2.1.1 Notations

Let F be a complete discrete valuation field with ring of integers $\mathcal {O}_F$ and perfect residue field k of characteristic $p>0$ . We assume that k is a finite field or an algebraic closure thereof. Fix a uniformizer $\varpi \in \mathcal {O}_F$ . If F has characteristic zero, for a k-algebra R, let $W(R)$ be its ring of Witt vectors, and let $W_{\mathcal {O}_F}(R) = W(R) \otimes _{W(k)} \mathcal {O}_F$ . Let $W_{\mathcal {O}_F, n}(R) = W(R) \otimes _{W(k)} \mathcal {O}_F/ \varpi ^n$ . If F has characteristic p, the choice of $\varpi $ induces an isomorphism $F \cong k ( \!(t) \!)$ . For a k-scheme X, we denote its perfection by

$$ \begin{align*}X_{\operatorname{perf}}:= \lim (\cdots \xrightarrow{\text{Fr}} X \xrightarrow{\text{Fr}} X ),\end{align*} $$

where $\text {Fr}$ is the absolute Frobenius morphism. The underlying topological spaces of X and $X_{\operatorname {perf}}$ are canonically isomorphic. Moreover, there is a natural equivalence of small étale sites $X_{\operatorname {\acute {e}t}} \cong (X_{\operatorname {perf}})_{\operatorname {\acute {e}t}}$ [Sta24, Tag 04DY]. In particular, the étale cohomology groups of X and $X_{\operatorname {perf}}$ are canonically isomorphic.

2.1.2 The affine flag variety

Let G be a connected reductive group over F and let $\mathscr {B}(G,F)$ be its Bruhat–Tits building. For each facet $\mathbf {f} \subset \mathscr {B}(G,F)$ , let $\mathcal {G} := \mathcal {G}_{\mathbf {f}}$ be the parahoric $\mathcal {O}_F$ -group scheme with generic fiber G as in [Reference Bruhat and TitsBT84, Définition 5.2.6 ff.]. Let $K := \mathcal {G}(\mathcal {O}_F)$ . If F has characteristic zero, following [Reference ZhuZhu17a], we define the following functors on perfect k-algebras:

(2.1) $$ \begin{align} LG \colon R \mapsto G(W_{\mathcal{O}_F}(R)[1/p]), \quad L^+\mathcal{G} \colon R \mapsto \mathcal{G}(W_{\mathcal{O}_F}(R)).\end{align} $$

The functor $LG$ is called the loop group, and $L^+\mathcal {G}$ is called the positive loop group. We also define the functor $L^n\mathcal {G} \colon R \mapsto \mathcal {G}(W_{\mathcal {O}_F, n}(R)).$ Then $L^+\mathcal {G} = \displaystyle \lim _{\longleftarrow } L^n\mathcal {G}$ . The affine flag variety of $\mathcal {G}$ is the functor on perfect k-algebras given by the étale-quotient

$$ \begin{align*}\mathcal{F}\!\ell_{\mathcal{G}} = LG/L^+\mathcal{G}.\end{align*} $$

By [Reference Bhatt and ScholzeBS17], $\mathcal {F}\!\ell _{\mathcal {G}}$ is represented by an increasing union of perfections of projective k-schemes.

If F has characteristic p, the loop group (resp. positive loop group) is $LG(R) = G(R(\!(t)\!))$ (resp. $L^+\mathcal {G}(R) = \mathcal {G}(R[\![t]\!])$ . We set $L^n\mathcal {G}(R) = G(R[t]/t^n)$ . These functors are usually defined on all k-algebras, but for uniformity of exposition, we restrict to perfect k-algebras. Then the étale-quotient $\mathcal {F}\!\ell _{\mathcal {G}} = LG/L^+\mathcal {G}$ is represented by the perfection of the ind-projective k-scheme constructed in [Reference Pappas and RapoportPR08].

2.1.3 Greenberg functors

If F has characteristic zero, there is no canonical ind-projective k-scheme whose perfection is $\mathcal {F}\!\ell _{\mathcal {G}}$ . The problem is that the natural extension of the functor $LG$ to non-perfect k-algebras is not well-behaved. (For example, if R is non-perfect, $W_{\mathcal {O}_F}(R)$ can have p-torsion, in which case $W_{\mathcal {O}_F}(R)$ is not a submodule of $W_{\mathcal {O}_F}(R)[1/p]$ .) However, as observed by Greenberg [Reference GreenbergGre61], the functor $L^+\mathcal {G}$ is well-behaved on all k-algebras. Following [Reference ZhuZhu17a], we define the following functors on all k-algebras:

$$ \begin{align*}L^+_p\mathcal{G} \colon R \mapsto G(W_{\mathcal{O}_F}(R)), \quad L^n_p\mathcal{G} \colon R \mapsto \mathcal{G}(W_{\mathcal{O}_F,n}(R)), \: n \geq 1.\end{align*} $$

Then $L^+_p\mathcal {G} = \displaystyle \lim _{\longleftarrow } L^n_p\mathcal {G}$ and $L^+\mathcal {G} = (L^+_p\mathcal {G})_{\operatorname {perf}}$ . The functor $L^n_p\mathcal {G}$ is represented by a k-scheme of finite-type. If F has characteristic p, let $L_pG$ , $L^+_p\mathcal {G}$ , and $L^n_p \mathcal {G}$ be the extensions of $LG$ , $L^+\mathcal {G}$ , and $L^n \mathcal {G}$ to all k-algebras. There are no issues with $L_pG$ because $R[\![t]\!]$ never has t-torsion. We will use these functors to construct deperfections of finite-dimensional subschemes of $\mathcal {F}\!\ell _{\mathcal {G}}$ , i.e., finite-type k-schemes whose perfections are subschemes of $\mathcal {F}\!\ell _{\mathcal {G}}$ .

2.1.4 Rational points

Let $\breve {F}$ be the completion of a maximal unramified extension of F and let $\overline {k}$ be its residue field. Since $\overline {k}$ is local for the étale topology,

$$ \begin{align*}\mathcal{F}\!\ell_{\mathcal{G}}(\overline{k}) = LG(\overline{k})/L^+\mathcal{G}(\overline{k}) = G(\breve{F})/\mathcal{G}(\mathcal{O}_{\breve{F}}).\end{align*} $$

