2.1 General notation

Throughout the article, we let $\breve {k}/k$ with integers $\mathcal {O}_k \subseteq \mathcal {O}$ , residue field extension $\overline {\mathbb {F}}_q/\mathbb {F}_q$ , and Frobenius F be as in the introduction. We denote by $\varpi $ a uniformizer of k.

Given a $\mathbb {F}_q$ -algebra R, let $\mathrm {Perf}_R$ be the category of perfect R-algebras. For $R \in \mathrm {Perf}_{\mathbb {F}_q}$ , let $W(R)$ be the ring of p-typical Witt vectors of R, and put $\mathbb {W}(R) = W(R) \otimes _{\mathbb {Z}_p} \mathcal {O}_k$ if $\mathrm {char}(k) = 0$ , resp. $\mathbb {W}(R) = R[\![\varpi ]\!]$ otherwise. In particular, $\mathbb {W}(\mathbb {F}_q) = \mathcal {O}_k$ and $\mathbb {W}(\overline {\mathbb {F}}_q) = \mathcal {O}$ . Let $[\cdot ] \colon R \rightarrow \mathbb {W}(R)$ be the Teichmüller lift if $\mathrm {char}(k) = 0$ , resp. $[x] = x$ if $\mathrm {char}(k)> 0$ .

Let $\mathcal {X}$ be any $\mathcal {O}$ -scheme and let X be any $\breve {k}$ -scheme. We will abbreviate

$$\begin{align*}\breve{\mathcal{X}} := \mathcal{X}(\mathcal{O}) \quad \text{ and } \quad \breve{X} = X(\breve{k}). \end{align*}$$

Suppose that $\mathcal {X}$ is affine and of finite type over $\mathcal {O}$ . We regard the set $\breve {\mathcal {X}}$ as a perfect affine scheme $\mathbb {X}$ over $\overline {\mathbb {F}}_q$ , so that $\mathbb {X}(\overline {\mathbb {F}}_q) = \breve {\mathcal {X}}$ . More precisely, one puts $\mathbb {X} = L^+\mathcal {X}$ , where $L^+\mathcal {X} \colon \mathrm {Perf}_{\overline {\mathbb {F}}_q} \rightarrow \mathrm {Sets}$ , $L^+\mathcal {X}(R) = \mathcal {X}(\mathbb {W}(R))$ is the functor of positive loops; see, for example, [Reference Chan and Ivanov7, §2.5] for details. We always will identify the scheme $\mathbb {X}$ with the set $\breve {\mathcal {X}}$ of its geometric points. If $\mathcal {X}$ is defined over $\mathcal {O}_k$ , $\mathbb {X}$ has a natural $\mathbb {F}_q$ -structure, corresponding to the F-action on $\breve {\mathcal {X}}$ . Moreover, the set

$$\begin{align*}\mathbb{X}(\mathbb{F}_q) = \breve{\mathcal{X}}^F = \mathcal{X}(\mathcal{O}_k) \end{align*}$$

has a natural structure of a profinite set. Similarly, if X is affine of finite type over $\breve {k}$ , then we regard $\breve {X}$ as an ind-(perfect affine scheme) over $\overline {\mathbb {F}}_q$ via the loop functor $LX \colon \mathrm {Perf}_{\overline {\mathbb {F}}_q} \ni R \mapsto X(\mathbb {W}(R)[p^{-1}])$ , and the same claim about $\mathbb {F}_q$ -structure holds, except that now $\breve {X}^F$ is only locally profinite.

2.2 Group-theoretic setup

We fix a reductive group G defined over k and split over $\breve {k}$ . We fix a k-rational, $\breve {k}$ -split maximal torus T of G, and we denote by $N_G(T)$ its normalizer. Its Weyl group $W= N_G(T)/T$ is a finite étale group scheme over k becoming constant over $\breve {k}$ . We identify W with the set of its $\breve {k}$ -points, endowed with the action of F. We denote by $X_\ast (T)$ , $X^\ast (T)$ the groups of (co)characters of $T_{\breve {k}}$ , equipped with natural F-actions, and by $\langle ,\rangle \colon X^\ast (T) \times X_\ast (T) \rightarrow \mathbb {Z}$ the natural W- and F-equivariant pairing. We denote by N the order of F as an automorphism of $X_\ast (T)$ .

We fix a Borel subgroup $T \subseteq B \subseteq G$ defined over $\breve {k}$ , and we denote by U the unipotent radical of B. Denote by $\Phi \subseteq X^\ast (T)$ the set of roots of T in G, and by $\Phi ^+$ resp. $\Phi ^-$ the subset of positive roots corresponding to U resp. $\overline {U}$ . For each $\alpha \in \Phi $ , let $U_\alpha \cong \mathbb {G}_{a,\breve {k}}$ denote the corresponding root subgroup.

2.3 Filtration of the torus and affine roots

Let $\mathcal {T}$ denote the connected Néron model of T. Let $\breve T^0$ be the maximal bounded subgroup of $\breve T$ . Then $\mathcal {T}(\mathcal {O}) = \breve T^0$ . Moreover, for $r \in \mathbb {Z}_{\geq 0}$ ,

$$\begin{align*}\breve T^r = \{t \in \breve T^0 \colon \operatorname{\mathrm{ord}}_{\varpi}(\chi(t) - 1) \geq r \, \forall \chi \in X^\ast(T) \} \end{align*}$$

defines a descending separated filtration on $\breve T$ . For each r, one has an isomorphism

$$\begin{align*}V := X_\ast(T) \otimes \overline{\mathbb{F}}_q \stackrel{\sim}{\longrightarrow} \breve T^r /\breve T^{r+1}, \quad \lambda \otimes x \mapsto \lambda(1 + [x]\varpi^r). \end{align*}$$

Fix some (e.g., hyperspecial) point $\textbf {x}_0$ in the apartment $\mathcal {A}_{T,\breve {k}}$ of T in the reduced affine building of G over $\breve {k}$ . Let

$$\begin{align*}\widetilde \Phi_{\mathrm{aff}} = \{\alpha+m: x \mapsto -\alpha(x - {\mathbf x}_0) + m; \alpha \in \Phi, m \in \mathbb{Z} \} \cong \Phi \times \mathbb{Z} \end{align*}$$

be the set of affine roots. Let $\widetilde \Phi = \Phi _{\mathrm {aff}} \sqcup \mathbb {Z}_{\geq 0}$ be the (enlarged) set of affine roots of T in G. For an affine root $\alpha +m$ , we have the corresponding subgroup $\breve U_{\alpha +m} \subseteq \breve U$ . For $m \in \mathbb {Z}_{\geq 0}$ , the corresponding root subgroup is $\breve T^m$ . There is an action of F on $\widetilde \Phi $ , such that $F\breve U_{\alpha +m} = \breve U_{F(\alpha +m)}$ .

2.4 Parahoric model and Moy–Prasad quotients

Assume that T is elliptic. Then the apartment of T in the reduced affine building of G over k consists of precisely one point ${\mathbf x}$ . We denote by $\mathcal {G}$ the parahoric $\mathcal {O}_k$ -model of G with connected special fiber attached to ${\mathbf x}$ , and by $\mathcal {G}^+$ its pro-unipotent radical.

If $H \subseteq G$ is a closed subgroup, then we denote by $\mathcal {H}$ the closure of H in $\mathcal {G}$ .Footnote ² Similarly, we denote by $\mathcal {T}^+$ the closure of T in $\mathcal {G}^+$ .

Note that $\breve {\mathcal {G}}$ (resp. $\breve {\mathcal {G}}^+$ ) is generated by all $\breve U_{f}$ with $f \in \widetilde \Phi $ satisfying $f({\mathbf x}) \geq 0$ (resp. $f({\mathbf x})> 0$ ), and that $\breve {\mathcal {G}}/\breve {\mathcal {G}}^+$ is naturally isomorphic to the reductive quotient of the special fiber of $\mathcal {G}$ .

For any $h = r$ or $h = r+$ with $r \in \mathbb {Z}_{\geq 0}$ , Moy–Prasad have defined in [Reference Moy and Prasad24] the normal F-stable subgroup $\breve {\mathcal {G}}^h \subseteq \breve {\mathcal {G}}$ generated by all $\breve U_f$ with $f \in \widetilde \Phi $ satisfying $f({\mathbf x}) \geq h$ . Note that $\breve {\mathcal {G}} = \mathcal {G}(\mathcal {O})^0$ and $\breve {\mathcal {G}}^+ = \mathcal {G}(\mathcal {O})^{0+}$ . There is a smooth $\mathbb {F}_q$ -group scheme $\mathbb {G}_r$ with

$$\begin{align*}\mathbb{G}_r(\overline{\mathbb{F}}_q) = \breve{\mathcal{G}}/\breve{\mathcal{G}}^r. \end{align*}$$

It has the subgroup $\mathbb {G}_r^+ = \breve {\mathcal {G}}^+/\breve {\mathcal {G}}^r$ , and the set of affine roots appearing in $\mathbb {G}_r^+$ is

$$\begin{align*}\widetilde \Phi_r^+ = \{f \in \widetilde\Phi \colon 0 < f({\mathbf x}) < r \}. \end{align*}$$

According with §2.1, we have also the $\mathbb {F}_q$ -groups $\mathbb {G}$ and $\mathbb {G}^+$ such that $\mathbb {G}(\overline {\mathbb {F}}_q) = \breve {\mathcal {G}}$ and $\mathbb {G}^+(\overline {\mathbb {F}}_q) = \breve {\mathcal {G}}^+$ . Note that and .

Note that any of the subgroups $H = T,B,U, \dots $ of G defines a closed subgroup $\mathbb {H}_r \subseteq \mathbb {G}_r$ (resp. $\mathbb {H} \subseteq \mathbb {G}$ ) by first taking the closure $\mathcal {H} \subseteq \mathcal {G}$ of H, and then letting $\mathbb {H}_r(\overline {\mathbb {F}}_q)$ be the image of the map $\mathcal {H}(\mathcal {O}) \rightarrow \mathcal {G}(\mathcal {O}) \rightarrow \mathbb {G}_r(\overline {\mathbb {F}}_q)$ . Similarly, H defines a closed subgroup $\mathbb {H}_r^+ \subseteq \mathbb {G}_r^+$ (and $\mathbb {H}^+ \subseteq \mathbb {G}^+$ ). Note that if $F^sH = H$ for some $s\geq 1$ , then $\mathbb {H}_r,\mathbb {H}_r^+$ are defined over $\mathbb {F}_{q^s}$ .

2.5 Coxeter pairs

Let $c \in W$ be the unique element such that $FB = {}^c B$ . Then for any lift $\dot c$ of c, ${\mathrm {Ad}}({\dot c})^{-1} \circ F: \breve G \to \breve G$ fixes the pinning $(T,B)$ , and hence defines an automorphism $\sigma _W$ of the Coxeter system $(W,S)$ . We call $(T,B)$ (and $(T,U)$ ) a Coxeter pair if c is a Coxeter element in the Coxeter triple $(W,S,\sigma _W)$ – that is, if a(ny) reduced expression of c contains precisely one element from each $\sigma _W$ -orbit on S. More generally, $(T, U)$ is called a minimal elliptic pair if c is of minimal length in its $\sigma _W$ -twisted conjugacy class. We have implications $(T,B)$ Coxeter $\Rightarrow $ $(T,B)$ minimal elliptic $\Rightarrow $ T is elliptic.

