2.1 Sheaves

We will work with usual $\infty $ -categories of constructible sheaves on prestacks, as discussed in various contexts and various degrees of precision in [Reference Gaitsgory27, Reference Liu and Zheng32, Reference Richarz and Scholbach48, Reference Arinkin, Gaitsgory, Kazhdan, Raskin, Rozenblyum and Varshavsky8, Reference Scholze51, Reference Bouthier, Kazhdan and Varshavsky17, Reference Zhu53, Reference Dhillon, Li, Yun and Zhu21], with a field of coefficients denoted $\Bbbk $ (which is not assumed to be of characteristic $0$ ). More specifically, we fix an algebraically closed field $\mathbb {F}$ , and consider one of the following settings:

1. (1) $\mathbb {F}$ is arbitrary, $\ell $ is a prime number invertible in $\mathbb {F}$ , $\Bbbk $ is either a finite extension or an algebraic closure of $\mathbb {F}_\ell $ or $\mathbb {Q}_\ell $ , and we consider étale $\Bbbk $ -sheaves on separated schemes of finite type over $\mathbb {F}$ ;

2. (2) $\mathbb {F}=\mathbb {C}$ , $\Bbbk $ is arbitrary, and we work with sheaves for the analytic topology on separated schemes of finite type over $\mathbb {F}$ .

The first setting is, e.g. discussed in detail in [Reference Dhillon, Li, Yun and Zhu21, §2.2], building on the more complete treatment in the setting of perfect prestacks in [Reference Zhu53, §10]. The second setting is unfortunately not yet discussed in the literature. We will not consider the adaptation of the constructions in this setting, as we are not capable of doing so; instead we will just assume that a sheaf theory as discussed in [Reference Dhillon, Li, Yun and Zhu21, §2.2] exists also in this setting, and possesses similar properties.

So, for any algebraic space X of finite presentation over $\mathbb {F}$ we have a $\Bbbk $ -linear small stable $\infty $ -category of bounded complexes of constructible sheaves on X, denoted $\mathsf {Shv}_{\mathrm {c}}(X)$ , and the $\Bbbk $ -linear stable presentable $\infty $ -category

$$\begin{align*}\mathsf{Shv}(X) := \mathrm{Ind}(\mathsf{Shv}_{\mathrm{c}}(X)) \end{align*}$$

(where $\mathrm {Ind}$ denotes the ind-completion). We have push/pull functors associated with morphisms of algebraic spaces of finite presentation for categories $\mathsf {Shv}_{\mathrm {c}}(-)$ , and we deduce similar functors for the categories $\mathsf {Shv}(-)$ by $\mathrm {Ind}$ -extension. In particular, all of our push/pull functors between such $\infty $ -categories will be continuous (i.e. preserve colimits) by design.

Next, one extends the definition of $\mathsf {Shv}_{\mathrm {c}}(X)$ and $\mathsf {Shv}(X)$ by left Kan extension to all quasi-compact and quasi-separated (qcqs) algebraic spaces. Concretely, if a (qcqs) algebraic space X is written as a cofiltered limit $\lim _i X_i$ with each $X_i$ of finite presentation, and with transition maps being affine morphisms, we have

$$\begin{align*}\mathsf{Shv}_{\mathrm{c}}(X) = \operatorname*{{\mathrm{colim}}}_i \mathsf{Shv}_{\mathrm{c}}(X_i), \quad \mathsf{Shv}(X) = \operatorname*{{\mathrm{colim}}}_i \mathsf{Shv}(X_i) \end{align*}$$

with transition functors given by $!$ -pullbacks. Then, these constructions are extended further to prestacks by right Kan extension. Concretely, for a prestack X, we have

(2.1) $$ \begin{align} \mathsf{Shv}_{\mathrm{c}}(X) = \lim_S \mathsf{Shv}_{\mathrm{c}}(S), \quad \mathsf{Shv}(X) = \lim_S \mathsf{Shv}(S) \end{align} $$

where S runs over the qcqs algebraic spaces mapping to X, with transition functors given by $!$ -pullbacks.