2.1.5 Perfect Schubert schemes

Choose a maximal F-split torus A such that $\mathbf {f}$ is contained in the apartment $\mathscr {A}(G,A,F)$ . Let $\mathbf {f}' \subset \mathscr {A}(G,A,F)$ be another facet. For $w \in L^+ \mathcal {G}_{\mathbf {f}'}(k) \backslash LG(k) / L^+ \mathcal {G}_{\mathbf {f}}(k)$ , define the orbit

$$ \begin{align*}\mathcal{F}\!\ell_{w}^\circ(\mathbf{f}', \mathbf{f}) := L^+ \mathcal{G}_{\mathbf{f}'} \cdot \widetilde{w} \cdot e \subset \mathcal{F}\!\ell_{\mathcal{G}},\end{align*} $$

where $\widetilde {w} \in LG(k)$ is any lift of w and e is the basepoint of $\mathcal {F}\!\ell _{\mathcal {G}_{\mathbf {f}}}$ . Let $\mathcal {F}\!\ell _{w}(\mathbf {f}', \mathbf {f})$ be the closure of $\mathcal {F}\!\ell _{w}^\circ (\mathbf {f}', \mathbf {f})$ . The stabilizer of $\widetilde {w} \cdot e$ in $L^+ \mathcal {G}_{\mathbf {f}'}$ is the positive loop group of the parahoric $\mathcal {O}_{F}$ -group scheme associated to $\mathbf {f}' \cup w \mathbf {f}$ (cf. [Reference Anschütz, Gleason, Lourenço and RicharzAGLR22, Proposition 3.7]). It follows that $\mathcal {F}\!\ell _{w}^\circ (\mathbf {f}', \mathbf {f})$ is canonically isomorphic to the perfection of a smooth quasi-projective k-scheme. Furthermore, $\mathcal {F}\!\ell _{w}(\mathbf {f}', \mathbf {f})$ is the perfection of a projective k-scheme by [Reference Bhatt and ScholzeBS17] (if F has characteristic zero) and [Reference Pappas and RapoportPR08] (if F has characteristic p). We call $\mathcal {F}\!\ell _{w}^\circ (\mathbf {f}', \mathbf {f})$ a perfect Schubert cell and $\mathcal {F}\!\ell _{w}(\mathbf {f}', \mathbf {f})$ a perfect Schubert scheme.

Following [Reference Haines and RapoportHR08, Reference Anschütz, Gleason, Lourenço and RicharzAGLR22], the perfect Schubert schemes can be parametrized as follows. Fix a maximal $\breve {F}$ -split torus $S \subset G$ containing A and let $T = C_G(S)$ . Then T is a maximal torus, and we obtain a chain of F-tori $A \subset S \subset T.$ The connected Néron model $\mathcal {T}$ of T over $\mathcal {O}_F$ is contained in $\mathcal {G}$ [Reference Kaletha and PrasadKP23, Proposition 8.2.4], and $\mathcal {T}(\mathcal {O}_{\breve {F}})$ is the unique parahoric subgroup of $T(\breve {F})$ . The Iwahori–Weyl group is

(2.2) $$ \begin{align} W := N_G(S)(\breve{F})/ \mathcal{T}(\mathcal{O}_{\breve{F}}). \end{align} $$

Choose an alcove $\mathbf {a}$ in the apartment $\mathscr {A}(G,A,F)$ . This determines a splitting $W = W_{\operatorname {af}} \rtimes \pi _1(G)_I$ . Here, $W_{\operatorname {af}} \subset W$ is the affine Weyl group, I is the inertia group of F, and $\pi _1(G)$ is the algebraic fundamental group. As $W_{\operatorname {af}}$ is a Coxeter group, we may use this splitting to extend the length function $\ell $ and partial Bruhat order to W by declaring elements of $\pi _1(G)_I$ to have length $0$ . The Iwahori group scheme over $\mathcal {O}_F$ associated to the alcove $\mathbf {a}$ will be denoted by $\mathcal {I} : = \mathcal {G}_{\mathbf {a}}.$

The facets $\mathbf {f}, \mathbf {f}'$ are contained in unique facets $\breve {\mathbf {f}}, \breve {\mathbf {f}}'$ in $\mathscr {A}(G,\breve {F},S)$ . Then $\mathcal {G}_{\breve {\mathbf {f}}} = \mathcal {G}_{\mathbf {f}} \times \operatorname {Spec}(\mathcal {O}_{\breve {F}})$ . Let $W_{\breve {\mathbf {f}}} = (N_G(S)(\breve {F}) \cap \mathcal {G}_{\mathbf {f}} (\mathcal {O}_{\breve {F}}))/\mathcal {T}(\mathcal {O}_{\breve {F}})$ . By the Bruhat decomposition (see [Reference Haines and RapoportHR08, Proposition 8]),

$$ \begin{align*}W_{\breve{\mathbf{f}}'} \backslash W / W_{\breve{\mathbf{f}}} = L^+ \mathcal{G}_{\mathbf{f}'}(\overline{k}) \backslash LG(\overline{k}) / L^+ \mathcal{G}_{\mathbf{f}}(\overline{k}).\end{align*} $$

Let $\sigma \in \operatorname {Gal}(\overline {k}/k)$ be a geometric Frobenius element (if $\overline {k} = k$ , then we let $\sigma $ be the identity) and let $W_{\mathbf {f}} := W_{\breve {\mathbf {f}}}^\sigma $ . By [Reference Haines and RapoportHR08, Remark 9], passing to $\sigma $ -invariants induces a bijection

(2.3) $$ \begin{align} W_{\mathbf{f}'} \backslash W^\sigma / W_{\mathbf{f}} = L^+ \mathcal{G}_{\mathbf{f}'}(k) \backslash LG(k) / L^+ \mathcal{G}_{\mathbf{f}}(k). \end{align} $$

2.1.6 Closure relations and dimensions

From now on, we assume that $\mathbf {f}$ and $\mathbf {f}'$ are contained in the closure of the alcove $\mathbf {a}$ . By the reduction steps in [Reference Haines and RicharzHR23, §3.1], over $\overline {k}$ , every $\mathcal {F}\!\ell _{w}(\mathbf {f}', \mathbf {f})$ is isomorphic to a perfect Schubert scheme for a pair $(\mathbf {f}', \mathbf {f})$ satisfying this assumption.

For the remainder of Section 2, we assume $k = \overline {k}$ unless otherwise stated. This implies $F= \breve {F}$ . The results still apply when k is finite if we restrict to perfect Schubert schemes defined over k. In this subsection, we recall one of the main results in [Reference RicharzRic13]. We note that the results in op. cit. are stated in the case where F has equal characteristic, but the proofs apply verbatim in mixed characteristic.

By [Reference RicharzRic13, Lemma 1.6], for $w \in W$ , there is a unique element $w ^{\mathbf {f}}$ of minimal length in $w W_{\mathbf {f}}$ , and a unique element $_{\mathbf {f}'} w ^{\mathbf {f}}$ of maximal length in $\{(vw)^{\mathbf {f}} \: : \: v \in W_{\mathbf {f}'}\}$ . Taking equivalence classes gives a natural bijection

$$ \begin{align*}_{\mathbf{f}'} W ^{\mathbf{f}} := \{ _{\mathbf{f}'} w ^{\mathbf{f}} \: : \: w \in W\} \cong W_{\mathbf{f}'} \backslash W / W_{\mathbf{f}}.\end{align*} $$

Lemma 2.2. The perfect Schubert scheme $\mathcal {F}\!\ell _{w}(\mathbf {f}', \mathbf {f})$ is set-theoretically the following union of locally closed perfect Schubert cells.

$$ \begin{align*}\mathcal{F}\!\ell_{w}(\mathbf{f}', \mathbf{f}) = \bigsqcup_{\substack{v \in _{\mathbf{f}'} W ^{\mathbf{f}} \\ v \leq _{\mathbf{f}'} w ^{\mathbf{f}}}} \mathcal{F}\!\ell_{v}^\circ(\mathbf{f}', \mathbf{f}).\end{align*} $$

The dimension of $\mathcal {F}\!\ell _{w}(\mathbf {f}', \mathbf {f})$ is $\ell (_{\mathbf {f}'} w ^{\mathbf {f}})$ .