We define

$$\begin{align*}\Delta := \Phi^- \cap F\Phi^+. \end{align*}$$

Note that if $(T,B)$ is Coxeter, then each F-orbit in $\Phi $ has length exactly N and intersects the set in precisely one element; see, for example, [Reference Steinberg28, §7]. In particular, $\#\Delta $ is equal to the semisimple rank of G, $\Phi / \langle c \sigma _W \rangle \cong \Delta $ and $\#\Phi = N \cdot \#\Delta $ .

2.6 A condition on p

Assume that T is elliptic. We will prove Theorem 5.5 under the following condition on the characteristic p of $\mathbb {F}_q$ , which is satisfied if p does not divide the order of the Weyl group of G.

Note the all torsion primes for $\Phi $ are $\leq 5$ . Note that the second part of this condition holds for all p when $G_{\mathrm {der}}$ is simply connected. Let $P = P(G,T)$ denote the set of primes, for which this condition does not hold. If G is of type $A_n$ , then $P \subseteq \{\ell \text { prime } \colon \ell \text { divides } n \}$ . If G is of type $B_n$ or $C_n$ with n even, then $P \subseteq \{2\}$ . If G is of type $B_n$ or $C_n$ with n odd, then $P \subseteq \{2\} \cup \{\ell \text { prime } \colon \ell \text { divides } n\}$ . If G is of type $D_n$ , then $P \subseteq \{\ell \text { prime } \colon \ell < n\}$ .

We will use this condition in the proof of Theorem 5.5 by applying the following lemma to derived subgroups of various F-stable unramified twisted Levi subgroups of G containing T. Recall $V = X_\ast (T) \otimes \overline {\mathbb {F}}_q$ from §2.3 and consider the following norm map:

$$\begin{align*}\mathrm{Nm}_N: V \to V, ~ v \mapsto v + F(v) + \cdots + F^{N-1}(v). \end{align*}$$

Proof. By assumption, we have $\mathbb {Z}\Phi ^\vee \otimes \mathbb {F}_{q^N} = X_*(T) \otimes \mathbb {F}_{q^N}$ . Hence,

$$\begin{align*}V^F = \mathrm{Nm}_N(V^{F^N}) = \mathrm{ Nm}_N(X_*(T) \otimes \mathbb{F}_{q^N}) = \mathrm{Nm}_N(\mathbb{Z}\Phi^\vee \otimes \mathbb{F}_{q^N}),\end{align*}$$

as desired.

2.7 Homology

For a morphism $Y \rightarrow Z$ of perfect $\mathbb {F}_p$ -schemes and a coefficient ring $\Lambda $ , which we assume to be either $\overline {\mathbb {Q}}_\ell $ or $\overline {\mathbb {F}}_\ell $ here, in [Reference Ivanov and Mann18] the left adjoint $f_\natural \colon D_{\blacksquare }(Y,\Lambda ) \rightarrow D_{\blacksquare }(Z, \Lambda )$ of $f^\ast $ on unramified solid sheaves is constructed. Readers feeling uncomfortable with the use $f_\natural $ , could just regard (2.2) as a definition (which is well-behaved because of (2.1)). Assume that $Z = \operatorname {\mathrm {Spec}} \overline {\mathbb {F}}_q$ , in which case we get the $\Lambda $ -module

$$\begin{align*}H_i(Y,\Lambda) := H^{-i}f_\natural\Lambda. \end{align*}$$

Assume now that with all $f_r \colon Y_r \rightarrow \operatorname {\mathrm {Spec}} \overline {\mathbb {F}}_q$ perfections of smooth morphisms of dimension $d_r$ . Assume that there are compatible actions of finite groups $\Gamma _r$ on $Y_r$ , inducing an action of on Y. Let $\chi \colon \Gamma \rightarrow \Lambda ^\times $ be a smooth character. There is some $r_\chi \geq 0$ such that for each $r\geq r_\chi $ , $\chi $ factors through a character of $\Gamma _r$ again denoted $\chi $ . Assume that for all $r \geq r_\chi $ , the map

(2.1) $$ \begin{align} f_{r !}\Lambda[\chi][2(d_r - d_{r_\chi})] \rightarrow f_{r_\chi !}\Lambda[\chi]. \end{align} $$

is an isomorphism. It then follows from the properties of $f_\natural $ – see [Reference Ivanov and Mann18] – that

where the second equality holds because $f_r$ is smooth (and hence $f_{r \natural } = f_{r!}[2d_r]$ ) and the last equality is by (2.1). With other words,

(2.2) $$ \begin{align} H_i(Y,\Lambda)[\chi] = H^{2d_r - i}_c(Y_{r_\chi},\Lambda)[\chi] \quad \text{for all } r \geq r_\chi. \end{align} $$

4.1 Comparison with the definition in [Reference Ivanov19]

Assume that T is elliptic. Assume that G admits a (necessarily unique) unramified inner form $G_0$ over k (the general case easily reduces to this by using a derived embedding of G into a group with connected center). Then one can choose

• • a k-rational pinning $(T_0, B_0 = T_0 U_0)$ of $G_0$ with Weyl group $(W_0,S_0)$ ,

• • an elliptic element $w \in W_0$ ,

• • a lift $\dot w \in N(T_0)(\breve {k})$ ,

such that there is a $\breve {k}$ -rational isomorphism $G \stackrel {\sim }{ \rightarrow } G_0$ , identifying $T,B,U,W$ with $T_0,B_0,U_0,W_0$ , and F with $Ad(\dot w) \circ \sigma $ as an automorphism of $\breve G \cong \breve G_0$ . Let $b \in \breve G$ . In [Reference Ivanov19, §8], the p-adic Deligne–Lusztig space attached to the datum $(G_0, \dot w, b)$ is defined as the arc-sheaf on perfect $\overline {\mathbb {F}}_q$ -schemes,

$$\begin{align*}\dot X_{\dot w}(b) = \{x \in L(G_0/U_0) \colon x^{-1}b\sigma(x) \in L(U_0 \dot w U_0) \}, \end{align*}$$

where $L(\cdot )$ is the perfect loop functor as in §2.1. Note that $(g,t) \colon x \mapsto gxt$ defines an action of the locally profinite group $G(k) \times T(k)$ on this arc-sheaf; see [Reference Ivanov19, §8] for details. This seems to be a natural definition, most similar to classical Deligne–Lusztig varieties.

Note that w equals the relative position of U with $FU$ . Thus, $(T,U)$ is a Coxeter (resp. minimal elliptic) pair if and only if w is a Coxeter (resp. minimal elliptic) element.

Identifying G with $G_0$ via the given isomorphism, we have the composition

(4.1) $$ \begin{align} \nonumber X_{T,U} & \rightarrow \{g \in \breve G_0 \colon g^{-1}\dot w\sigma(g) \in \dot w \breve U_0 \}/(\breve U_0 \cap {}^{w}\breve U_0)\\ &\stackrel{\sim}{ \rightarrow } \dot X_{\dot w}(\dot w), \end{align} $$

given by $g \mapsto g (\breve U_0 \cap {}^{w}\breve U_0) \mapsto g\breve U_0$ . Just as was done in [Reference Ivanov19, Proposition 5.12] for classical groups, we deduce from Proposition 3.1:

4.2 Integral decomposition of $X_{T,U}$

A priori, $X_{T,U}$ is a huge ind-scheme, which is hard to control. However, in the Coxeter case, it has the following decomposition. Let X be as in (1.2) and note that $X \subseteq X_{T,U}$ is a closed subscheme. Surprisingly, it is also open and the following holds.

Theorem 4.2 [Reference Ivanov20],[Reference Nie25].

Suppose $(T,U)$ is a Coxeter pair and let X be as in (1.2). Then there is a decomposition

$$\begin{align*}X_{T,U} = \bigsqcup_{g \in G(k)/\mathcal{G}(\mathcal{O}_k)} g X. \end{align*}$$

In particular, $X_{T,U}$ is a disjoint union of affine perfect $\overline {\mathbb {F}}_q$ -schemes.

This reduces the study of the cohomology of $X_{T,U}$ to that of X.

4.3 Drinfeld stratification

In this subsection, the ellipticity assumption on T can be dropped. Note that the projection $\mathbb {G} \rightarrow \mathbb {G}_{0+}$ restricts to a projection

$$\begin{align*}X \rightarrow X_{0+} = \{g \in \mathbb{G}_{0+} \colon g^{-1}F(g) \in \overline{\mathbb{U}}_{0+} \cap F\mathbb{U}_{0+}\}\end{align*}$$

over (a variant of) a classical Deligne–Lusztig variety. Let $\mathfrak {L}$ denote the set of all twisted Levi subgroups of $\mathbb {G}_{0+}$ containing $\mathbb {T}_{0+}$ . For any $\mathbb {L}_{0+} \in \mathfrak {L}$ , we have the locally closed $\mathbb {G}_{0+}^F \times \mathbb {T}_{0+}^F$ -stable closed subscheme

$$\begin{align*}X_{0+}^{(\mathbb{L}_{0+})} = \{ g \in \mathbb{G}_{0+} \colon g^{-1}F(g) \in \mathbb{L}_{0+} \cap \overline{\mathbb{U}}_{0+} \cap F\mathbb{U}_{0+} \}. \end{align*}$$

Pulling back to X, we obtain a closed subscheme $X^{(\mathbb {L}_{0+})} \subseteq X$ . Following [Reference Chan and Ivanov8] and [Reference Chan and Oi10, §6.2], we then call

$$\begin{align*}X^{(\mathbb{L}_{0+})} {\,\setminus\,} \bigcup_{\mathbb{L}_{0+}' \subseteq \mathbb{L}_{0+} \in \mathfrak{L}} X^{(\mathbb{L}'_{0+})} \end{align*}$$

a Drinfeld stratum of X. This defines a finite and locally closed stratification of X. Its has a unique minimal/closed stratum $X^{(\mathbb {T}_{0+})}$ .

In the rest of the article, we consider Y and its cohomology. To approximate Y, consider for any $r \in \mathbb {Z}_{>0}$ the affine perfect $\overline {\mathbb {F}}_q$ -scheme

$$\begin{align*}Y_r = \{ g \in \mathbb{G}_r^+ \colon g^{-1}F(g) \in \overline{\mathbb{U}}_r\cap F\mathbb{U}_r^+ \}, \end{align*}$$

equipped with $(\mathbb {G}_r^+)^F \times (\mathbb {T}_r^+)^F$ -action, so that . Similarly, we have the schemes $X_r^{(\mathbb {L}_{0+})} \subseteq \mathbb {G}_r$ approximating $X^{(\mathbb {L}_{0+})}$ .

Recall the set $\Delta $ from §2.5. Let $\Phi ^{\mathrm {red}}$ denote the set of those $\alpha \in \Phi $ for which $\alpha ({\mathbf x}) \in \mathbb {Z}$ , and let $\Delta ^{\mathrm {red}} = \Phi ^{\mathrm {red}} \cap \Delta $ .