By construction, for any prestack Y we have a canonical fully faithful functor

$$\begin{align*}\mathsf{Shv}_{\mathrm{c}}(Y) \to \mathsf{Shv}(Y). \end{align*}$$

This functor induces a canonical functor

$$ \begin{align*} \mathrm{Ind}(\mathsf{Shv}_{\mathrm{c}}(Y)) \to \mathsf{Shv}(Y) \end{align*} $$

commuting with colimits. If Y is an algebraic space of finite presentation over $\mathbb {F}$ , this functor is an equivalence, but for more general prestacks this property is not true (see Remark 2.2 below). In case Y is an algebraic stack of finite type over $\mathbb {F}$ then the homotopy category of $\mathsf {Shv}_{\mathrm {c}}(Y)$ is the familiar bounded derived category of constructible complexes on Y.

2.2 Functorialities

As explained in the references discussed in §2.1, the $\infty $ -categories $\mathsf {Shv}(X)$ are equipped with highly structured functorialities, which make them a sheaf theory (on an appropriate $\infty $ -category of correspondences) in the sense of [Reference Zhu53, §8.2]. In this subsection we spell out the consequences we will use below. First, for any morphism of prestacks $f : X \to Y$ we have a continuous $!$ -pullback functor

$$\begin{align*}f^! : \mathsf{Shv}(Y) \to \mathsf{Shv}(X). \end{align*}$$

In particular this allows us, for any prestack X, to define the dualizing complex $\omega _X$ as the $!$ -pullback of $\Bbbk $ , considered as a complex of sheaves on $\mathrm {pt}:=\mathrm {Spec}(\mathbb {F})$ , under the structure morphism $X \to \mathrm {pt}$ .

Next, for any morphism of prestacks $f : X \to Y$ which is indfp in the sense of [Reference Dhillon, Li, Yun and Zhu21, §2.2], we have a continuous $*$ -pushforward functor

$$\begin{align*}f_* : \mathsf{Shv}(X) \to \mathsf{Shv}(Y). \end{align*}$$

The functors $(-)^!$ and $(-)_*$ satisfy appropriate forms of the projection formula and base change theorem, as discussed (in the setting of general sheaf theories) in [Reference Zhu53, §8.2.1]. Considering adjoints to these functors (which exist, and are well behaved, in some special cases) one also obtains $*$ -pullback and $!$ -pushforward functors, which satisfy projection formulas and base change theorems obtained by adjunction.

In practice, the prestacks we will consider will arise from ind-schemes of ind-finite type over $\mathbb {F}$ . (For the basics of the general theory of ind-schemes, see, e.g. [Reference Richarz47].) If Y is such an object, which we write as

$$\begin{align*}Y = \operatorname*{{\mathrm{colim}}}_{i \in I} Y_i \end{align*}$$

where I is a filtered poset, each $Y_i$ is a scheme of finite type over $\mathbb {F}$ , and the transition morphisms $Y_i \to Y_j$ are closed immersions, then we have

$$\begin{align*}\mathsf{Shv}(Y) = \lim_{i \in I} \mathsf{Shv}(Y_i) \end{align*}$$

where for $i \leq j$ the transition functor $\mathsf {Shv}(Y_j) \to \mathsf {Shv}(Y_i)$ is the $!$ -pullback functor. Now each of these functors admits a left adjoint, namely the (fully faithful) pushforward functor under the closed immersion $Y_i \to Y_j$ . By the usual result of Lurie allowing one to transform a limit into a colimit (see [Reference Gaitsgory and Rozenblyum29, Part I, Chap. 1, Proposition 2.5.7]), this implies that we have

(2.2) $$ \begin{align} \mathsf{Shv}(Y) = \operatorname*{{\mathrm{colim}}}_{i \in I} \mathsf{Shv}(Y_i), \end{align} $$

where now for $i \leq j$ the transition functor $\mathsf {Shv}(Y_i) \to \mathsf {Shv}(Y_j)$ is the pushforward functor, and the colimit is computed in the $\infty $ -category of presentable $\infty $ -categories.

If $X,Y$ are ind-schemes of ind-finite type over $\mathbb {F}$ , with respective presentations

$$\begin{align*}X = \operatorname*{{\mathrm{colim}}}_{i \in I} X_i, \quad Y=\operatorname*{{\mathrm{colim}}}_{j \in J} Y_j \end{align*}$$

as above, and if $f : X \to Y$ is a morphism, then we have a continuous functor $f_*$ by the comments above. But as in [Reference Richarz and Scholbach48, Proposition 2.3.3] we also have a (continuous) functor $f_!$ obtained as the left adjoint of $f^!$ . Concretely, there exists an increasing function $a : I \to J$ and a system of morphisms $(f_i)_{i \in I}$ compatible with the structure morphisms, where $f_i$ is a morphism from $X_i$ to $Y_{a(i)}$ , such that $f=\operatorname *{{\mathrm {colim}}}_{i \in I} f_i$ . Then the continuous pushforward functors

$$\begin{align*}f_*, f_! : \mathsf{Shv}(X) \to \mathsf{Shv}(Y) \end{align*}$$

are obtained by passage to colimits from the case of schemes of finite type.