2.1.7 Deperfections

An integral domain A is said to be p-closed if for every $a \in \operatorname {Frac}(A)$ such that $a^p \in A$ , we have $a \in A$ . An integral scheme is p-closed if it has a covering by open affines which are p-closed. We recall the following special case of the construction in [Reference ZhuZhu17a, Proposition A.15].

In [Reference ZhuZhu17a, §B.2], Zhu constructed canonical deperfections of Witt vector Schubert schemes when $\mathcal {G}$ is a split reductive group and conjectured that these are normal and Cohen-Macaulay. Canonical deperfections for general parahorics were constructed in [Reference Anschütz, Gleason, Lourenço and RicharzAGLR22]. In [Reference Anschütz, Gleason, Lourenço and RicharzAGLR22, Theorem 1.10], it was shown that canonical deperfections occurring in the $\mu $ -admissible locus of a minuscule cocharacter $\mu $ (with a minor technical assumption when $p=2$ ) satisfy Zhu’s conjecture. We now show that every p-closed deperfection of a perfect Schubert scheme is normal (this result is also implicit in [Reference Anschütz, Gleason, Lourenço and RicharzAGLR22]).

2.2.1 Demazure resolutions

Let $w \in W$ and write $w = w_{\operatorname {af}} \tau $ for $w_{\operatorname {af}} \in W_{\operatorname {af}}$ and $\tau \in \pi _1(G)_I$ . Choose a (not necessarily reduced) decomposition $\dot {w} = s_1 \cdots s_n$ of $w_{\operatorname {af}}$ as a product of simple reflections along the walls of $\mathbf {a}$ . For each i, let $\mathcal {P}_i$ be the minimal parahoric group scheme attached to $s_i$ , so that $\mathcal {P}_i(\mathcal {O}_F) = \mathcal {I}(\mathcal {O}_F) \sqcup \mathcal {I}(\mathcal {O}_F) s_i \mathcal {I}(\mathcal {O}_F)$ for any lift of $s_i$ to $N_G(S)(F)$ . The perfect Demazure scheme associated to $\dot {w}$ is

(2.4) $$ \begin{align} D_{\dot{w}} := L^+\mathcal{P}_1 \times^{L^+ \mathcal{I}} \cdots \times^{L^+ \mathcal{I}} L^+\mathcal{P}_n / L^+ \mathcal{I}. \end{align} $$

Here, we have taken étale quotient of $\prod _{i=1}^n L^+\mathcal {P}_i$ by the right action of $\prod _{i=1}^n L^+\mathcal {I}$ given by

$$ \begin{align*}(i_1, \cdots, i_n) \cdot (p_1, \cdots, p_n) = (p_1i_1, i_1^{-1} p_2 i_2, \cdots, i_{n-1}^{-1}p_n i_n ).\end{align*} $$

Since any lift of $\tau $ to $N_G(S)(F)$ normalizes $L^+\mathcal {I}$ , the multiplication map $(p_1, \cdots , p_n) \mapsto p_1 \cdots p_n \tau $ induces a map $D_{\dot {w}} \rightarrow \mathcal {F}\!\ell _{\mathcal {G}_{\mathbf {f}}}$ . We define $\pi _{\dot {w}}(D_{\dot {w}})$ to be the image of this map, which is isomorphic to $\mathcal {F}\!\ell _{w'}(\mathbf {a}, \mathbf {f})$ for some $w' \in W$ . Let

(2.5) $$ \begin{align} \pi_{\dot{w}} \colon D_{\dot{w}} \rightarrow \pi_{\dot{w}}(D_{\dot{w}}) \end{align} $$

be the resulting surjective map.

The following result is well-known in equal characteristic [Reference FaltingsFal03, Lemma 9], [Reference Pappas and RapoportPR08, Proposition 9.7] and in mixed-characteristic [Reference He and ZhouHZ20, Proposition 3.8], [Reference Anschütz, Gleason, Lourenço and RicharzAGLR22, Proposition 3.7]. Below we assemble the arguments in one place for the reader’s convenience.

Lemma 2.8. (1) The perfect Demazure scheme $D_{\dot {w}}$ is the perfection of a smooth, projective k-scheme. The multiplication map $\pi _{\dot {w}}$ satisfies

$$ \begin{align*}R\pi_{\dot{w}, *}(\mathcal{O}_{D_{\dot{w}}}) \cong \mathcal{O}_{\pi_{\dot{w}}(D_{\dot{w}})}[0] \quad \text{and} \quad R\pi_{\dot{w}, !}(\mathbb{F}_p) \cong \mathbb{F}_p[0].\end{align*} $$

(2) Furthermore, if $w \in {}_{\mathbf {f}'} W ^{\mathbf {f}}$ and $\dot {w}$ is a reduced expression, then $\pi _{\dot {w}}(D_{\dot {w}})= \mathcal {F}\!\ell _{w}(\mathbf {f'}, \mathbf {f})$ and $\pi _{\dot {w}}$ is an isomorphism over $\mathcal {F}\!\ell _{w}^\circ (\mathbf {f}', \mathbf {f})$ .

Proof. By the arguments in [Reference Pappas and RapoportPR08, Proposition 8.7] (which apply verbatim in mixed characteristic), $L^n_p\mathcal {P}_i/ L_p^n \mathcal {I} \cong \mathbb {P}^1_k$ for $n \gg 0$ , and this quotient admits sections Zariski-locally. As in [Reference He and ZhouHZ20, Proposition 3.5], this implies that $D_{\dot {w}}$ is the perfection of an iterated $\mathbb {P}^1_k$ -bundle which is smooth and projective. By [Reference Bhatt and ScholzeBS17, Lemma 6.9], to show that $R\pi _{\dot {w}, *}(\mathcal {O}_{D_{\dot {w}}}) \cong \mathcal {O}_{\pi _{\dot {w}}(D_{\dot {w}})}[0]$ , it suffices to check this on geometric fibers.