5.1 A total order on affine roots

For $f \in \widetilde \Phi $ , we write $\alpha _f \in \Phi \sqcup \{0\}$ and $m_f \in \mathbb {Z}$ such that $f = \alpha _f + m_f$ . Let $\mathcal {O}_f$ be the F-orbit of f.

Let $\Phi _{\mathrm {aff}}^+$ (resp. $\widetilde \Phi ^+$ ) be the set of affine roots $f \in \Phi _{\mathrm {aff}}$ (resp. $f \in \widetilde \Phi $ ) such that $f({\mathbf x})> 0$ . Note that $\widetilde \Phi ^+ = \Phi _{\mathrm {aff}}^+ \sqcup \mathbb {Z}_{\geqslant 1}$ and $\widetilde \Phi = \Phi _{\mathrm {aff}}^+ \sqcup \widetilde \Phi ^0 \sqcup -\widetilde \Phi _{\mathrm {aff}}^+$ . Here, $\widetilde \Phi ^0 = \{f \in \widetilde \Phi; f({\mathbf x}) = 0\}$ .

Recall that $\Delta = \Phi ^- \cap F\Phi ^+$ . Set $\Delta _{\mathrm {aff}}^+ = (\Delta \times \mathbb {Z}) \cap \Phi _{\mathrm {aff}}^+$ and $\widetilde \Delta ^+ = \Delta _{\mathrm {aff}}^+ \sqcup \mathbb {Z}_{\geqslant 1}$ .

Let $f \in \widetilde \Delta ^+$ . We denote by $f+$ and $f-$ the descendant and the ascendant of f in $\widetilde \Delta ^+ \sqcup \{0\}$ , respectively, such that $0 = f-$ if $f = \min \widetilde \Delta ^+$ . Set $\widetilde \Phi ^f = \{f' \in \widetilde \Phi ^+; f' \geqslant f\}$ .

5.2 The variety $Y^A_B$

We fix an integer $r \in \mathbb {Z}_{\geqslant 1}$ . Let $\mathbb {G}_r^+ = \breve {\mathcal {G}}^{0+} / \breve {\mathcal {G}}^r$ and let $\widetilde \Phi _r^+ = \{f \in \widetilde \Phi; 0 < f({\mathbf x}) < r\}$ be the set of affine roots appearing in $\mathbb {G}_r^+$ .

Let $f \in \widetilde \Phi ^+$ . If $f \in \Phi _{\mathrm {aff}}^+$ , we take $\mathbb {A}_f = \mathbb {G}_a$ and define $u_f: \mathbb {A}^1 \to \mathbb {G}_r^+$ by $x \mapsto U_{\alpha _f}([x]\varpi ^{m_f})$ for $x \in \overline {\mathbb {F}}_q$ . If $f \in \mathbb {Z}_{\geqslant 1}$ , we take $\mathbb {A}_f = V := X_*(T) \otimes \overline {\mathbb {F}}_q$ and define $u_f: \mathbb {A}_f \to \mathbb {G}_r^+$ by $\lambda \otimes x \mapsto \lambda (1 + [x] \varpi ^{n_f})$ for $\lambda \in X_*(T)$ and $x \in \overline {\mathbb {F}}_q$ .

We define an abelian group $\mathbb {A}[r] = \prod _{f \in \widetilde \Phi _r^+} \mathbb {A}_f$ . Then we have an isomorphism of varieties

$$\begin{align*}u: \mathbb{A}[r] \overset \sim \to \mathbb{G}_r^+, \quad (x_f)_f \mapsto \prod_f u_f(x_f),\end{align*}$$

where the product is taking with respect to the linear order $\leqslant $ on $\widetilde \Phi ^+$ .

Let $E \subseteq \widetilde \Phi _r^+$ . We set $\mathbb {A}_E = \prod _{f \in E} \mathbb {A}_f$ , which is viewed as a subgroup group of $\mathbb {A}[r]$ in the natural way. Denote by $p_E: \mathbb {A}[r] \to \mathbb {A}_E$ the natural projection. Using the identification $u: \mathbb {A}[r] \cong \mathbb {G}_r^+$ , we define

$$\begin{align*}\mathrm{pr}_E = u \circ p_E \circ u^{-1}: \mathbb{G}_r^+ \to u(\mathbb{A}_E).\end{align*}$$

For $f \in \widetilde \Phi _r^+$ , we put $p_f = p_{\{f\}}$ and $\mathrm {pr}_f = \mathrm {pr}_{\{f\}}$ . By abuse of notation, we will identify $\mathrm {pr}_f: \mathbb {G}_r^+ \to u(\mathbb {A}_f)$ with $u^{-1} \circ \mathrm { pr}_f: \mathbb {G}_r^+ \to \mathbb {A}_f$ freely according to the context.

Let $A, B \subseteq \widetilde \Phi ^+$ be two subsets. We set

$$\begin{align*}A + B = \{f + f' \in \widetilde \Phi; f \in A, f' \in B\}.\end{align*}$$

We say A is closed if $A + A \subseteq A$ and $A + \mathbb {Z}_{\geqslant 0} = A$ . In this case, we denote by $\mathbb {G}_r^A \subseteq \mathbb {G}_r^+$ the subgroup generated by $u(\mathbb {A}_f)$ for $f \in A$ .

Suppose that $\widetilde \Phi ^r \subseteq A, B \subseteq \widetilde \Phi ^+$ are F-stable and closed such that $B \subseteq A$ and $A + B \subseteq B$ . Then $\mathbb {G}_r^B$ is a normal subgroup of $\mathbb {G}_r^A$ . The isomorphism $u: \mathbb {A}[r] \overset \sim \to \mathbb {G}_r^+$ restricts to an isomorphism $u_{A:B}: \mathbb {A}_{A \setminus B} \overset \sim \to \mathbb {G}_r^A / \mathbb {G}_r^B$ . So we get an embedding

$$\begin{align*}s_{A:B} = u \circ u_{A:B}^{-1} : \mathbb{G}_r^A/\mathbb{G}_r^B \to \mathbb{G}_r^+.\end{align*}$$

We define

$$\begin{align*}Y_B^A = \{g \in \mathbb{G}_r^A; g^{-1} F(g) \in (\overline{\mathbb{U}}_r \cap F\mathbb{U}_r) \mathbb{G}_r^B\} / \mathbb{G}_r^B \subseteq \mathbb{G}_r^A / \mathbb{G}_r^B,\end{align*}$$

which admits a natural action by $(\mathbb {G}_r^A)^F \times (\mathbb {T}_r^+ \cap \mathbb {G}_r^A)^F$ . Let $\chi : (\mathbb {T}_r^+ \cap \mathbb {G}_r^A)^F \to \overline {\mathbb {Q}}_\ell ^\times $ be a character. We denote by $H_c^i(Y_B^A, \overline {\mathbb {Q}}_\ell )[\chi ]$ the $\chi $ -weight space of the $(\mathbb {T}_r^+)^F$ -action on $H_c^i(Y_B^A, \overline {\mathbb {Q}}_\ell )$ . For $f \in \widetilde \Phi _r^+$ , we define

$$\begin{align*}\pi^{A:B}_f = u^{-1} \circ \mathrm{ pr}_{\mathcal{O}_f} \circ L \circ s_{A:B}: \mathbb{G}_r^A/\mathbb{G}_r^B \to \mathbb{A}_{\mathcal{O}_f}.\end{align*}$$

Here, for any F-stable sub-quotient group of $\mathbb {G}_r^+$ , we always denote by L the Lang’s self-map given by $g \mapsto g^{-1} F(g)$ .

Proposition 5.3. Let $\widetilde \Phi ^r \subseteq A, B \subseteq \widetilde \Phi ^+$ be F-stable and closed. Let $f \in B$ and $C = B \setminus \mathcal {O}_f$ . Suppose that $\widetilde \Phi ^r \subseteq C$ is closed, $C + A \subseteq C$ and $\mathcal {O}_f + A \subseteq C$ . Then

(1) if $f \in \Delta _{\mathrm {aff}}^+$ , then the map $\psi = (q_f, \mathrm {pr}_f): Y_C^A \cong Y_B^A \times \mathbb {A}_f$ is an isomorphism;

(2) if $f \in \mathbb {Z}_{\geqslant 1}$ (in which case $\mathbb {A}_{\mathcal {O}_f} = \mathbb {A}_f = V$ ), then there is a Cartesian diagram

Here, $q_f$ denotes the natural projection.

Proof. By assumption, the map u induces an identification $\mathbb {A}_{\mathcal {O}_f} \cong \mathbb {G}_r^B/\mathbb {G}_r^C$ as abelian groups. Moreover,

(a) $\mathbb {G}_r^B/\mathbb {G}_r^C$ lies in the center of $\mathbb {G}_r^A/\mathbb {G}_r^C$ .

Assume that $f \in \Delta _{\mathrm {aff}}^+$ . We define a morphism $\phi : Y_B^A \times \mathbb {A}_f \to Y_C^A$ as follows. Let $(g, y) \in Y_B^A \times \mathbb {A}_f$ . Write $\pi ^{A:B}_f(g) = (z_i)_{1 \leqslant i \leqslant N} \in \mathbb {A}_{\mathcal {O}_f}$ with each $z_i \in \mathbb {A}_{F^i(f)}$ . We define

$$\begin{align*}\phi(g, y) = s_{A:B}(g) u(y) F(u(y)) \cdots F^{N-1}(u(y)) \prod_{1 \leqslant i \leqslant N-1} u(z_i) F(u(z_i)) \cdots F^{N-i-1}(u(z_i)).\end{align*}$$

By (a), one checks that

$$\begin{align*}\phi(Y_B^A \times \mathbb{A}_f) \subseteq Y_C^A\end{align*}$$

and $\psi \circ \phi = \mathrm {id}$ . Let $g \in Y_C^A$ and set $g' = \phi (\psi (g)) \in Y_C^A$ . Then $\psi (g) = \psi (g')$ ; that is, $g^{-1} g' \in \mathbb {A}_{\mathcal {O}_f \setminus \{f\}} \subseteq \mathbb {G}_r^B / \mathbb {G}_r^C$ . As $g, g' \in Y_C^A$ , it follows by (a) that

$$\begin{align*}L(g^{-1} g') = L(g)^{-1} L(g') \in \mathbb{A}_f \subseteq \mathbb{G}_r^B / \mathbb{G}_r^C.\end{align*}$$

Hence, $g = g'$ by Lemma 5.4. So $\phi \circ \psi = \mathrm {id}$ and (1) is proved.