Let us now assume that the morphism f as above is representable in schemes of finite type, in the sense that for any $\mathbb {F}$ -scheme of finite type S and for any morphism $S \to Y$ the fibre product $S \times _Y X$ is a scheme of finite type. In this case, as in [Reference Richarz and Scholbach48, Theorem 2.4.2] the functor $f_*$ admits a (continuous) left adjoint

$$\begin{align*}f^* : \mathsf{Shv}(Y) \to \mathsf{Shv}(X). \end{align*}$$

In fact, one can choose the presentations above such that $J=I$ and $X_i=Y_i \times _Y X$ for any i, and choose $a=\mathrm {id}_I$ and $f_i$ to be the morphism induced by f. Then $f^*$ is defined on the subcategory $\mathsf {Shv}(Y_i)$ by the usual theory for schemes of finite type, and we deduce that it is defined everywhere.

The property considered above is satisfied in particular when f is represented by a closed immersion or by an open immersion; in the former case we have $f_*=f_!$ , and in the latter case we have $f^* = f^!$ . If $Z,X$ are ind-schemes of ind-finite type over $\mathbb {F}$ and if $i : Z \to X$ is a morphism which is representable by a closed immersion, with complementary open immersion $j : U \to X$ , then the categories

$$\begin{align*}\mathsf{Shv}(Z), \quad \mathsf{Shv}(X), \quad \mathsf{Shv}(U) \end{align*}$$

and the functors $i_*$ , $i^*$ , $i^!$ , $j_*$ , $j^!$ , $j^*$ satisfy the ‘recollement’ formalism of [Reference Beĭlinson, Bernstein, Deligne and Gabber11, §1.4], or more specifically its immediate translation in the world of stable $\infty $ -categories studied e.g. in [Reference Loregian33, §5.2]. In particular, for any $\mathcal {F} \in \mathsf {Shv}(X)$ we have fibre sequences

(2.3) $$ \begin{align} i_! i^! \mathcal{F} \to \mathcal{F} \to j_* j^* \mathcal{F}, \qquad j_! j^! \mathcal{F} \to \mathcal{F} \to i_*i^* \mathcal{F}. \end{align} $$

In this setting we will say that a complex $\mathcal {F}$ is supported on Z if it belongs to the essential image of the fully-faithful functor $i_*$ , or equivalently if $j^* \mathcal {F} = 0$ .

Below we will use the following fact. Consider an ind-scheme of ind-finite type $X = \operatorname *{{\mathrm {colim}}}_i X_i$ over $\mathbb {F}$ , and a collection $(U_n : n \in \mathbb {Z}_{\geq 0})$ of open sub-ind-schemes such that $U_n \subset U_{n+1}$ for any n, and $X = \bigcup _n U_n$ . Denote, for $n \in \mathbb {Z}_{\geq 0}$ , by $j_n : U_n \to X$ the open embedding. For any $\mathcal {F}$ in $\mathsf {Shv}(X)$ , we have a canonical functor $\mathbb {Z}_{\geq 0} \to \mathsf {Shv}(X)$ sending n to $(j_n)_! (j_n)^* \mathcal {F}$ , and a canonical morphism

(2.4) $$ \begin{align} \operatorname*{{\mathrm{colim}}}_n \, (j_n)_! (j_n)^* \mathcal{F} \to \mathcal{F}. \end{align} $$

We claim that this morphism is an isomorphism. In fact, by (2.2) we can assume that $\mathcal {F}$ is obtained from a complex on $X_i$ , and then the claim follows from the fact that $U_n \cap X_i = X_i$ for $n \gg 0$ since $X_i$ is noetherian.