We first suppose $\mathbf {f} = \mathbf {f}' = \mathbf {a}$ , and we induct on the length of $\dot {w}$ . Let $\dot {v} = s_2 \cdots s_{n} \tau $ . Since $\pi _{\dot {v}}(D_{\dot {v}})$ is irreducible and $L^+\mathcal {I}$ -stable, it is of the form $\mathcal {F}\!\ell _{v'}(\mathbf {a}, \mathbf {a})$ for some $v' \in W$ . As in [Reference FaltingsFal03, Lemma 9], by performing the multiplication in two steps $(p_1, \cdots , p_n) \mapsto (p_1, p_2 \cdots p_n \tau ) \mapsto p_1 \cdots p_n \tau $ , we may factor $\pi _{\dot {w}}$ as

$$ \begin{align*}D_{\dot{w}} \longrightarrow L^+\mathcal{P}_1 \times^{L^+ \mathcal{I}}\mathcal{F}\!\ell_{v'}(\mathbf{a}, \mathbf{a}) \longrightarrow \pi_{\dot{w}}(D_{\dot{w}}). \end{align*} $$

By induction, pushforward along the first map preserves the structure sheaf. By properties of Tits systems [Reference Bruhat and TitsBT72, (1.2.6)] (especially T 3), if $s_1 v' < v'$ , then $\pi _{\dot {w}}(D_{\dot {w}}) = \mathcal {F}\!\ell _{v'}(\mathbf {a}, \mathbf {a})$ , and the second map has fibers isomorphic to $(\mathbb {P}^1)_{\operatorname {perf}}$ . Otherwise, if $s_1 v'> v'$ , then $\pi _{\dot {w}}(D_{\dot {w}}) = \mathcal {F}\!\ell _{s_1v'}(\mathbf {a}, \mathbf {a})$ . To describe the fibers in this case, consider the union $Z = \cup _{v" < v'} \mathcal {F}\!\ell _{v"}(\mathbf {a}, \mathbf {a})$ of those perfect Schubert schemes in $\mathcal {F}\!\ell _{v'}(\mathbf {a}, \mathbf {a})$ for which $s_1 v" < v"$ . Then the second map is an isomorphism away from Z and has fibers isomorphic to $(\mathbb {P}^1)_{\operatorname {perf}}$ over Z. Thus, in either case, the result follows since $R\Gamma ((\mathbb {P}^1_L)_{\operatorname {perf}}, \mathcal {O}_{(\mathbb {P}^1_L)_{\operatorname {perf}}}) = L[0]$ for any perfect field L.

For general $\mathbf {f}, \mathbf {f}'$ , we factor $\pi _{\dot {w}}$ as

(2.6) $$ \begin{align} D_{\dot{w}} \longrightarrow \mathcal{F}\!\ell_{\mathcal{I}} \longrightarrow \mathcal{F}\!\ell_{\mathcal{G}_{\mathbf{f}}}, \end{align} $$

where the first map is a Demazure map with $\mathbf {f} = \mathbf {f}' = \mathbf {a}$ , and the second map is the quotient. The image of the first map is of the form $\mathcal {F}\!\ell _{w'}(\mathbf {a}, \mathbf {a})$ for some $w' \in W$ . Thus, it suffices to show that pushforward along the natural surjection $\pi \colon \mathcal {F}\!\ell _{w'}(\mathbf {a}, \mathbf {a}) \rightarrow \pi _{\dot {w}}({D_{\dot {w}}})$ preserves the structure sheaf. For this, let $\overline {P}^{\operatorname {red}}$ be the maximal reductive quotient of the special fiber of $\mathcal {G}_{\mathbf {f}}$ . By [Reference RicharzRic13, Remark 2.9], the image of the special fiber of $\mathcal {I} \rightarrow \mathcal {G}_{\mathbf {f}}$ in $\overline {P}^{\operatorname {red}}$ is a Borel subgroup $\overline {B}$ , and we have

(2.7) $$ \begin{align} L_p^+\mathcal{G}_{\mathbf{f}}/L_p^+\mathcal{I} \cong \overline{P}^{\operatorname{red}} / \overline{B}. \end{align} $$

Furthermore, the maximal F-split torus S has a natural $\mathcal {O}_F$ -structure whose special fiber $\overline {S}$ is a maximal torus in $\overline {P}^{\operatorname {red}}$ . The group $W_{\mathbf {f}}$ identifies with the Weyl group of $(\overline {P}^{\operatorname {red}}, \overline {S})$ . Thus, since $\pi $ is $L^+\mathcal {I}$ -equivariant, the fibers of $\pi $ are isomorphic to perfections of unions of $\overline {B}$ -orbit closures in $\overline {P}^{\operatorname {red}} / \overline {B}$ . Such a union is connected, so by normality of $\pi _{\dot {w}}({D_{\dot{w}}})$ this implies that $\pi _{*}(\mathcal {O}_{\mathcal {F}\!\ell _{w'}(\mathbf {a}, \mathbf {a})}) \cong \mathcal {O}_{\pi _{\dot {w}}({D_{\dot {w}}})}$ . For the higher direct images, it is well-known that the structure sheaves of $\overline {B}$ -orbit closures (i.e, Schubert varieties), have vanishing higher cohomology. (For example, this follows because the Demazure resolution is a rational resolution by an iterated $\mathbb {P}^1$ -bundle [Reference Brion and KumarBK05, Theorem 3.3.4]). If our fiber in $\overline {P}^{\operatorname {red}} / \overline {B}$ is not irreducible, we induct on its support in $\overline {P}^{\operatorname {red}} / \overline {B}$ . Write an arbitrary union of $\overline {B}$ -orbit closures in terms of its irreducible components as $X = X_1 \cup \cdots \cup X_m$ . If $Y = X_2 \cup \cdots \cup X_m$ , we conclude by induction and the exact sequenceFootnote ⁴

$$ \begin{align*}0 \longrightarrow \mathcal{O}_{X_1 \cup Y} \longrightarrow \mathcal{O}_{X_1} \oplus \mathcal{O}_{Y} \longrightarrow \mathcal{O}_{X_1 \cap Y} \longrightarrow 0. \end{align*} $$

This proves that $R\pi _{\dot {w}, *}(\mathcal {O}_{D_{\dot {w}}}) \cong \mathcal {O}_{\pi _{\dot {w}}(D_{\dot {w}})}$ for general $\mathbf {f}, \mathbf {f}'$ .Footnote ⁵

To compute $\pi _{\dot {w}, !}(\mathbb {F}_p)$ , we first note that since $\pi _{\dot {w}}$ is the perfection of a proper map, we can replace $\pi _{\dot {w}, !}$ by $\pi _{\dot {w}, *}$ . Then we apply $R\pi _{\dot {w}, *}$ to the Artin–Schreier sequence

Finally, the proof of [Reference RicharzRic13, Proposition 2.8] shows that

$$ \begin{align*}\mathcal{F}\!\ell_{w}^\circ(\mathbf{f}', \mathbf{f}) = \mathcal{F}\!\ell^\circ_{_{\mathbf{f}'} w ^{\mathbf{f}}}(\mathbf{a}, \mathbf{f}).\end{align*} $$

Thus, if $w \in {}_{\mathbf {f}'} W ^{\mathbf {f}}$ and $\dot {w}$ is a reduced expression, by tracing through our proof that $R\pi _{\dot {w}, *}(\mathcal {O}_{D_{\dot {w}}}) \cong \mathcal {O}_{\pi _{\dot {w}}(D_{\dot {w}})}$ , it follows that $\pi _{\dot {w}}(D_{\dot {w}})= \mathcal {F}\!\ell _{w}(\mathbf {f'}, \mathbf {f})$ and $\pi _{\dot {w}}$ and is an isomorphism over $\mathcal {F}\!\ell _{w}^\circ (\mathbf {f}', \mathbf {f})$ .