Assume that $f \in \mathbb {Z}_{\geqslant 1}$ . As both vertical maps in the diagram are finite étale $V^F$ -torsors, it suffices to show that the square commutes. Let $g \in Y_C^A$ . Write $s_{A:C}(g) = u(x) u(y)$ with $x \in \mathbb {A}_{A \setminus B}$ and $y \in \mathbb {A}_f$ . Then $\mathrm {pr}_f(g) = y$ and $q_f(g) = u(x)$ . As $f \in \mathbb {Z}_{\geqslant 1}$ and $g \in Y_C^A$ , we have $\mathrm {pr}_f (L(s_{A:C}(g)) = 0 \in \mathbb {A}_f$ . Using (a), one computes that

$$ \begin{align*} \pi_f^{A:B}(q_f(g)) &= \mathrm{pr}_f(L(u(x))) \\ &= \mathrm{pr}_f(L(s_{A:C}(g) u(y)^{-1})) \\ &= \mathrm{pr}_f(L(s_{A:C}(g)) L(u(y)^{-1})) \\ &= \mathrm{pr}_f(L(s_{A:C}(g))) \mathrm{pr}_f(L(u(y)^{-1})) \\ &= L(y^{-1}) \\ &= - L(y). \end{align*} $$

So (2) is proved.

Proof. By definition, we have

$$\begin{align*}L(x) = F(x) - x = \sum_{i=0}^{N-1}F(x_{i-1}) - x_i \in \mathbb{A}_{\mathcal{O}_f},\end{align*}$$

from which the lemma follows.

5.3 Main result

For $f' \leqslant f \in \widetilde \Delta ^+$ , we set $\mathbb {G}_f^+ = \mathbb {G}_r^+ / \mathbb {G}_r^{\widetilde \Phi ^f}$ , $Y_f = Y^{\widetilde \Phi ^+}_{\widetilde \Phi ^f}$ , $\mathbb {T}_f = \mathbb {T}_{\lceil f \rceil }$ and $\mathbb {T}_f^{f'} = \ker (\mathbb {T}_f \to \mathbb {T}_{f'})$ , where $\widetilde \Phi ^f = \{f' \in \widetilde \Phi ^+; f' \geqslant f\}$ and $\lceil f \rceil = \min \{n \in \mathbb {Z}_{\geqslant 1}, n \geqslant f\}$ . Note that $\mathbb {T}_{f+}^{f}$ is nontrivial if and only if $f \in \mathbb {Z}_{\geqslant 1}$ , in which case $\mathbb {T}_{f+}^f \cong V = X_*(T) \otimes \overline {\mathbb {F}}_q$ .

Theorem 5.5. Assume that p satisfies Condition 2.1. Let $f \in \widetilde \Delta ^+$ and let $\chi : (\mathbb {T}_f^+)^F \to \overline {\mathbb {Q}}_\ell ^\times $ be a character. Then there exists $s = s_{f, \chi } \in \mathbb {Z}_{\geqslant 0}$ such that

$$\begin{align*}H_c^i(Y_f, \overline{\mathbb{Q}}_\ell)[\chi] \neq 0 \Longleftrightarrow i = s,\end{align*}$$

on which $F^N$ acts by multiplication by $(-1)^s q^{sN/2}$ .

After necessary preparations, we prove Theorem 5.5 in §5.7. We compute the cohomological degree $s_{\chi ,r}$ explicitly in terms of the Howe factorization of $\chi $ in §5.8. We will make use of the following well-known result.

Theorem 5.6. Let $f: Z \to Y$ be an étale $\Gamma $ -torsor, where $\Gamma $ is a finite group. Let $\Lambda $ be a ring. Assume that either $\Lambda $ is finite, or $Z, Y$ are irreducible and geometrically unibranch. Then

$$\begin{align*}f_! (\Lambda) = \bigoplus_\rho \rho \otimes \mathcal{E}_\rho,\end{align*}$$

where $\rho $ ranges over irreducible representations of $\Gamma $ and $\mathcal {E}_\rho $ is a local system on Y.

Proof. It suffices to show $\pi _!(\overline {\mathbb {Q}}_\ell ) \cong \overline {\mathbb {Q}}_\ell [-2]$ as $\Gamma $ -equivariant sheaves. Indeed, the adjunction map gives an isomorphism

$$\begin{align*}\pi_!(\overline{\mathbb{Q}}_\ell) \cong \pi_!\pi^*(\overline{\mathbb{Q}}_\ell) \cong \pi_!\pi^!(\overline{\mathbb{Q}}_\ell [-2]) \cong \overline{\mathbb{Q}}_\ell [-2]\end{align*}$$

as $\Gamma $ -equivariant sheaves.

5.4 Multiplicative local systems

Let P be a commutative unipotent algebraic group defined over $\mathbb {F}_q$ . Then the map $\mathcal {L} \mapsto t_{\mathcal {L}}$ induces a bijection from the isomorphism classes of multiplicative local systems on P to the set $\operatorname {\mathrm {Hom}}(H(\mathbb {F}_q), \overline {\mathbb {Q}}_\ell ^\times )$ of characters of $P(\mathbb {F}_q)$ . Here, $t_{\mathcal {L}}: P(\mathbb {F}_q) \to \overline {\mathbb {Q}}_\ell ^\times $ is the trace-of-Frobenius function for $\mathcal {L}$ . See [Reference Boyarchenko and Drinfeld1, §1.8] for details. For $\theta \in \operatorname {\mathrm {Hom}}(P(\mathbb {F}_q), \overline {\mathbb {Q}}_\ell ^\times )$ , we denote by $\mathcal {L}_\theta $ the multiplicative local system corresponding to $\theta $ .

For a character $\chi $ of $(\mathbb {T}_{f+}^+)^F$ , we denote by $\chi _{f+}^f$ the restriction of $\chi $ to $(\mathbb {T}_{f+}^f)^F$ . Proposition 5.3 has the following consequence:

Corollary 5.9. Let $f \in \widetilde \Delta ^+$ and let $\chi $ be a character of $(\mathbb {T}_{f+}^+)^F$ .

(1) if $f \in \Delta _{\mathrm {aff}}^+$ , then $H_c^i(Y_{f+}, \overline {\mathbb {Q}}_\ell )[\chi ] \cong H_c^{i-2}(Y_f, \overline {\mathbb {Q}}_\ell )[\chi ]$ ;

(2) if $f \in \mathbb {Z}_{\geqslant 1}$ , then $H_c^i(Y_{f+}, \overline {\mathbb {Q}}_\ell )[\chi _{f+}^f] \cong H_c^i(Y_f, \pi ^*(\mathcal {L}_{\chi _{f+}^f}))$ , and hence,

$$\begin{align*}H_c^i(Y_{f+}, \overline{\mathbb{Q}}_\ell)[\chi] \cong H_c^i(Y_f, \pi^*(\mathcal{L}_{\chi_{f+}^f}))[\chi].\end{align*}$$

Here, $\pi = \pi ^{\widetilde \Phi ^+:\widetilde \Phi ^f}_f$ and $H_c^i(Y_{f+}, \overline {\mathbb {Q}}_\ell )[\chi _{f+}^f]$ is the $\chi _{f+}^f$ -weight space of $(\mathbb {T}_{f+}^f)^F$ .

Proof. If $f \in \Delta _{\mathrm {aff}}^+$ , by Lemma 5.3 (1), we have $Y_{f+} \cong Y_f \times \mathbb {G}_a$ , and the natural projection $q_f: Y_{f+} \to Y_f$ respects the right actions of $(\mathbb {T}_{f+}^+)^F = (\mathbb {T}_f^+)^F$ on $Y_{f+}$ and $Y_f$ . So the statement (1) follows from Proposition 5.7.

Now assume that $f \in \mathbb {Z}_{\geqslant 1}$ . Note that the Lang’s map $L: \mathbb {T}_{f+}^f \to \mathbb {T}_{f+}^f$ is an étale $(\mathbb {T}_{f+}^f)^F$ -torsor. It follows from Theorem 5.6 that

$$\begin{align*}L_! (\overline{\mathbb{Q}}_\ell) = \bigoplus_\theta \mathcal{L}_\theta,\end{align*}$$

where $\theta $ ranges over characters of $(\mathbb {T}_{f+}^f)^F$ , and $\mathcal {L}_\theta $ is the multiplicative local system corresponding to $\theta $ . By the Cartesian diagram in Proposition 5.3 (2), it follows from the base change theorem that

$$\begin{align*}(q_f)_!(\overline{\mathbb{Q}}_\ell) = (q_f)_! \mathrm{pr}_f^* (\overline{\mathbb{Q}}_\ell) \cong \pi_f^* L_!(\overline{\mathbb{Q}}_\ell) = \bigoplus_\theta \pi_f^* \mathcal{L}_\theta.\end{align*}$$

So the statement (2) follows by noticing that $(\mathbb {T}_{f+}^f)^F$ acts on the sheaf $\mathcal {L}_\theta $ via the character  $\theta $ .

From this we deduce the following:

5.5 Reduction to the semisimple case

Let $G' \subseteq G$ be the derived subgroup. Let $T'$ be a maximal torus of $G'$ contained in T. One can define the objects $Y_f' = Y_f$ for $G'$ in a similar way.

Lemma 5.11. For $f \in \widetilde \Delta ^+$ , we have

$$\begin{align*}Y_f = \bigsqcup_{x \in (\mathbb{T}_f^+)^F/ (\mathbb{T}_f^{\prime +})^F} x Y_f' = \bigsqcup_{x \in (\mathbb{T}_f^+)^F / (\mathbb{T}_f^{\prime +})^F} Y_f' x^{-1}.\end{align*}$$

In particular, $H_c^i(Y_f, \overline {\mathbb {Q}}_\ell ) \cong \mathrm { ind}_{(\mathbb {T}_f^{\prime +})^F}^{(\mathbb {T}_f^+)^F} H_c^i(Y_f', \overline {\mathbb {Q}}_\ell )$ as $(\mathbb {T}_f^+)^F$ -modules.

Proof. Let $g \in Y_f$ . Then $g^{-1} F(g) \in \overline {\mathbb {U}}_f^+ \cap F\mathbb {U}_f^+ \subseteq \mathbb {G}_f^{\prime +}$ . By Lang’s theorem, there exists $g' \in \mathbb {G}_f^{\prime +}$ such that ${g'}^{-1} F(g') = g^{-1} F(g)$ . So $g = (g {g'}^{-1}) g' \in (\mathbb {G}_f^+)^F Y_f'$ , and hence, $Y_f = (\mathbb {G}_f^+)^F Y_f'$ .

However, there is a natural isomorphism

$$\begin{align*}(\mathbb{T}_f^+)^F / (\mathbb{T}_f^{\prime +})^F \cong (\mathbb{G}_f^+)^F / (\mathbb{G}_f^{\prime +})^F.\end{align*}$$

Now it follows that

$$\begin{align*}Y_f^+ = (\mathbb{G}_f^+)^F Y_f' = \bigsqcup_{x \in (\mathbb{T}_f^+)^F/ (\mathbb{T}_f^{\prime +})^F} x Y_f^{\prime +} = \bigsqcup_{x \in (\mathbb{T}_f^+)^F / (\mathbb{T}_f^{\prime +})^F} Y_f^{\prime +} x^{-1},\end{align*}$$

where the last equality follows from the observation that $(\mathbb {T}_f^+)^F$ normalizes $Y_f'$ .