In case Y is an ind-scheme of ind-finite type over $\mathbb {F}$ as above and H is an affine $\mathbb {F}$ -group scheme of finite type acting on Y and preserving each $Y_i$ , we also have

(2.5) $$ \begin{align} \mathsf{Shv}(H \backslash Y) = \lim_{i \in I} \mathsf{Shv}(H \backslash Y_i) = \operatorname*{{\mathrm{colim}}}_{i \in I} \mathsf{Shv}(H \backslash Y_i) \end{align} $$

where in the first case the limit is taken with respect to $!$ -pullbacks, and in the second case the colimit is taken with respect to pushforward functors (and in the $\infty $ -category of presentable $\infty $ -categories). We also have the functors $f_*, f_!, f^*, f^!$ as above for morphisms induced by equivariant morphisms of ind-schemes satisfying the appropriate assumptions.

We also have a natural ‘forgetful’ functor

$$\begin{align*}\mathrm{oblv}^H : \mathsf{Shv}(H \backslash Y) \to \mathsf{Shv}(Y), \end{align*}$$

formally defined as the $!$ -pullback under the projection map $Y \to H \backslash Y$ . To avoid cluttered notation, the dualizing complex $\omega _{H \backslash Y}$ will also be denoted $\omega _Y$ ; its image under $\mathrm {oblv}^H$ is $\omega _Y$ . Note that $\mathrm {oblv}^H$ is conservative, since $\mathsf {Shv}(H \backslash Y)$ can be described in terms of the usual ‘Čech nerve’ of the projection $Y \to H \backslash Y$ .

For X an $\mathbb {F}$ -ind-scheme of ind-finite type with a presentation $X=\operatorname *{{\mathrm {colim}}}_{i \in I} X_i$ as above, it turns out that the $\infty $ -category $\mathsf {Shv}_{\mathrm {c}}(X)$ does not model what one often calls ‘constructible sheaves’ in this context; namely, there exists a fully faithful functor

$$\begin{align*}\operatorname*{{\mathrm{colim}}}_{i \in I} \mathsf{Shv}_{\mathrm{c}}(X_i) \to \mathsf{Shv}_{\mathrm{c}}(X) \end{align*}$$

(where the left-hand side is the colimit in small stable $\infty $ -categories, or equivalently in all $\infty $ -categories, and with respect to pushforward functors), but this functor is not an equivalence in general (see [Reference Bouthier, Kazhdan and Varshavsky17, Remark 5.3.4] or the discussion at the beginning of [Reference Zhu53, §10.6.2]). To remedy this, Zhu develops in [Reference Zhu53, §10] a formalism $\mathsf {Shv}_{\mathrm {fg}}$ of ‘finitely generated sheaves’ on appropriate prestacks (taking values in small stable $\infty $ -categories). When X is an ind-scheme as above, then it immediately follows from the definition that

(2.6) $$ \begin{align} \mathsf{Shv}_{\mathrm{fg}}(X) = \operatorname*{{\mathrm{colim}}}_{i \in I} \mathsf{Shv}_{\mathrm{c}}(X_i), \end{align} $$

see [Reference Zhu53, Lemma 10.161], and we have

$$\begin{align*}\mathsf{Shv}(X) = \mathrm{Ind}(\mathsf{Shv}_{\mathrm{fg}}(X)), \end{align*}$$

see [Reference Zhu53, Remark 10.162]. In fact an equivalence similar to (2.6) holds for what Zhu calls ‘ind-placid stacks’, which includes the Hecke stack. Details will be discussed below in the special cases we want to consider.

2.3 Affine flag varieties

Let $\mathbb {F}$ be as in §2.1, and let G be a connected reductive algebraic group over $\mathbb {F}$ . Fix a Borel subgroup $B \subset G$ and a maximal torus $T \subset B$ , and denote by U the unipotent radical of B. We will denote by

$$\begin{align*}\mathbf{Y} = X_*(T), \quad \text{resp.} \quad \mathbf{X}=X^*(T), \end{align*}$$

the cocharacter, resp. character, lattice of T. Let $\mathfrak {R} \subset \mathbf {X}$ be the set of roots for $(G,T)$ , and let $\mathfrak {R}_+ \subset \mathfrak {R}$ be the set of positive roots determined by B, i.e. the set of nonzero weights of T on $\mathrm {Lie}(B)$ . We will denote by $\rho $ the halfsum of the positive roots. We will also denote by $W_{\mathrm {f}}$ the Weyl group of $(G,T)$ , which we endow with the Coxeter structure determined by our choice of B, and by $w_\circ $ its longest element.