2.2.2 Convolution

In this subsection, we let $\mathcal {G} = \mathcal {G}_{\mathbf {f}}$ and $\mathcal {F}\!\ell _w = \mathcal {F}\!\ell _w(\mathbf {f}, \mathbf {f})$ . The convolution diagram is as follows:

(2.8)

Here, p is the quotient map on the first factor and the identity on the second, q is the quotient by the diagonal action of $L^+ \mathcal {G}$ , and m is the multiplication map. The map $LG \times ^{L^+ \mathcal {G}} \mathcal {F}\!\ell _{\mathcal {G}} \rightarrow \mathcal {F}\!\ell _{\mathcal {G}} \times \mathcal {F}\!\ell _{\mathcal {G}}$ , $(g_1, g_2) \mapsto (g_1, g_1g_2)$ is an isomorphism, so the convolution Grassmannian $LG \times ^{L^+ \mathcal {G}} \mathcal {F}\!\ell _{\mathcal {G}}$ is represented by an increasing union of perfections of projective k-schemes.

Lemma 2.9. If k is a finite field or an algebraic closure thereof and $x \in \mathcal {F}\!\ell _{\mathcal {G}}(k)$ ,

$$ \begin{align*}m^{-1}(x) = \{(y, y^{-1}x) \: : \: y \in G(F)/K\} \subset G(F)/K \times G(F)/K.\end{align*} $$

We now define the following finite-dimensional perfect subschemes (2.9) of the convolution Grassmannian. Fix $w_1, w_2 \in W$ . Let $n \gg 0 $ be an integer such that $L^+\mathcal {G}$ acts on $\mathcal {F}\!\ell _{w_2}$ through the quotient $L^+ \mathcal {G} \rightarrow L^n\mathcal {G}$ . Let $\pi ^{(n)} \colon L^{(n)}G \rightarrow \mathcal {F}\!\ell _{\mathcal {G}}$ be the reduction of the $L^+\mathcal {G}$ -torsor $LG \rightarrow \mathcal {F}\!\ell _{\mathcal {G}}$ to an $L^n\mathcal {G}$ -torsor. Then we define

(2.9) $$ \begin{align} \mathcal{F}\!\ell_{w_1} \times^{L^+\mathcal{G}} \mathcal{F}\!\ell_{w_2} : = (\pi^{(n)})^{-1}(\mathcal{F}\!\ell_{w_1}) \times^{L^n\mathcal{G}} \mathcal{F}\!\ell_{w_2}.\end{align} $$

This is a closed subscheme of $LG \times \mathcal {F}\!\ell _{\mathcal {G}}$ , isomorphic to the perfection of a projective k-scheme, and independent of $n \gg 0$ .

Proof. Any lift $\widetilde \tau _1 \in N_G(S)(F)$ acts on $\mathcal {F}\!\ell _{\mathcal {G}}$ by multiplication on the left, and this action preserves $L^+\mathcal {I}$ -orbits, so $\tau _1 \cdot \pi _{\dot {w}}(D_{\dot {w}})$ is well-defined. Since m is $LG$ -equivariant, after multiplying on the left by $\widetilde \tau _1$ , we can assume $w_1 \in W_{\operatorname {af}}$ . Now consider the following commutative diagram.

Here, we have factored $\pi _{\dot {w}}$ as the composition

$$ \begin{align*}(p_1, \cdots, p_n, q_1, \cdots, q_m) \mapsto (p_1, \cdots, p_n, q_1 \cdots q_m \tau_2) \mapsto (p_1 \cdots p_n, q_1 \cdots q_m\tau_2) \mapsto p_1 \cdots q_m\tau_2.\end{align*} $$

By Lemma 2.8, each map (except possibly m) is surjective, and pushforward preserves the structure sheaf. For the bottom two maps, we also use flat base change and the fact that both properties can be checked étale-locally. It follows that m maps $\mathcal {F}\!\ell _{w_1} \times ^{L^+\mathcal {I}} \mathcal {F}\!\ell _{w_2}$ onto $\pi _{\dot {w}}(D_{\dot {w}})$ , and $Rm_*$ preserves the structure sheaf. It also follows that $Rm_!(\mathbb {F}_p) \cong \mathbb {F}_p[0]$ .

2.3.1 Attractors and fixed points

Let R be a ring, and let $\mathbb {G}_{m}$ be the multiplicative group over $\operatorname {Spec}(R)$ . For an R-scheme X with an action of $\mathbb {G}_m$ , we have the following functors on R-algebras (cf. [Reference DrinfeldDri13, Reference RicharzRic19]).

(2.10) $$ \begin{align} X^0 \colon R' \mapsto & \operatorname{Hom}_R^{\mathbb{G}_m}(\operatorname{Spec}(R'), X), \end{align} $$

(2.11) $$ \begin{align} X^+ \colon R' \mapsto & \operatorname{Hom}_R^{\mathbb{G}_m}(\mathbb{A}^1_{R'}, X). \end{align} $$

Here, $\operatorname {Spec}(R')$ has the trivial $\mathbb {G}_m$ -action and $\mathbb {A}^1_{R'}$ has the usual (multiplicative) action. These are the functors of fixed-points and attractors, respectively (we will not need the repellers). There are natural maps

Here, $q^+$ is induced by the zero section $\operatorname {Spec}(R') \rightarrow \mathbb {A}^1_{R'}$ . If X is an (ind)-perfect k-scheme with an action of $(\mathbb {G}_m)_{\operatorname {perf}}$ , we also use the notation $X^0$ and $X^+$ to denote the restriction of these functors to perfect k-algebras. The domain of these functors will be clear from the context, and their formation commutes with the passage from X to $X_{\operatorname {perf}}$ .

For the rest of subsection 2.3.1, we assume $R=k$ and that X is a separated k-scheme of finite-type.

We write the connected components of $X^+$ as

$$ \begin{align*}X^+ = \bigsqcup_{i \in \pi_0(X^0)} X^+_i.\end{align*} $$

Similarly, let $\{X_i^0\}_{i \in \pi _0(X^0)}$ be the connected components of $X^0$ . Let $q^+_i \colon X^+_i \rightarrow X^0_i$ be the restriction of $q^+$ to the corresponding connected component.