5.6 Handling jumps in the Howe factorization of $\chi $

We fix a positive integer $h \leqslant r$ and a character $\chi $ of $(\mathbb {T}_{h+}^+)^F $ . Recall that $\mathbb {T}_{h+}^h \cong \mathbb {A}_h = V = X_*(T) \otimes \overline {\mathbb {F}}_q$ , and recall from §2.6 the norm map

$$\begin{align*}\mathrm{Nm}_N: V \to V, ~ v \mapsto v + F(v) + \cdots + F^{N-1}(v). \end{align*}$$

Using the character $\chi $ , we define a root system

$$\begin{align*}\Phi_\chi = \{\alpha \in \Phi; \chi \circ \mathrm{Nm}_N (\alpha^\vee \otimes \mathbb{F}_{q^N}) = \{1\}\}.\end{align*}$$

Note that $\Phi _\chi $ is F-stable. By [Reference Kaletha21, Lemma 3.6.1], it is a Levi subsystem of $\Phi $ (note that by Condition 2.1, p is not a torsion prime for $\Phi $ ).

Let $M = M_\chi \subseteq G$ be the twisted Levi subgroup generated by T and $U_\alpha $ for $\alpha \in \Phi _\chi $ . Let $\widetilde \Phi _M$ be the set of affine roots of M. We set

$$\begin{align*}D = (\Delta_{\mathrm{aff}}^+ \cap \Phi_h^+) \setminus \widetilde \Phi_M = \{f \in \Delta_{\mathrm{aff}}^+ \setminus \widetilde \Phi_M; f < h\}.\end{align*}$$

By Lemma 5.1, the map $f \mapsto \mathcal {O}_f$ gives a natural bijection

$$\begin{align*}D \overset \sim \to (\widetilde \Phi_h^+ \setminus \widetilde \Phi_M) / \langle F \rangle.\end{align*}$$

Let $f \in D$ . As $f < h$ , it follows by Definition 5.2 that $0 < f({\mathbf x}) < h$ , and hence, $h-f \in \widetilde \Phi _h^+ \setminus \widetilde \Phi _M$ . Hence, there exists a unique affine root $f^\flat \in \Delta _{\mathrm {aff}}^+$ such that $-f+h \in \mathcal {O}_{f^\flat }$ . In particular, $f^\flat \in D$ and $f({\mathbf x}) + f^\flat ({\mathbf x}) = h$ . We label all the affine roots in D by

$$\begin{align*}f_1, \dots, f_{m-1}, f_m = f_m^\flat, \dots, f_n = f_n^\flat, f_{m-1}^\flat, \dots, f_1^\flat\end{align*}$$

such that

$$\begin{align*}f_1({\mathbf x}) \leqslant \cdots \leqslant f_{m-1}(x) \leqslant \frac{h}{2} = f_m({\mathbf x}) = \cdots = f_n({\mathbf x}) = \frac{h}{2} \leqslant f_{m-1}^\flat({\mathbf x}) \leqslant \cdots \leqslant f_1^\flat({\mathbf x}),\end{align*}$$

$f_i < f_i^\flat $ for $1 \leqslant i \leqslant m-1$ and $f_{m-1}^\flat < \cdots < f_1^\flat $ .

Let $1 \leqslant i \leqslant m$ . We set $D_i^\flat = \{f_j^\flat \in D; 1 \leqslant j \leqslant i\}$ if $1 \leqslant i \leqslant m-1$ and $D_i^\flat = \{f_j^\flat; 1 \leqslant j \leqslant n\}$ if $i=m$ . Define

$$\begin{align*}A_i = \widetilde \Phi^+ \setminus \cup_{j=1}^{i-1} \mathcal{O}_{f_j}, \quad B_i = \widetilde \Phi^h \cup \bigcup_{f \in D_i^\flat} \mathcal{O}_f, \quad C_{i-1} = B_{i-1} \setminus \{h\}.\end{align*}$$

Moreover, we set $A_0 = A_1 = \widetilde \Phi ^+$ , $B_0 = \widetilde \Phi ^h$ and $C_0 = B_0 \setminus \{h\}$ . Note that $A_m = B_m \cup \widetilde \Phi _M^+$ with $\widetilde \Phi _M^+ = \widetilde \Phi _M \cap \widetilde \Phi ^+$ .

Proof. We only show the second and the third inclusions. The others can be proved similarly. Let $f \in A_i$ and $f' \in B_i$ such that $f + f' \in \widetilde \Phi $ .

First, we assume that $f \in \widetilde \Phi _M^+$ . Then $f+f' \notin \widetilde \Phi _M^+$ since $f' \notin \widetilde \Phi _M^+$ . As

$$\begin{align*}(f+f')({\mathbf x})> f'({\mathbf x}) \geqslant f_i^\flat({\mathbf x}) \geqslant h/2,\end{align*}$$

we have $f + f' \in \cup _{f" \in D_{i-1}^\flat } \mathcal {O}_{f"} \subseteq C_{i-1} \subseteq B_{i-1}$ , as desired.

Now we assume that $f \notin \widetilde \Phi _M^+$ . Then $f({\mathbf x}) \geqslant f_i({\mathbf x})$ and

$$\begin{align*}(f+f')({\mathbf x}) \geqslant f_i({\mathbf x}) + f_i^\flat({\mathbf x}) = h.\end{align*}$$

By Definition 5.2, we have $f + f' \in \widetilde \Phi ^h \subseteq B_{i-1}$ , as desired.

Let $g \in \mathbb {G}_r^+$ , $x \in \mathbb {A}[r]$ and $E \subseteq \widetilde \Phi _r^+$ . We set $g_E = \mathrm {pr}_E(g) \in u(\mathbb {A}_E)$ , $x_E = p_E(x) \in \mathbb {A}_E$ and $\hat x = u(x) \in \mathbb {G}_r^+$ . For $f \in \widetilde \Phi _r^+$ , we will set $x_f = x_{\{f\}}$ and $x_{\geqslant f} = x_{\widetilde \Phi ^f}$ . We can define $g_f$ and $g_{\geqslant f} \in \mathbb {G}_r^+$ in a similar way. By abuse of notation, we will identify $g_f \in u(\mathbb {A}_f)$ with $u^{-1}(g_f) \in \mathbb {A}_f$ according to the context.

Lemma 5.13. Let $A, B \subseteq \widetilde \Phi ^+$ such that $A + B \subseteq \widetilde \Phi ^h$ . Let $x \in \mathbb {A}_A$ and $y \in \mathbb {A}_B$ . Then

$$\begin{align*}(\hat y \hat x)_h = -\sum_f \alpha_f^\vee \otimes y_{h-f} x_f + y_h + x_h \in \mathbb{A}_h = V,\end{align*}$$

where f ranges over A such that $f < h-f$ .

Proof. As $A + B \subseteq \widetilde \Phi ^h$ , for any $f \in A$ and $f' \in B$ , we have $[\hat y_{f'}, \hat x_f] = \hat y_{f'} \hat x_f \hat y_{f'}^{-1} \hat x_f^{-1} \in \mathbb {G}_r^h$ . Moreover, one computes that

(a) $[\hat y_{f'}, \hat x_f]_h = \alpha _f^\vee \otimes y_{f'} x_f$ if $f + f' = h$ , and $[\hat y_{f'}, \hat x_f]_h= 0$ otherwise.

Assume that $y = y_{\leqslant f'}$ and $x = x_{\geqslant f}$ for some $f' \in B$ and $f \in A$ . We argue by induction on f. If $f \geqslant f'$ , the statement is trivial. Suppose that $f < f'$ . Then we have

$$ \begin{align*} (\hat y \hat x)_h &= (\hat y_{< f'} \hat y_{f'} \hat x_f \hat x_{> f})_h \\ &= (\hat y_{< f'} \hat x_f \hat y_{f'} [\hat y_{f'}^{-1}, \hat x_f^{-1}] \hat x_{> f})_h \\ &=(\hat y_{< f'} \hat x_f \hat y_{f'} \hat x_{>f} )_h + [\hat y_{f'}^{-1}, \hat x_f^{-1}]_h \\ &~\vdots \\ &= (\hat y_{\leqslant f} \hat x_f \hat y_{[f+, f']} \hat x_{>f})_h + \sum_{f" \in [f+, f']} [\hat y_{f"}^{-1}, \hat x_f^{-1}]_h \\ &= (\hat y_{\leqslant f} \hat x_f)_h + (\hat y_{[f+, f']} \hat x_{>f})_h + \sum_{f" \in [f+, f']} [\hat y_{f"}^{-1}, \hat x_f^{-1}]_h, \end{align*} $$

where $[f+, f'] = \{f" \in \widetilde \Phi ^+; f+ \leqslant f" \leqslant f'\}$ . Now the statement follows from (a) and induction hypothesis.

Proof. Assume that $x = x_{\leqslant f}$ and $y = y_{\geqslant f'}$ for some $f \in A_i$ and $f' \in B_i$ . We argue by induction on $f'$ . If $f \leqslant f'$ , the statement is trivial. Assume that $f> f'$ . We claim that

(a) $f + f'> h$ if $f + f' \in \widetilde \Phi $ .

First, note that $f({\mathbf x}) + f'({\mathbf x}) \geqslant 2f'({\mathbf x}) \geqslant 2f_i^\flat ({\mathbf x}) \geqslant h$ . Suppose that (a) does not hold. Then $f + f' = h$ . Assume $x \in \mathbb {A}_{A_i \cap \widetilde \Phi _M}$ . Then $f \in \widetilde \Phi _M^+$ and $f + f' \in C_{i-1}$ by Lemma 5.12, which is a contradiction. Assume $1 \leqslant i \leqslant m-1$ . If $f \in \mathcal {O}_{f_i}$ , then $f' \in \mathcal {O}_{f_i^\flat }$ and hence $f < f'$ by our choice that $f_i < f_i^\flat $ , which is a contradiction. So $f \in A_{i+1}$ and $f + f' \in A_{i+1} + B_i \subseteq C_i$ by Lemma 5.12, which is also a contradiction. So (a) is proved.

By (a), we have $[\hat x_f^{-1}, \hat y_{f'}^{-1}] \in \mathbb {G}_r^{h+}$ . Hence,

$$ \begin{align*} (\hat x \hat y)_h &= ((\hat x_{< f} \hat y_{f'} \hat x_f [\hat x_f^{-1}, \hat y_{f'}^{-1}]) \hat y_{>f'})_h \\ &= (\hat x_{<f} \hat y_{f'} \hat x_f \hat y_{>f'})_h \\ &~\vdots \\ &= (\hat x_{\leqslant f'} \hat y_{f'} \hat x_{[f'+, f]} \hat y_{>f'})_h \\ &= (\hat x_{\leqslant f'} \hat y_{f'})_h + (\hat x_{[f'+, f]} \hat y_{>f'})_h \\ &= (\hat x_{\leqslant f'})_h + (\hat y_{f'})_h + (\hat x_{[f'+, f]} \hat y_{>f'})_h. \end{align*} $$

Now the statement follows by induction hypothesis.

We set $\pi = \pi _h^{\widetilde \Phi ^+: \widetilde \Phi ^h}: \mathbb {G}_h^+ = \mathbb {G}_r^+/\mathbb {G}_r^h \to \mathbb {A}_h \cong V$ .