For any affine scheme X over $\mathbb {F}$ we will denote by

$$\begin{align*}\mathcal{L} X, \quad \text{resp.} \quad \mathcal{L}^+ X, \end{align*}$$

the ind-affine ind-scheme, resp. affine scheme, representing the functor sending an $\mathbb {F}$ -algebra R to $X(R( \hspace {-1pt} (z) \hspace {-1pt} ))$ , resp. $X(R[ \hspace {-1pt} [z] \hspace {-1pt} ])$ , where z is an indeterminate. There exist canonical morphisms $\mathcal {L}^+ X \to \mathcal {L} X$ and $\mathcal {L}^+ X \to X$ . In case X is a group scheme, $\mathcal {L} X$ , resp. $\mathcal {L}^+ X$ , admits a natural structure of group ind-scheme, resp. group scheme, and the morphisms $\mathcal {L}^+ X \to \mathcal {L} X$ and $\mathcal {L}^+ X \to X$ are morphisms of group (ind-)schemes. In this case the kernel of the morphism $\mathcal {L}^+ X \to X$ will be denoted $\mathcal {L}^{++} X$ .

In particular we will consider the groups $\mathcal {L} G$ , $\mathcal {L}^+ G$ , $\mathcal {L}^+ T$ and $\mathcal {L}^{++} T$ . Each $\lambda \in \mathbf {Y}$ defines an $\mathbb {F}$ -point $z^\lambda \in (\mathcal {L} T)(\mathbb {F})$ ; we will denote similarly its image in $(\mathcal {L} G)(\mathbb {F})$ .

Let $\mathrm {I} \subset \mathcal {L}^+G$ be the Iwahori subgroup attached to B, i.e. the preimage of B under the canonical morphism $\mathcal {L}^+G \to G$ , and let $\mathrm {I}_{\mathrm {u}}$ be the pro-unipotent radical of $\mathrm {I}$ , i.e. the preimage of U under this morphism. We will consider in addition the ‘semiinfinite Iwahori subgroup’

$$\begin{align*}\mathrm{I}^{{\frac{\infty}{2}}} := \mathcal{L}^+T \ltimes \mathcal{L} U \end{align*}$$

and its ‘pro-unipotent radical’

$$\begin{align*}\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} := \mathcal{L}^{++} T \ltimes \mathcal{L} U. \end{align*}$$

Here $\mathrm {I}$ and $\mathrm {I}_{\mathrm {u}}$ are group schemes over $\mathbb {F}$ , but $\mathrm {I}^{{\frac {\infty }{2}}}$ and $\mathrm {I}_{\mathrm {u}}^{{\frac {\infty }{2}}}$ are only group ind-schemes over $\mathbb {F}$ . Since U is unipotent, we can write

$$\begin{align*}\mathcal{L} U = \operatorname*{{\mathrm{colim}}}_{n \geq 0} \mathcal{L}_{(n)} U \end{align*}$$

where each $\mathcal {L}_{(n)} U$ is a group scheme over $\mathbb {F}$ and each transition morphism $\mathcal {L}_{(n)} U \to \mathcal {L}_{(n+1)} U$ is a closed immersion of group schemes. These group schemes can be chosen to be stable under the conjugation action of $\mathcal {L}^+ T$ , so that we obtain presentations

(2.7) $$ \begin{align} \mathrm{I}^{{\frac{\infty}{2}}} = \operatorname*{{\mathrm{colim}}}_{n \geq 0} \mathrm{I}^{{\frac{\infty}{2}}}_{(n)}, \quad \mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} = \operatorname*{{\mathrm{colim}}}_{n \geq 0} \mathrm{I}^{{\frac{\infty}{2}}}_{\mathrm{u},(n)} \end{align} $$

where

$$\begin{align*}\mathrm{I}^{{\frac{\infty}{2}}}_{(n)} := \mathcal{L}^+T \ltimes \mathcal{L}_{(n)} U, \quad \mathrm{I}^{{\frac{\infty}{2}}}_{\mathrm{u},(n)} = \mathcal{L}^{++} T \ltimes \mathcal{L}_{(n)} U \end{align*}$$

are group schemes.

