. In particular, $\mathrm {IH}^{2i}(S_{\theta })$ contains a subspace isomorphic to $\mathrm H^{2i}(S_{\theta })$ . Thus, we have

$$ \begin{align*}\mathrm H^{2i}(S_{\theta}) \hookrightarrow \mathrm{IH}^{2i}(S_{\theta}) \simeq \mathrm{IH}^{2(\theta-i)}(S_{\theta})(\theta-2i),\end{align*} $$

where the isomorphism follows from the hard Lefschetz theorem for intersection cohomology. Since $2(\theta -i)> \theta $ , the RHS is isomorphic to $A_0^{\theta -i,\theta -i}\oplus B_0^{\theta -i+1,\theta -i-1}$ by Proposition 8.1. But $A_0^{\theta -i,\theta -i} \simeq A_0^{i,i}$ and $B_0^{\theta -i+1,\theta -i-1} \simeq B_0^{i+1,i-1}$ as $\mathrm {Sp}(2\theta ,\mathbb F_q)$ -modules. As $\mathrm H^{2i}(S_{\theta })$ already contains $A_0^{i,i} \oplus B_0^{i+1,i-1}$ , we actually have isomorphisms $\mathrm {IH}^{2i}(S_{\theta }) = \mathrm H^{2i}(S_{\theta }) \simeq A_0^{i,i} \oplus B_0^{i+1,i-1}$ . At this stage, we have proved statement (2) of Theorem 6.1. According to Lemma 6.6, the proof is over.

As a by-product, we have proved the following statement.