Proposition 5.15. Let $1 \leqslant i \leqslant m$ . Then there is an isomorphism

$$\begin{align*}\psi_i: Y^{A_i}_h \cong Y^{A_i}_{B_i} \times \mathbb{A}_{D_i^\flat}.\end{align*}$$

Moreover, for $(\hat x, y) \in Y^{A_i}_{B_i} \times \mathbb {A}_{D_i^\flat }$ with $x \in \mathbb {A}_{A_i \setminus B_i}$ we have

(1) if $1 \leqslant i \leqslant m-1$ , then

$$\begin{align*}\pi(\psi_i^{-1}(\hat x, y)) = \alpha_{f_i}^\vee \otimes (x_{f_i}^{q^N} - x_{f_i}) y_{f_i^\flat}^{q^{n_i}} + \pi(\psi_i^{-1}(\hat x, 0)) \in V,\end{align*}$$

where $0 \leqslant n_i \leqslant N-1$ such that $F^{n_i}(f_i^\flat ) = -f_i + h$ ;

(2) if $i=m$ , then $\pi (\psi _i^{-1}(\hat x, y)) = -\sum _{j=m}^n \alpha _{f_j}^\vee \otimes y_{f_j}^{q^{N/2}+1} + \pi (\psi _i^{-1}(\hat x, 0))$ .

Proof. Without loss of generality, we may assume that $m = n$ . In particular, $B_i = \widetilde \Phi ^h \cup \mathcal {O}_{f_1^\flat } \cup \cdots \cup \mathcal {O}_{f_i^\flat }$ for $0 \leqslant i \leqslant m$ .

By Lemma 5.12, we have $A_i + \mathcal {O}_{f_j^\flat } \subseteq A_j + B_j \subseteq B_{j-1}$ for $1 \leqslant j \leqslant i \leqslant m$ . Thus, by applying Proposition 5.3 (1) repeatedly, we obtain an isomorphism

$$\begin{align*}\psi_i: Y^{A_i}_h = Y^{A_i}_{B_0} \cong Y^{A_i}_{B_1} \times \mathbb{A}_{f_1^\flat} \cong \cdots \cong Y^{A_i}_{B_i} \times \mathbb{A}_{D_i^\flat}.\end{align*}$$

Let $z = s_{\widetilde \Phi ^+: \widetilde \Phi ^h} \circ \psi _i^{-1}(\hat x, y)$ . We claim that

(a) $z = \hat x \hat w$ for some $w \in \mathbb {A}_{B_i \setminus \widetilde \Phi ^h}$ such that $w_{F^j(f_i^\flat )} = y_{f_i^\flat }^{q^j} + P_j(x)$ for $0 \leqslant j \leqslant N-1$ , where each $P_j$ is a polynomial function on $\mathbb {A}_{A_i \setminus B_i}$ . Moreover, $P_j = 0$ if $i = m$ .

Indeed, the first claim follows from the Proposition 5.3. Moreover, if $i = m$ , then $A_i \setminus B_i \subseteq \widetilde \Phi _M$ and $L(\hat x) \in \mathbb {M}_r^+$ . Hence, $P_j = 0$ for $1 \leqslant j \leqslant N-1$ by Proposition 5.3. So (a) is proved.

Then we claim that

(b) $x_{F^j(f_i)} = x_{f_i}^{q^j}$ for $1 \leqslant i \leqslant m-1$ and $0 \leqslant j \leqslant N-1$ .

Indeed, Let $v = x_{\mathcal {O}_{f_i}} \in \mathbb {A}_{\mathcal {O}_{f_i}}$ . As $\hat x \in Y^{A_i}_{B_i}$ , we have $\hat v \in Y^{A_i}_{A_{i+1}} \subseteq \mathbb {G}_r^{A_i} / \mathbb {G}_r^{A_{i+1}} \cong \mathbb {A}_{\mathcal {O}_{f_i}}$ ; that is, $L(\hat v) \in \mathbb {A}_{f_i} \subseteq \mathbb {A}_{\mathcal {O}_{f_i}}$ . Now (b) follows from Lemma 5.4.

Assume that $1 \leqslant i \leqslant m-1$ . By (b), we have

(c) $L(\hat x)_f = 0 $ if $f \in \mathcal {O}_{f_i} \setminus \{f_i\}$ and $L(\hat x)_f = x_{f_i}^{q^N} - x_{f_i}$ if $f = f_i$ .

Note that $\hat w, F(\hat w) \in \mathbb {G}_r^{B_i}$ , $\mathcal {O}_{f_i} < \mathcal {O}_{f_i^\flat }$ and $B_i + B_i \subseteq \widetilde \Phi ^{h+}$ . Moreover, $L(\hat x) \in \mathbb {G}_r^{A_i}$ and $[\hat w, (L(\hat x)_{<f_i})^{-1}] \in [\mathbb {G}_r^{B_i}, \mathbb {G}_r^{A_{i+1}}] \subseteq \mathbb {G}_r^{C_{i-1}}$ . It follows from Lemma 5.13 and Lemma 5.14 that $(\hat w^{-1})_h = 0$ and

$$ \begin{gather*} (L(\hat x)_{\geqslant f_i} F(\hat w))_h = (L(\hat x)_{\geqslant f_i})_h + F(\hat w)_h = L(\hat x)_h; \\ (\hat w^{-1} [\hat w, (L(\hat x)_{< f_i})^{-1}])_h = (\hat w^{-1})_h + ([\hat w, (L(\hat x)_{< f_i})^{-1}])_h = 0.\end{gather*} $$

Then one computes that

$$ \begin{align*} \pi(\psi_i^{-1}(\hat x, y)) &= (\hat w^{-1} L(\hat x) F(\hat w))_h \\ &=(L(\hat x)_{<f_i} \hat w^{-1} [\hat w, (L(\hat x)_{<f_i})^{-1}] L(\hat x)_{\geqslant f_i} F(\hat w))_h \\ &= (\hat w^{-1} [\hat w, (L(\hat x)_{<f_i})^{-1}] L(\hat x)_{\geqslant f_i} F(\hat w))_h \\ &= \sum_{f \in \mathcal{O}_{f_i}} ((\hat w^{-1})_{h-f} L(\hat x)_f)_h + (\hat w^{-1} [\hat w, (L(\hat x)_{< f_i})^{-1}])_h + (L(\hat x)_{\geqslant f_i} F(\hat w))_h \\ &= ((\hat w^{-1})_{h-f_i} L(\hat x)_{f_i})_h + L(\hat x)_h \\ &= \alpha_{f_i}^\vee \otimes ((x_{f_i}^{q^N} - x_{f_i}) (y_{f_i^\flat}^{q^{n_i}} + P_{n_i}(x))) + L(\hat x)_h \\ &= \alpha_{f_i}^\vee \otimes (x_{f_i}^{q^N} - x_{f_i}) y_{f_i^\flat}^{q^{n_i}} + \pi(\psi_i^{-1}(\hat x, 0), \end{align*} $$

where the third equality follows from that $f_i < B_i$ , the fourth equality follows from Lemma 5.13, and the fifth equality follows from (c).

Assume $i = m$ . Then $\hat x, L(\hat x) \in \mathbb {M}_r^+$ and $F^{N/2}(f_i) = h - f_i$ . Moreover, by the second statement of (a), we have

$$ \begin{align*} \hat w_{\mathcal{O}_{f_i}}^{-1} F(\hat w_{\mathcal{O}_{f_i}}) &= (\hat w_{f_i} \cdots F^{N-1}(\hat w_{f_i}))^{-1} F(\hat w_{f_i}) \cdots F^{N-1}(\hat w_{f_i}) \\ &\equiv \hat w_{f_i}^{-1} F^N(\hat w_{f_i}) [\hat w_{f_i}, F^{N/2}(\hat w_{f_i})^{-1}] \quad\mod \mathbb{G}_r^{h+}. \end{align*} $$

Thus,

$$\begin{align*}(\hat w^{-1} F(\hat w))_h = (\hat w_{\mathcal{O}_{f_i}}^{-1} F(\hat w_{\mathcal{O}_{f_i}}))_h = [\hat w_{f_i}, F^{N/2}(\hat w_{f_i})^{-1}]_h = -\alpha_{f_i}^\vee \otimes y_{f_i}^{q^{N/2} + 1}.\end{align*}$$

As $L(\hat x) \in \mathbb {M}_r^+$ , we have $[\hat w, L(\hat x)^{-1}] \in \mathbb {G}_r^{C_{i-1}}$ . In particular, $[\hat w, L(\hat x)^{-1}]_h = 0$ and $[[\hat w, L(\hat x)^{-1}], F(\hat w)] \in \mathbb {G}_r^{h+}$ . Now we have

$$ \begin{align*} \pi(\psi_i^{-1}(\hat x, y)) &= (\hat w^{-1} L(\hat x) F(\hat w))_h \\ &= (L(\hat x) \hat w^{-1} [\hat w, L(\hat x)^{-1}] F(\hat w))_h \\ &= (\hat w^{-1} [\hat w, L(\hat x)^{-1}] F(\hat w))_h + L(\hat x)_h \\ &= (\hat w^{-1} F(\hat w) [\hat w, L(\hat x)^{-1}])_h + L(\hat x)_h \\ &= (\hat w^{-1} F(\hat w))_h + [\hat w, L(\hat x)^{-1}]_h + L(\hat x)_h \\ &= -\alpha_i^\vee \otimes y_{f_j}^{q^{N/2} + 1} + L(\hat x)_h \\ &= -\alpha_j^\vee \otimes y_{f_j}^{q^{N/2} + 1} + \pi(\psi_i^{-1}(\hat x, 0)), \end{align*} $$

where the third (resp. the fifth) equality follows from Lemma 5.14 (resp. Lemma 5.13). The proof is finished.

Recall that $\mathcal {L}_{\chi _{h+}^h}$ is the multiplicative local system on $V = X_*(T) \otimes \overline {\mathbb {F}}_q$ corresponding to the character $\chi _{h+}^h: V^F \to \overline {\mathbb {Q}}_\ell ^\times $ .

Proposition 5.16. Let $\alpha \in \Phi \setminus \Phi _M$ and let $\kappa : \mathbb {G}_a \to V$ be the map given by $x \mapsto \alpha ^\vee \otimes x$ for $x \in \overline {\mathbb {F}}_q$ .

(1) $\kappa ^* \mathcal {L}_{\chi _{h+}^h}$ is nontrivial, and hence, $H_c^i(\mathbb {G}_a, \kappa ^* \mathcal {L}_{\chi _{h+}^h}) = 0$ for $i \in \mathbb {Z}$ ;

(2) if N is even and $F^{N/2}(\alpha ) = -\alpha $ , then

$$ \begin{align*} \dim H_c^i(\mathbb{G}_a, \tau^* \mathcal{L}_{\chi_{h+}^h}) = \begin{cases} q^{N/2} &\text{ if } i = 1; \\ 0, &\text{otherwise,} \end{cases} \end{align*} $$

where $\tau : \mathbb {G}_a \to V$ is given by $x \mapsto \alpha ^\vee \otimes x^{q^{N/2} + 1}$ . Moreover, in this case, $F^N$ acts on $H_c^1(\mathbb {G}_a, \tau ^* \mathcal {L}_{\chi _{h+}^h})$ via $-q^{N/2}$ .