We will consider the affine flag variety

$$\begin{align*}\mathrm{Fl} = \mathcal{L} G/\mathrm{I}, \end{align*}$$

the ‘extended affine’ flag variety

$$\begin{align*}\widetilde{\mathrm{Fl}} = \mathcal{L} G/\mathrm{I}_{\mathrm{u}}, \end{align*}$$

and the affine Grassmannian

$$\begin{align*}\mathrm{Gr} = \mathcal{L} G/\mathcal{L}^+G. \end{align*}$$

In the three cases the quotient can be taken either for the étale topology or for the fppf topology (giving rise to the same object), and we obtain separated ind-schemes of ind-finite type over $\mathbb {F}$ , endowed with actions of the group scheme $\mathcal {L}^+ G$ . Moreover, there exist $\mathcal {L}^+ G$ -equivariant morphisms

$$ \begin{align*} \widetilde{\mathrm{Fl}} \xrightarrow{q} \mathrm{Fl} \xrightarrow{p} \mathrm{Gr}. \end{align*} $$

2.4 Sheaves on affine flag varieties

Following the constructions in §§2.1–2.2 we can consider the stable presentable $\infty $ -categories

$$\begin{align*}\mathsf{Shv}(\widetilde{\mathrm{Fl}}), \quad \mathsf{Shv}(\mathrm{Fl}), \quad \mathsf{Shv}(\mathrm{Gr}), \quad \mathsf{Shv}(T \backslash \widetilde{\mathrm{Fl}}), \quad \mathsf{Shv}(T \backslash \mathrm{Fl}), \quad \mathsf{Shv}(T \backslash \mathrm{Gr}), \end{align*}$$

and we have continuous functors

$$ \begin{align*}\begin{gathered} p^!, p^* : \mathsf{Shv}(\mathrm{Gr}) \to \mathsf{Shv}(\mathrm{Fl}), \quad p_!, p_* : \mathsf{Shv}(\mathrm{Fl}) \to \mathsf{Shv}(\mathrm{Gr}) \\ q^!, q^* : \mathsf{Shv}(\mathrm{Fl}) \to \mathsf{Shv}(\widetilde{\mathrm{Fl}}), \quad q_!, q_* : \mathsf{Shv}(\widetilde{\mathrm{Fl}}) \to \mathsf{Shv}(\mathrm{Fl}) \end{gathered} \end{align*} $$

and similarly for the equivariant versions. These functors satisfy

$$\begin{align*}q^* \cong q^![-2\dim(T)], \quad p^* \cong p^![-2\ell(w_\circ)], \quad p_* \cong p_!, \end{align*}$$

and we have adjoint pairs $(q^*, q_*)$ , $(q_!, q^!)$ , $(p^*, p_*)$ , $(p_!, p^!)$ . We will also set

$$\begin{align*}p^\dagger = p^![-\ell(w_\circ)] \cong p^*[\ell(w_\circ)]. \end{align*}$$

2.5 Iwahori-equivariant sheaves on affine flag varieties

Let X be either $\mathrm {Gr}$ , $\mathrm {Fl}$ , or $\widetilde {\mathrm {Fl}}$ , and let J be either $\mathrm {I}$ or $\mathrm {I}_{\mathrm {u}}$ . Then we have the corresponding ‘Hecke stack’ $J \backslash X$ , i.e. the étale stackification of the naive quotient stack $J \backslash X$ . The associated $\infty $ -category of $\Bbbk $ -sheaves $\mathsf {Shv}(J \backslash X)$ admits the following concrete description in terms of $\infty $ -categories of sheaves on algebraic stacks of finite type.

There exists a presentation

(2.9) $$ \begin{align} X = \operatorname*{{\mathrm{colim}}}_{n \geq 0} X_n \end{align} $$

of X as an ind-scheme where each $X_n$ is separated of finite type over $\mathbb {F}$ and stable under the action of J, and an identification

(2.10) $$ \begin{align} J = \lim_{m \geq 0} J_{m} \end{align} $$

where each $J_{m}$ is an affine group scheme of finite type over $\mathbb {F}$ and the transition morphisms $J_{m} \to J_{m-1}$ are quotient morphisms with smooth connected unipotent kernels, such that for any $n \geq 0$ there exists $M_n \in \mathbb {Z}_{\geq 0}$ such that the action of J on $X_n$ factors through an action of $J_{M_n}$ . Then for $n \geq 0$ the $\infty $ -category

$$\begin{align*}\mathsf{Shv}(J_{m} \backslash X_n) \end{align*}$$

does not depend on $m \geq M_n$ up to canonical equivalence; in fact the $!$ -pullback functor $\mathsf {Shv}(J_{m} \backslash X_n) \to \mathsf {Shv}(J \backslash X_n)$ is an equivalence, see [Reference Richarz and Scholbach48, Proposition 2.2.11]. We have natural fully faithful functors

$$\begin{align*}\mathsf{Shv}(\mathrm{I}_{\mathrm{u}} \backslash X_n) \to \mathsf{Shv}(X_n), \quad \mathsf{Shv}(\mathrm{I} \backslash X_n) \to \mathsf{Shv}(T \backslash X_n) \end{align*}$$

(again, defined via $!$ -pullback).