A sequence $Z_0 \subset Z_1 \subset \cdots \subset Z_n = X$ of closed subschemes is called a filtration of X, and the schemes $Z_i \setminus Z_{i-1}$ are the cells of the filtration. With additional assumptions, we can say more about $X^+$ as in the following lemma.

For applications to mod p Hecke algebras, we will apply the following result to a deperfection of the Demazure resolution (2.5) in Proposition 2.17 when F has characteristic p. When F has characteristic zero, we will verify directly that the conclusion of the lemma is still true for the perfect Witt vector Demazure resolution.

Lemma 2.13. In addition to the assumptions of Lemma 2.12, suppose moreover that there exists a smooth projective k-scheme $\widetilde X$ equipped with a $\mathbb {G}_m$ -action and a $\mathbb {G}_m$ -equivariant surjection $\pi \colon \widetilde X \rightarrow X$ . (1) There is exactly one $i_0 \in \pi _0(X^+)$ such that $X_{i_0}^+$ is closed in X, and it is characterized by the property that $q^+_{i_0}$ is a universal homeomorphism. (2) Furthermore, if $R\pi _!(\mathbb {F}_p) = \mathbb {F}_p[0]$ , we have

$$ \begin{align*}Rq_{i!}^+(\mathbb{F}_p)= \begin{cases} \mathbb{F}_p[0], & i= i_0 \\ 0, & \text{otherwise.} \end{cases}\end{align*} $$

2.3.2 Parabolic subgroups

For the rest of the paper, we assume $\mathbf {f} = \mathbf {f}'$ . We let $\mathcal {G} = \mathcal {G}_{\mathbf {f}}$ and $\mathcal {F}\!\ell _w = \mathcal {F}\!\ell _w(\mathbf {f}, \mathbf {f})$ .

Fix a semi-standard F-Levi subgroup $M \subset G$ , i.e., the centralizer of a subtorus of A. Let $\lambda \colon \mathbb {G}_m \rightarrow T$ be a cocharacter defined over F. By conjugation, this gives rise to a $\mathbb {G}_m$ -action on G. Then the attractor $P:=G^+$ as in (2.11) is a parabolic F-subgroup of G. We may choose $\lambda $ so that $G^0 = M$ is the fixed point functor as in (2.10) for this $\mathbb {G}_m$ -action defined by $\lambda $ . The k-points of the unipotent radical $U \subset P$ are

(2.12) $$ \begin{align} U(k) = \{g \in G(k) \:: \: \lim_{t \longrightarrow 0} \lambda(t) \cdot g \cdot \lambda(t)^{-1} = 1\}. \end{align} $$

If F has characteristic zero, the Teichmüller map induces a homomorphism $(\mathbb {G}_m)_{\operatorname {perf}} \rightarrow L^+\mathbb {G}_m$ . The cocharacter $\lambda \colon \mathbb {G}_{m, \mathcal {O}_F} \rightarrow \mathcal {T}$ also induces a homomorphism $L^+\mathbb {G}_m \rightarrow L^+\mathcal {T}$ . Since $\mathcal {T} \subset \mathcal {I}$ and $L^+\mathcal {I}$ acts on $\mathcal {F}\!\ell _{\mathcal {G}}$ , this induces a $(\mathbb {G}_m)_{\operatorname {perf}}$ -action on $\mathcal {F}\!\ell _{\mathcal {G}}$ . We may then define the fixed points $(\mathcal {F}\!\ell _{\mathcal {G}})^0$ and attractors $(\mathcal {F}\!\ell _{\mathcal {G}})^+$ as functors on perfect k-algebras. If F has characteristic p, we replace the Teichmüller map with natural inclusion of units $R^{\times } = \mathbb {G}_m(R) \rightarrow \mathbb {G}_m(R[\![t]\!]) = L^+\mathbb {G}_m(R)$ for a k-algebra R.

2.3.3 Equivariant resolutions

Proof. First, assume F has characteristic p. Then $D_{\dot {w}}$ has a canonical deperfection

$$ \begin{align*}D_{\dot{w}}^{\operatorname{dep}} : = L^+\mathcal{P}_{1,p} \times^{L^+_p \mathcal{I}} \cdots \times^{L^+_p \mathcal{I}} L^+\mathcal{P}_{n,p} / L^+_p \mathcal{I}.\end{align*} $$

This is a smooth, projective, iterated $\mathbb {P}^1$ -bundle. By considering the birational map $\pi _{\dot {w}}$ , we can view $k(D_{\dot {w}}^{\operatorname {dep}})$ as a subfield of $k(\mathcal {F}\!\ell _{w})$ . By Proposition 2.3, this gives us a particular deperfection $\mathcal {F}\!\ell _{w}^{\operatorname {dep}}$ of $\mathcal {F}\!\ell _{w}$ , equipped with a birational map

(2.13) $$ \begin{align} \pi_{\dot{w}}^{\operatorname{dep}} \colon D_{\dot{w}}^{\operatorname{dep}} \rightarrow \mathcal{F}\!\ell_{w}^{\operatorname{dep}}, \end{align} $$

which is a deperfection of (2.5). In fact, $\mathcal {F}\!\ell _{w}^{\operatorname {dep}}$ is the seminormalization of the usual affine Schubert variety associated to w as in [Reference Pappas and RapoportPR08]. Moreover, $\pi _{\dot {w}}^{\operatorname {dep}}$ is $L_p^+ \mathcal {I}$ -equivariant, so it is also $\mathbb {G}_m$ -equivariant for the action induced by

(2.14) $$ \begin{align} \mathbb{G}_m \rightarrow L_p^+ \mathbb{G}_m \xrightarrow{L_p^+ \lambda} L_p^+ \mathcal{T} \rightarrow L^+_p \mathcal{I}.\end{align} $$

If F has characteristic zero, the maps ${L^+_p \mathcal {I}} \rightarrow L^+\mathcal {P}_{i,p}$ have non-reduced kernels, so we cannot define $D_{\dot {w}}^{\operatorname {dep}}$ as above (the following method also works if F has characteristic p). Instead, let n be an integer large enough so that $L^+\mathcal {I}$ acts on $D_{\dot w}$ and $\mathcal {F}\!\ell _{w}$ through the quotient $L^n\mathcal {I}$ . Let $H \subset L^n_p\mathcal {I}$ be the unique reduced closed subgroup whose perfection is the stabilizer of a chosen point in the open orbit in $D_{\dot {w}}$ (and hence also in $\mathcal {F}\!\ell _{w}$ because $\pi _{\dot w}$ is an isomorphism over the open orbit). Then H is a smooth affine k-group by [Reference ZhuZhu17a, Lemma A.26]. Let $D_{\dot {w}}^{\operatorname {dep}}$ and $\mathcal {F}\!\ell _{w}^{\operatorname {dep}}$ be the deperfections associated to $k(L^n_p\mathcal {I}/H)$ by Proposition 2.3. Since these deperfections are normal, they admit $L^n_p\mathcal {I}$ -actions which extend the action over the open orbit (cf. [Reference Anschütz, Gleason, Lourenço and RicharzAGLR22, Proposition 3.3]). The map $\pi _{\dot w}$ also deperfects to an $L^n_p\mathcal {I}$ -equivariant map as in (2.13) because $\pi _{\dot w}$ is $L^n\mathcal {I}$ -equivariant and birational. Then we get the desired $\mathbb {G}_m$ -action from the composition (2.14).