Proof. Consider the composition of maps

$$\begin{align*}\theta: \mathbb{F}_{q^N} \overset \kappa \to V^{F^N} \overset {\mathrm{Nm}_N} \to V^F \overset {\chi_{h+}^h} \to \overline{\mathbb{Q}}_\ell^\times,\end{align*}$$

where $\mathrm {Nm}_N: V \to V$ is given by $v \mapsto v + \cdots + F^{N-1}(v)$ . As $\kappa $ is a homomorphism defined over $\mathbb {F}_{q^N}$ , we have $\kappa ^* \mathcal {L}_{\chi _{h+}^h} = \mathcal {L}_\theta $ by Lemma 5.8. Moreover, since $\alpha \in \Phi \setminus \Phi _M$ , $\theta $ is nontrivial by definition. Hence, $\mathcal {L}_\theta $ is nontrivial, and the statement (1) follows from [Reference Boyarchenko2, Lemma 9.4].

Assume that N is even and $F^{N/2}(\alpha ) = -\alpha $ . Then for $x \in \mathbb {F}_{q^{N/2}}$ , we have

$$ \begin{align*} \mathrm{Nm}_N(\alpha^\vee \otimes x) &= \sum_{i=0}^{N-1} F^i(\alpha^\vee) \otimes x^{q^i} \\ &= \sum_{i=0}^{N/2 - 1}(F^i(\alpha^\vee)\otimes x^{q^i} + F^{N/2 + i}(\alpha) \otimes x^{q^{N/2 + i}}) \\ &= \sum_{i=0}^{N/2 - 1}(F^i(\alpha^\vee)\otimes x^{q^i} + F^i(-\alpha^\vee) \otimes x^{q^i}) \\ &=0.\end{align*} $$

Hence, the (nontrivial) character $\theta $ of $\mathbb {F}_{q^N}$ restricts to a trivial character of $\mathbb {F}_{q^{N/2}}$ . Now the statement (2) follows from [Reference Boyarchenko and Weinstein5, Proposition 6.6.1].

Let Z be a locally closed subvariety of $\mathbb {G}_h^+$ with the natural embedding map $i_Z: Z \hookrightarrow \mathbb {G}_h^+$ . For a local system $\mathcal {F}$ on $\mathbb {G}_h^+$ , we write $H_c^j(Z, \mathcal {F}) = H_c^j(Z, i_Z^*\mathcal {F})$ for simplicity. We set $\pi = \pi _h^{\widetilde \Phi ^+:\widetilde \Phi ^h}: \mathbb {G}_h^+ = \mathbb {G}_r^+/\mathbb {G}_r^h \to \mathbb {A}_h \cong V$ .

Proof. By Proposition 5.15, we have an isomorphism

$$\begin{align*}\psi_i: Y^{A_i}_h \cong Y^{A_i}_{B_i} \times \mathbb{A}_{D_i^\flat}.\end{align*}$$

Let $p: Y^{A_i}_h \to Y^{A_i}_{B_i}$ be the natural projection. Set $\mathcal {L} = \mathcal {L}_{\chi _{h+}^h}$ .

Assume $1 \leqslant i \leqslant m-1$ . Let $Y^i = \{\hat x \in Y^{A_i}_h; x_{f_i}^{q^N} - x_{f_i} = 0\}$ . Then $\psi _i$ restricts to an isomorphism

$$\begin{align*}Y^i \cong Y_i \times \mathbb{A}_{D_i^\flat},\end{align*}$$

where $Y_i = \{\hat x \in Y^{A_i}_{B_i}; x_{f_i}^{q^N} - x_{f_i} = 0\}$ . In view of Proposition 5.15 (1), the restriction of $\pi $ to $Y^{A_i}_{B_i} \times \mathbb {A}_{D_i^\flat }$ is given by

$$\begin{align*}\pi(\psi_i^{-1}(\hat x, y)) = \pi_i(\psi_i^{-1}(\hat x, y)) = \eta(\hat x, y) + \pi_0(\hat x),\end{align*}$$

where $\eta (\hat x, y) = \alpha _{f_i}^\vee \otimes (x_{f_i}^{q^N}- x_{f_i}) y_{f_i}^{q^{n_i}}$ with $1 \leqslant n_i \leqslant N-1$ such that $F^{n_i}(f_i^\flat ) = h - f_i$ , and $\pi _0$ is the restriction of $\pi $ to $Y^{A_i}_{B_i} \times \{0\} \subseteq Y^{A_i}_{B_i} \times \mathbb {A}_{D_i^\flat }$ . As $\mathcal {L}$ is a multiplicative local system, we have $\pi ^* \mathcal {L} \cong \eta ^* \mathcal {L} \otimes p^* \pi _0^* \mathcal {L}$ . Hence, by the projection formula,

$$\begin{align*}p_! \pi^* \mathcal{L} \cong p_! \eta^* \mathcal{L} \otimes \pi_0^* \mathcal{L}.\end{align*}$$

For $\hat x \in Y^{A_i}_h$ , we define $\eta _{\hat x}: \mathbb {A}_{D_i^\flat } \to V$ to be the homomorphism given by $\eta _{\hat x}(y) = \eta (\hat x, y)$ . As $\alpha _{f_i} \in \Phi \setminus \Phi _M$ , it follows by Proposition 5.16 that $\eta _{\hat x}^* \mathcal {L}$ is a trivial multiplicative local system if and only if $x_{f_i}^{q^N} - x_{f_i} =0$ ; that is, $\hat x \in Y_i$ . Thus, $p_! \pi ^* \mathcal {L}$ is supported on $Y_i \times \mathbb {A}_{D_i^\flat } \cong Y^i$ , and hence, $p_! \pi ^* \mathcal {L} \cong p_! (\pi |_{Y^i})^* \mathcal {L}$ . Noticing that

$$\begin{align*}Y^i = \sqcup_{g \in (\mathbb{G}_h^{A_i})^F / (\mathbb{G}_h^{A_{i+1}})^F} g Y^{A_{i+1}}_h\end{align*}$$

and that $\#((\mathbb {G}_h^{A_i})^F / (\mathbb {G}_h^{A_{i+1}})^F) = \#(\mathbb {G}_h^{A_i} / \mathbb {G}_h^{A_{i+1}})^F = q^N$ , we have

$$\begin{align*}H_c^j(Y^{A_i}_h, \pi^* \mathcal{L}) \cong H_c^j(Y^i, \pi^* \mathcal{L}) \cong H_c^j(Y^{A_{i+1}}_h, \pi^* \mathcal{L})^{\oplus q^N},\end{align*}$$

and the first statement is proved.

By Proposition 5.15 (2), for $(\hat x, y) \in Y^{A_m}_{B_m} \times \mathbb {A}_{D_m^\flat } =Y^M_h \times \mathbb {A}_{D_m^\flat }$ , we have

$$\begin{align*}\pi(\hat x, y) = \tau(y) + \pi_M(\hat x),\end{align*}$$

where $\tau (y) = \sum _{j=m}^n \alpha _{f_j}^\vee \otimes y_{f_j}^{q^{N/2}+1}$ . Thus, $\pi ^* \mathcal {L} \cong \pi _M^* \mathcal {L} \boxtimes \tau ^* \mathcal {L}$ . By Künneth formula, we have

$$ \begin{align*} H_c^j(Y^{A_m}_h, \pi^* \mathcal{L}) &\cong \oplus_s H_c^s(\mathbb{A}_{D_m^\flat}, \tau^* \mathcal{L}) \otimes H_c^{j-s}(Y^M_h, \pi_M^* \mathcal{L}) \\ &\cong \otimes_{i=m}^n H_c^1(\mathbb{A}_{f_i}, \tau_i^* \mathcal{L}) \otimes \otimes_{i=1}^{m-1} H_c^2(\mathbb{A}_{f_i^\flat}, \overline{\mathbb{Q}}_\ell) \otimes H_c^{j-n-m+1}(Y^M_h, \pi_M^* \mathcal{L}) \\ &\cong H_c^{j-n-m+1}(Y^M_h, \pi_M^* \mathcal{L})^{q^{(n-m+1)N/2}}((-q^{N/2})^{m+n-1}), \end{align*} $$

where $\tau _i : \mathbb {G}_a \cong \mathbb {A}_{f_i} \to V$ is given by $x \mapsto \alpha _{f_i}^\vee \otimes x^{q^{N/2}+1}$ , and the last isomorphism follows from Proposition 5.16 (2). This finishes the proof of the second statement.

5.7 Proof of Theorem 5.5

Let $G'$ , $T'$ , $Y_f'$ be as in §5.5. Let $\chi '$ be the restriction of $\chi $ to $(\mathbb {T}^{\prime +}_f)^F$ . By Lemma 5.11, we have

$$\begin{align*}H_c^j(Y_f, \overline{\mathbb{Q}}_\ell)[\chi] \cong (\mathrm{ind}_{(\mathbb{T}^{\prime +}_f)^F}^{(\mathbb{T}_f^+)^F} (H_c^j(Y_f', \overline{\mathbb{Q}}_\ell)[\chi']))[\chi].\end{align*}$$

So it suffices to prove the theorem for semisimple reductive groups $G = G'$ .

We argue by induction on $f \in \widetilde \Delta ^+$ and $\# \Phi $ . Indeed, if $f = \min \widetilde \Delta ^+$ , then $(\mathbb {T}_f^+)^F = Y_f = \{1\}$ and the statement is trivial. However, if $\Phi $ is empty – that is, $G = T$ – then $Y_f = (\mathbb {T}_f^+)^F$ is a finite set and the statement is also true. Now we assume the theorem holds for all reductive groups L such that $\# \Phi _L < \# \Phi _G$ , and for all $Y_{f'}$ with $f' \leqslant f \in \widetilde \Delta ^+$ .

If $f \in \Delta _{\mathrm {aff}}^+$ , by Corollary 5.9 (1), we have a $(\mathbb {T}_f^+)^F$ -equivariant isomorphism

$$\begin{align*}H_c^i(Y_{f+}, \overline{\mathbb{Q}}_\ell) = H_c^{i-2}(Y_f, \overline{\mathbb{Q}}_\ell)(-q^N).\end{align*}$$

Then the statement follows by induction hypothesis.

Now we assume $f = h \in \mathbb {Z}_{\geqslant 1}$ . Let notation be as in §5.6. By Corollary 5.9 (2),

$$\begin{align*}H_c^i(Y_{h+}, \overline{\mathbb{Q}}_\ell)[\chi] \cong H_c^i(Y_h, \pi^* \mathcal{L}_{\chi_{h+}^h})[\chi].\end{align*}$$

If $\chi _{h+}^h$ is trivial, then $\mathcal {L}_{\chi _{h+}^h} = \overline {\mathbb {Q}}_\ell $ and $H_c^i(Y_{h+}, \overline {\mathbb {Q}}_\ell )[\chi ] \cong H_c^i(Y_h, \overline {\mathbb {Q}}_\ell )[\chi ]$ . Hence, the statement also follows by induction hypothesis. Assume $\chi _{h+}^h$ is nontrivial and let notation be as in §5.6. By Condition 2.1 and Lemma 2.2, we have $M = M_\chi \neq G$ . By Proposition 5.17 and Corollary 5.9 (2), we have

$$ \begin{align*} &\quad\ H_c^i(Y_h, \pi^* \mathcal{L}_{\chi_{h+}^h}) [\chi] = H_c^i(Y^{A_1}_h, \pi^* \mathcal{L}_{\chi_{h+}^h})[\chi] \\ &\cong (H_c^{i-m-n+1}(Y^M_h, \pi_M^* \mathcal{L}_{\chi_{h+}^h})[\chi])^{\oplus q^{(m+n-1)N/2}}((-q)^{(m+n-1)N/2}) \\ &\cong (H_c^{i-m-n+1}(Y_{h+}^M, \overline{\mathbb{Q}}_\ell) [\chi])^{\oplus q^{(m+n-1)N/2}}((-q)^{(m+n-1)N/2}), \end{align*} $$

so the statement follows by induction hypothesis. The proof is finished.