For any $n \geq 1$ we have a canonical functor $\mathsf {Shv}(J \backslash X_n) \to \mathsf {Shv}(J \backslash X_{n-1})$ corresponding under the identifications above to the $!$ -pullback functor $\mathsf {Shv}(J_m \backslash X_n) \to \mathsf {Shv}(J_m \backslash X_{n-1})$ for $m \gg 0$ , and since $J \backslash X = \operatorname *{{\mathrm {colim}}}_n J \backslash X_n$ and $\mathsf {Shv}$ transforms colimits into limits we have

$$\begin{align*}\mathsf{Shv}(J \backslash X) = \lim_{n \geq 0} \mathsf{Shv}(J \backslash X_n). \end{align*}$$

Here again we have fully faithful functors

$$\begin{align*}\mathsf{Shv}(\mathrm{I}_{\mathrm{u}} \backslash X) \to \mathsf{Shv}(X), \quad \mathsf{Shv}(\mathrm{I} \backslash X) \to \mathsf{Shv}(T \backslash X). \end{align*}$$

As in (2.2) and (2.5), we also have

(2.11) $$ \begin{align} \mathsf{Shv}(J \backslash X) = \operatorname*{{\mathrm{colim}}}_{n \geq 0} \mathsf{Shv}(J \backslash X_n) \end{align} $$

where the colimit is taken with respect to pushforward functors.

We have canonical exact and continuous ‘forgetful’ functors

(2.12) $$ \begin{align} \mathsf{Shv}(\mathrm{I} \backslash X) \xrightarrow{\mathrm{oblv}^{\mathrm{I} | \mathrm{I}_{\mathrm{u}}}} \mathsf{Shv}(\mathrm{I}_{\mathrm{u}} \backslash X) \xrightarrow{\mathrm{oblv}^{\mathrm{I}_{\mathrm{u}}}} \mathsf{Shv}(X), \end{align} $$

the second of which is fully faithful. (Explicitly, these functors are $!$ -pullback functors associated with the natural maps $\mathrm {I}_{\mathrm {u}} \backslash X \to \mathrm {I} \backslash X$ and $X \to \mathrm {I}_{\mathrm {u}} \backslash X$ respectively.) The functors $p^!$ , $p^*$ , $p^\dagger $ , $q^*$ , $q^!$ of §2.4 restrict to functors

$$\begin{align*}p^!, p^*, p^\dagger : \mathsf{Shv}(J \backslash \mathrm{Gr}) \to \mathsf{Shv}(J \backslash \mathrm{Fl}), \quad q^*, q^! : \mathsf{Shv}(J \backslash \mathrm{Fl}) \to \mathsf{Shv}(J \backslash \widetilde{\mathrm{Fl}}) \end{align*}$$

which are compatible in the obvious way (via the functors $\mathrm {oblv}^{\mathrm {I} | \mathrm {I}_{\mathrm {u}}}$ and $\mathrm {oblv}^{\mathrm {I}_{\mathrm {u}}}$ ) with their non-equivariant counterparts. We also have pushforward functors

$$\begin{align*}p_*, p_! : \mathsf{Shv}(J \backslash \mathrm{Fl}) \to \mathsf{Shv}(J \backslash \mathrm{Gr}), \quad q_*, q_! : \mathsf{Shv}(J \backslash \widetilde{\mathrm{Fl}}) \to \mathsf{Shv}(J \backslash \mathrm{Fl}), \end{align*}$$

which are compatible in the obvious way with their non-equivariant counterparts, and the appropriate adjunctions.