2.3.4 Description of the attractors

Since the partial order on a Coxeter group is directed, we may write $\mathcal {F}\!\ell _{\mathcal {G}} = \operatorname {colim} \mathcal {F}\!\ell _w,$ where the colimit is over $w \in {}_{\mathbf {f}} W ^{\mathbf {f}}$ , and the transition maps are closed immersions. The formation of fixed points and attractors is compatible with filtered colimits of closed immersions [Reference Haines and RicharzHR21, Theorem 2.1]. Thus, $(\mathcal {F}\!\ell _G)^0 = \operatorname {colim} (\mathcal {F}\!\ell _w)^0$ and $(\mathcal {F}\!\ell _G)^+ = \operatorname {colim} (\mathcal {F}\!\ell _w)^+.$ Here, $(\mathcal {F}\!\ell _G)^0$ is a closed sub-ind-scheme of $\mathcal {F}\!\ell _{\mathcal {G}}$ , and $(\mathcal {F}\!\ell _G)^+$ is a disjoint union of locally closed sub-ind-schemes. The natural map $(\mathcal {F}\!\ell _G)^+ \rightarrow (\mathcal {F}\!\ell _G)^0$ induces a bijection $\pi _0((\mathcal {F}\!\ell _G)^+) \cong \pi _0((\mathcal {F}\!\ell _G)^0)$ .

By the definition of $\mathcal {M}:=\mathcal {G}^0$ , the closed immersion $\mathcal {F}\!\ell _{\mathcal {M}} \rightarrow \mathcal {F}\!\ell _{\mathcal {G}}$ factors through $(\mathcal {F}\!\ell _{\mathcal {G}})^0$ . By the proof of [Reference Anschütz, Gleason, Lourenço and RicharzAGLR22, Theorem 5.2], the map $\mathcal {F}\!\ell _{\mathcal {M}} \rightarrow (\mathcal {F}\!\ell _{\mathcal {G}})^0$ identifies $\mathcal {F}\!\ell _{\mathcal {M}}$ with a disjoint union of connected components of $(\mathcal {F}\!\ell _{\mathcal {G}})^0$ (see also [Reference Haines and RicharzHR21, Proposition 4.7, Lemma 4.11]). Furthermore, by [Reference Anschütz, Gleason, Lourenço and RicharzAGLR22, Theorem 5.2], the connected components of $(\mathcal {F}\!\ell _{\mathcal {G}})^0$ can be described as follows. Let $W_{M}$ be the Iwahori–Weyl group of M and let $W_{M, \operatorname {af}}$ be its affine Weyl group. Then the inclusion $\pi _0(\mathcal {F}\!\ell _{\mathcal {M}}) \hookrightarrow \pi _0((\mathcal {F}\!\ell _{\mathcal {G}})^0)$ identifies with the inclusion

$$ \begin{align*}W_{M, \operatorname{af}} \backslash W_M \hookrightarrow W_{M,\operatorname{af}} \backslash W / W_{\mathbf{f}}.\end{align*} $$

For $c \in W_{M,\operatorname {af}} \backslash W / W_{\mathbf {f}}$ , let $S_c$ be the corresponding connected componentFootnote ⁶ of $(\mathcal {F}\!\ell _{\mathcal {G}})^+$ . This is a locally closed sub-ind-scheme of $\mathcal {F}\!\ell _{\mathcal {G}}$ . Then

$$ \begin{align*}\mathcal{F}\!\ell_{\mathcal{G}} = \bigsqcup_{c \in W_{M,\operatorname{af}} \backslash W / W_{\mathbf{f}}} S_c.\end{align*} $$

The irreducible components of the $S_c \cap \mathcal {F}\!\ell _w$ generalizeFootnote ⁷ the Mirković–Vilonen cycles in [Reference Mirković and VilonenMV07, Theorem 3.2]. The k-points of $S_c$ can be described as follows. Let $M_{\operatorname {sc}}$ be the simply connected cover of the derived group of M and let $P_{\operatorname {sc}} = M_{\operatorname {sc}} \ltimes U$ . The following lemma also appears in the proof of [Reference Anschütz, Gleason, Lourenço and RicharzAGLR22, Theorem 5.2].

If $c \in W_{M, \operatorname {af}} \backslash W_M = \pi _1(M)_I$ , let $(\mathcal {F}\!\ell _{\mathcal {M}})^c$ be the corresponding connected component of $\mathcal {F}\!\ell _{\mathcal {M}}$ . Let

$$ \begin{align*}\pi_c \colon S_c \rightarrow (\mathcal{F}\!\ell_{\mathcal{M}})^c\end{align*} $$

be the resulting map obtained by restriction from the map $(\mathcal {F}\!\ell _{\mathcal {G}})^+ \rightarrow (\mathcal {F}\!\ell _{\mathcal {G}})^0$ .

Lemma 2.20. Let $m \in \mathcal {F}\!\ell _{\mathcal {M}}(k)$ and let $\widetilde {m} \in M(F)$ be a representative. If m lies in $(\mathcal {F}\!\ell _{\mathcal {M}})^c$ , then

$$ \begin{align*}\pi_c^{-1}(m) = \{[\widetilde{m} \cdot u] \: : \: u \in U(F)/(U(F) \cap K)\} \subset G(F)/K.\end{align*} $$

In the above, $[\widetilde {m} \cdot u]$ is the equivalence class of $\widetilde {m} \cdot u$ in $\mathcal {F}\!\ell _{\mathcal {G}}(k)$ for $u \in U(F)$ , and $[\widetilde {m} \cdot u_1] = [\widetilde {m} \cdot u_2]$ if and only if $u_1 \cdot (U(F) \cap K) = u_2 \cdot (U(F) \cap K)$ .

2.3.5 Attractors over finite fields

In this subsection, we assume that k is finite, and we fix an element $w \in W_{\mathbf {f}} \backslash W^\sigma / W_{\mathbf {f}}$ . By (2.3), this gives rise to a perfect Schubert scheme $\mathcal {F}\!\ell _{w}$ defined over k which admits a k-point. Also, fix a connected component $(\mathcal {F}\!\ell _{\mathcal {M}})^c$ of $\mathcal {F}\!\ell _{\mathcal {M}}$ which is defined over k and admits a k-point (and is therefore geometrically connected). Such connected components of $\mathcal {F}\!\ell _{\mathcal {M}}$ are indexed by $c \in \pi _1(M)_I^\sigma $ .