5.8 Computation of cohomological degree

Let $\chi $ be a smooth character $\mathcal {T}^+(\mathcal {O}_k)$ , which factors through $(\mathbb {T}_r^+)^F$ . We have the Howe factorization of an arbitrary lift of $\chi $ to a smooth character of $T(k)$ from [Reference Kaletha21, §3.6]. We use notation from loc. cit. In particular, we have the integers $(r \geq ) r_d \geq r_{d-1}> r_{d-2} > \dots > r_0 > 0$ at which the breaks happen and the increasing subsets $R_i:= R_{r_i} \subseteq \Phi $ (which are the roots systems of the twisted Levi subgroups $M_\chi $ appearing in §5.7). Moreover, $r_{-1} = 0$ , $R_d = \Phi $ by definition. Let $R_i^{\mathrm {red}} = R_i \cap \Phi ^{\mathrm {red}}$ , where $\Phi ^{\mathrm {red}}$ is as in §4.3.

Proposition 5.18. We have

$$\begin{align*}N s_{\chi,r} = 2r\cdot\#\Phi - \#\Phi^{\mathrm{red}} - \#R_0^{\mathrm{red}} - \sum_{i=0}^{d-1} r_i(\#R_{i+1} - \#R_i). \end{align*}$$

Proof. We can argue by induction on $\# \Phi $ (or on the number of jumps d). If $\Phi = \varnothing $ , the statement is trivial. Suppose it is true for all reductive groups L with $\#\Phi _L < \#\Phi $ . Then in view of §5.7 (where we can assume that $\chi $ is trivial over $\mathbb {T}_r^{h+1}$ with $h = r_{d-1}$ ), we have

$$\begin{align*}s_{\chi,r} = 2(r-r_{d-1}) \cdot \#\Delta +(m+n-1) + s_{\chi, r_{d-1}}^M,\end{align*}$$

where $s_{\chi ,r_{d-1}}^M$ is the unique integer i such that $H_c^i(Y_{r_{d-1}}^M, \overline {\mathbb {Q}}_\ell )[\chi ] \neq 0$ . Now,

$$ \begin{align*} m+n-1 &= \#D \\ &= \#\{ f \in \Delta_{\mathrm{aff}} \colon f(\textbf{x})> 0, f < r_{d-1} \} \cap (\widetilde R_d {\,\setminus\,} \widetilde R_{d-1})) \\ &= \frac{1}{N} \left( r_{d-1} (\#R_d - \#R_{d-1}) - (\#R_d^{\mathrm{red}} - \# R_{d-1}^{\mathrm{red}}) \right), \end{align*} $$

where $ \widetilde R_{d-1} \subseteq \widetilde \Phi $ is the preimage of $R_i$ under the natural projection $\widetilde \Phi \to \Phi \sqcup \{0\}$ . Note that $N \cdot \#\Delta = \#\Phi =\#R_d$ . The statement now follows by induction hypothesis.

This generalizes the formula from [Reference Chan and Ivanov8, Theorem 6.1.1]

Corollary 5.19. For the integer $s_\chi $ from Theorem 1.1, we have

$$\begin{align*}Ns_\chi = -\#\Phi^{\mathrm{red}} + \#R_0^{\mathrm{red}} + \sum_{i=0}^{d-1} r_i(\#R_{i+1} - \#R_i). \end{align*}$$

Note that when $\chi $ is sufficiently generic, this, combined with Corollary 1.3, gives a formula for the formal degree of the corresponding supercuspidal representation. Moreover, note that the essential parts of the formulas of Corollary 5.19 and of [Reference Schwein26, Theorem A] agree.

We now explicate the formula from Corollary 5.19 in two special cases.

Example 5.20. Assume that $\chi $ is toral; that is, $d = 1$ and the Howe filtration of $\chi $ has exactly one jump happening at some integer $r_0$ satisfying $0 < r_0 \leq r_1 := \mathrm {depth}(\chi )$ . Then $R_1 = \Phi $ and $R_0 = R_{-1} = \varnothing $ , and the formula from Corollary 5.19 simplifies to

$$\begin{align*}Ns_{\chi} = (r_0-1)\#\Phi. \end{align*}$$

Example 5.21. Assume that $G = \mathrm {GL}_n$ and $\mathcal {G}$ is hyperspecial, so that $\Phi = \Phi ^{\mathrm {red}}$ and $\#\Phi =n(n-1)$ . We can take $T \subseteq G$ to be the diagonal torus so that $\Phi = \{\alpha _{i,j}\}_{1\leq i\neq j \leq n}$ with $\alpha _{i,j}$ corresponding to $(i,j)$ th matrix entry. Moreover, we may take $c = (1,2,\dots ,n)$ ; then the (root systems of the) twisted Levi subgroups containing T are in bijection with divisors $0<k|n$ , with k corresponding to $R(k) = \{\alpha _{i,j}\}_{i-j \equiv 0\ \mod \frac {n}{k}}$ , so that $R(k) \subseteq R(k')$ if and only if $k|k'$ and $\#R(k) = k(k-1)$ (in particular, $R(1) = \varnothing $ corresponds to T itself). Given the integers $r_i$ ( $-1 \leq i \leq d$ ) as above, along with a corresponding sequence of divisors $1 = k_{-1}|k_0 | \dots |k_{d-1}|k_d=n$ of n, all of which are mutually different (except possibly $k_0 = 1$ ), formula from Corollary 5.19 then becomes

$$\begin{align*}Ns_\chi = -n(n-1) + k_0(k_0-1) + \sum_{i=0}^{d-1}r_i(k_{i+1}(k_{i+1}-1) - k_i(k_i-1)). \end{align*}$$

8.1 Review of the orbit method

The orbit method was originally developed by Kirillov [Reference Kirillov22] and extended later to various related setups. We briefly recall it in the two setups relevant for our article. We refer to [Reference Boyarchenko and Sabitova4] (in particular, §2 therein), [Reference Boyarchenko and Drinfeld1, §2] and [Reference Dixon, du Satoy, Mann and Segal11] and references therein for more detailed discussions.

Assume that $p>2$ .Footnote ³ For the first setup, recall that a uniform Lie algebra is a (topological) Lie algebra $\mathfrak {g}$ over $\mathbb {Z}_p$ , which is free of finite rank as a $\mathbb {Z}_p$ -module and satisfies $[\mathfrak {g},\mathfrak {g}] \subseteq p\mathfrak {g}$ . Following Lazard, there is a pro-p group $\Gamma = \exp \mathfrak {g}$ attached to $\mathfrak {g}$ , whose underlying topological space is $\mathfrak {g}$ and on which the group law is defined (via $\exp $ and $\log $ ) by the Campbell–Hausdorff series. For $\Gamma = \exp \mathfrak {g}$ , one has mutually inverse homeomorphisms $\exp \colon \Gamma \rightarrow \mathfrak {g}$ and $\log \colon \mathfrak {g} \rightarrow \Gamma $ . Set up appropriately, the functor $\mathfrak {g} \mapsto \exp \mathfrak {g}$ even defines an isomorphism of categories. We denote the inverse functor by $\Gamma \mapsto \log \Gamma $ . A profinite group $\Gamma $ is called uniform (short for uniformly powerful) if there is a uniform Lie-algebra $\mathfrak {g}$ with $\Gamma \cong \exp \mathfrak {g}$ . There is a similar isomorphism of categories between finite p-groups $\Gamma $ of nilpotence class $<p$ and finite nilpotent Lie rings $\mathfrak {g}$ of p-power order and nilpotence class $<p$ . We use the same notation as in the uniform pro-p case.

For the moment, let $\Gamma $ be either

1. (i) a uniform pro-p group, or

2. (ii) a finite p-group of nilpotence class $<p$ .

Let $\mathfrak {g} = \log \Gamma $ denote the corresponding uniform Lie $\mathbb {Z}_p$ -algebra resp. finite Lie ring. Let $\widehat \Gamma $ denote the set of isomorphism classes of smooth irreducible $\overline {\mathbb {Q}}_\ell $ -representations of $\Gamma $ . Note that there is an adjoint action of $\Gamma $ on $\mathfrak {g}$ . More precisely, for any $g \in \Gamma $ , we have the automorphism ${\mathrm {Ad}}\, g \colon \mathfrak {g} \rightarrow \mathfrak {g}$ given by $x \mapsto \log (g\exp (x)g^{-1})$ . Let

$$\begin{align*}\mathfrak{g}^\ast = \operatorname{\mathrm{Hom}}_{\mathrm{cont}}(\mathfrak{g},\overline{\mathbb{Q}}_\ell^\times) \end{align*}$$

be the dual of $\mathfrak {g}$ . The adjoint action of $\Gamma $ on $\mathfrak {g}$ induces an action of $\Gamma $ on $\mathfrak {g}^\ast $ . Kirillov’s orbit method, in the present setup established in [Reference Boyarchenko and Sabitova4], describes a natural bijection between $\widehat \Gamma $ and the set of $\Gamma $ -orbits in $\mathfrak {g}^\ast $ .

Theorem 8.1 (Theorem 2.6 in [Reference Boyarchenko and Sabitova4]).

Assume $p\geq 3$ and $\Gamma $ is either a uniform pro-p-group or a p-group of nilpotence class $<p$ and let $\mathfrak {g} = \mathrm {Lie}\, \Gamma $ . Then there exists a bijection $\Omega \leftrightarrow \rho _\Omega $ between $\Gamma $ -orbits $\Omega \subseteq \mathfrak {g}^\ast $ and $\widehat \Gamma $ , characterized by

$$\begin{align*}\operatorname{\mathrm{tr}}(g,\rho_\Omega) = \frac{1}{\#\Omega^{1/2}} \cdot \sum_{f\in \Omega} f(\log(g)). \end{align*}$$

Groups of the form $\Gamma = (\mathbb {G}_r^+)^F$ may or may not satisfy the assumptions of Theorem 8.1, as the following examples show.

8.2 Cohomological induction vs. the orbit method

For brevity, we write $\Gamma = (\mathbb {G}_r^+)^F$ and . Note that satisfies condition (i) or (ii) in §8.1 and let denote its Lie algebra. As is abelian, is not only a homeomorphism, but also an isomorphism of groups with inverse . Also, as is abelian, we may identify with . By Theorem 1.1, we get a map