The ind-schemes $(\mathcal {F}\!\ell _{\mathcal {G}})^0$ and $(\mathcal {F}\!\ell _{\mathcal {G}})^+$ are defined over k, as are the maps $\mathcal {F}\!\ell _{\mathcal {M}} \rightarrow (\mathcal {F}\!\ell _{\mathcal {G}})^0 \rightarrow \mathcal {F}\!\ell _{\mathcal {G}}$ . Let $S_c$ be the geometrically connected component of $(\mathcal {F}\!\ell _{\mathcal {G}})^+$ which maps to $(\mathcal {F}\!\ell _{\mathcal {M}})^c$ under the map $(\mathcal {F}\!\ell _{\mathcal {G}})^+ \rightarrow (\mathcal {F}\!\ell _{\mathcal {G}})^0$ .

Let $\pi _{c,w} \colon S_c \cap \mathcal {F}\!\ell _{w} \rightarrow (\mathcal {F}\!\ell _{\mathcal {M}})^c \cap \mathcal {F}\!\ell _w$ be the restriction of the map $\pi _c \colon S_c \rightarrow (\mathcal {F}\!\ell _{\mathcal {M}})^c$ .

Theorem 2.22. Let $c \in \pi _1(M)_I^\sigma $ and $w \in W_{\mathbf {f}} \backslash W^{\sigma } / W_{\mathbf {f}}$ . Let m be a k-point of $(\mathcal {F}\!\ell _{\mathcal {M}})^c \cap \mathcal {F}\!\ell _w$ and let $\widetilde m \in M(F)$ be a representative.

(1) At the level of k-points, we have

$$ \begin{align*}\pi_{c,w}^{-1}(m) = \mathcal{F}\!\ell_w(k) \cap \{[\widetilde m \cdot u] \: : \: u \in U(F)/(U(F) \cap K)\}.\end{align*} $$

In the above, $[\widetilde {m} \cdot u]$ is the equivalence class of $\widetilde {m} \cdot u$ in $\mathcal {F}\!\ell _{\mathcal {G}}(k)$ for $u \in U(F)$ , and $[\widetilde {m} \cdot u_1] = [\widetilde {m} \cdot u_2]$ if and only if $u_1 \cdot (U(F) \cap K) = u_2 \cdot (U(F) \cap K)$ .

(2) Furthermore, we have

$$ \begin{align*}R(\pi_{c,w})_{!}(\mathbb{F}_p)= \begin{cases} \mathbb{F}_p[0], & (\mathcal{F}\!\ell_{\mathcal{M}, \overline{k}})^c \cap \mathcal{F}\!\ell_{w, \overline{k}} \text{ is the unique closed attractor in } (\mathcal{F}\!\ell_{w, \overline{k}})^+ \\ 0, & \text{otherwise.} \end{cases}\end{align*} $$

3.1.1 The function-sheaf dictionary

In this subsection, we recall the function-sheaf dictionary following [Reference DeligneDel77, Rapport]. Let X be a separated scheme of finite-type over k. Let $\mathcal {F}$ be a constructible sheaf of $\mathbb {F}_p$ -vector spaces on X (we will only need the case where $\mathcal {F}$ is a constant sheaf). For each point $x \in X(k)$ , the stalk $\mathcal {F}_x$ is a finite-dimensional representation of $\operatorname {Gal}(\overline {k}/k)$ . Taking the trace of the action of the geometric Frobenius element $\gamma \in \operatorname {Gal}(\overline {k}/k)$ gives a value $\operatorname {Tr}(\gamma , \mathcal {F}_x) \in \mathbb {F}_p$ . In this way, we get a function

(3.1) $$ \begin{align} \mathcal{F}^{\operatorname{Tr}} \colon X(k) \rightarrow \mathbb{F}_p \quad x \mapsto \operatorname{Tr}(\gamma, \mathcal{F}_x).\end{align} $$

Moreover, if $X_{\overline {k}} = X \times \operatorname {Spec}(\overline {k})$ , there is a natural action of $\operatorname {Gal}(\overline {k}/k)$ on the étale cohomology with compact support $H_c^i(X_{\overline {k}}, \mathcal {F})$ . The function-sheaf dictionary is the following relationship between these actions.

Lemma 3.1. We have

$$ \begin{align*}\sum_{x \in X(k)} \operatorname{Tr}(\gamma, \mathcal{F}_x) = \sum_i (-1)^i \operatorname{Tr}(\gamma, H_c^i(X_{\overline{k}}, \mathcal{F})).\end{align*} $$

In particular, if $\mathcal {F} = \mathbb {F}_p$ is the constant sheaf,

$$ \begin{align*}|X(k)| \equiv \sum_i (-1)^i \operatorname{Tr}(\gamma, H_c^i(X_{\overline{k}}, \mathbb{F}_p)) \quad\pmod{p}.\end{align*} $$

3.1.2 Definitions and bases

The mod p Hecke algebra associated to K is the algebra $\mathcal {H}_{K}$ of compactly supported functions $K \backslash G(F) / K \rightarrow \mathbb {F}_p$ . The multiplication is convolution:

(3.2) $$ \begin{align} (f_1 * f_2)(g) = \sum_{h \in G(F)/K} f_1(h) f_2(h^{-1}g), \quad f_1, f_2 \in \mathcal{H}_K, \: g \in G(F). \end{align} $$

The algebra $\mathcal {H}_{K}$ has a basis consisting of the characteristic functions of the double cosets $K \backslash G(F) / K$ . By (2.3), these double cosets are indexed by $ w \in W_{\mathbf {f}} \backslash W^\sigma / W_{\mathbf {f}}$ . Let be the characteristic function of the corresponding double coset.

There is another basis of $\mathcal {H}_K$ more suitable for geometric arguments. For $w \in W_{\mathbf {f}} \backslash W^\sigma / W_{\mathbf {f}}$ , let , where the sum runs over those $v \in W_{\mathbf {f}} \backslash W^\sigma / W_{\mathbf {f}}$ such that $_{\mathbf {f}} v ^{\mathbf {f}} \leq {}_{\mathbf {f}} w ^{\mathbf {f}}$ . The basis $\phi _{w}$ arises from the function-sheaf dictionary as follows. Let $(\underline {\mathbb {F}}_p)_w$ be the constant étale sheaf supported on $\mathcal {F}\!\ell _w$ . Then by the closure relations in Lemma 2.2, we have

(3.3) $$ \begin{align} ((\underline{\mathbb{F}}_p)_w)^{\operatorname{Tr}} = \phi_w. \end{align} $$

3.1.3 The Satake transform

Recall the Levi decomposition $P = MU$ arising from the cocharacter $\lambda $ . The Satake transform with respect to P is the following map.

(3.4) $$ \begin{align} \mathcal{S} \colon \mathcal{H}_{\mathcal{G}} \rightarrow \mathcal{H}_{M(F) \cap K}, \quad \mathcal{S}(f)(m) = \sum_{u \in U(F) / U(F) \cap K} f(mu), \: \: m \in M(F). \end{align} $$