It is a standard fact that the $\infty $ -category $\mathsf {Shv}(\mathrm {I} \backslash \mathrm {Fl})$ admits a canonical structure of algebra object in the monoidal $\infty $ -category of $\Bbbk $ -linear stable presentable $\infty $ -categories. For discussions of this construction in various sheaf-theoretic contexts, see, e.g. [Reference Anschütz, Lourenço, Wu and Yu7, §4.2], [Reference Cass, van den Hove and Scholbach19, §2.2.8], [Reference Dhillon, Li, Yun and Zhu21, §4.1.1]. The associated bifunctor on $\mathsf {Shv}(\mathrm {I} \backslash \mathrm {Fl})$ is the usual convolution product $\star ^{\mathrm {I}}$ , and the unit object is the skyscraper sheaf $\delta _{\mathrm {Fl}}$ at the base point of $\mathrm {Fl}$ . The $\infty $ -category $\mathsf {Shv}(\mathrm {I} \backslash \mathrm {Gr})$ is naturally a left module over $\mathsf {Shv}(\mathrm {I} \backslash \mathrm {Fl})$ , and the $\infty $ -categories $\mathsf {Shv}(\mathrm {Fl})$ and $\mathsf {Shv}(T \backslash \mathrm {Fl})$ are naturally right modules over $\mathsf {Shv}(\mathrm {I} \backslash \mathrm {Fl})$ , with the full subcategories $\mathsf {Shv}(\mathrm {I}_{\mathrm {u}} \backslash \mathrm {Fl}) \subset \mathsf {Shv}(\mathrm {Fl})$ , $\mathsf {Shv}(\mathrm {I} \backslash \mathrm {Fl}) \subset \mathsf {Shv}(T \backslash \mathrm {Fl})$ being stable under this action.

In case $X=\mathrm {Gr}$ or $\mathrm {Fl}$ , we have the usual perverse t-structure (for the middle perversity) on the $\infty $ -category $\mathsf {Shv}(X)$ , and there exist unique t-structures on the $\infty $ -categories $\mathsf {Shv}(\mathrm {I} \backslash X)$ and $\mathsf {Shv}(\mathrm {I}_{\mathrm {u}} \backslash X)$ such that the functors in (2.12) are t-exact. (For an extensive study of these constructions, see, e.g. [Reference Bouthier, Kazhdan and Varshavsky17, §6].) All these t-structures will be called ‘perverse’. Their hearts are the Ind-completions of the usual categories of perverse sheaves; they will be denoted $\mathsf {Shv}(\mathrm {I} \backslash X)^\heartsuit $ , $\mathsf {Shv}(\mathrm {I}_{\mathrm {u}} \backslash X)^\heartsuit $ and $\mathsf {Shv}(X)^\heartsuit $ respectively. It is a standard fact also that the forgetful functor

$$\begin{align*}\mathrm{oblv}^{\mathrm{I} | \mathrm{I}_{\mathrm{u}}} : \mathsf{Shv}(\mathrm{I} \backslash X)^\heartsuit \to \mathsf{Shv}(\mathrm{I}_{\mathrm{u}} \backslash X)^\heartsuit \end{align*}$$

is fully faithful, and that its essential image is stable under subquotients.

For $X=\widetilde {\mathrm {Fl}}$ , we will only consider the category $\mathsf {Shv}(\mathrm {I} \backslash \widetilde {\mathrm {Fl}})$ . In this case, we will shift the usual perverse t-structure, in such a way that the functor

$$\begin{align*}q^! : \mathsf{Shv}(\mathrm{I} \backslash \mathrm{Fl}) \to \mathsf{Shv}(\mathrm{I} \backslash \widetilde{\mathrm{Fl}}) \end{align*}$$

becomes t-exact. Again this t-structure will be called the perverse t-structure, and its heart will be denoted $\mathsf {Shv}(\mathrm {I} \backslash \widetilde {\mathrm {Fl}})^\heartsuit $ . (The main point of this normalization is that, applying similar conventions for the $\infty $ -categories considered in Remark 2.5(2) below, the equivalences in (2.13) are t-exact.)

In some statements below we also consider finitely generated sheaves in the sense discussed in §2.2. With the notation used above we have

$$\begin{align*}\mathsf{Shv}_{\mathrm{fg}}(J \backslash X) = \operatorname*{{\mathrm{colim}}}_{n \geq 0} \mathsf{Shv}_{\mathrm{c}}(J\backslash X_n). \end{align*}$$

(In fact this follows from [Reference Zhu53, Lemma 10.161], since $J \backslash X$ is an ind-very placid stack, see [Reference Zhu53, Example 10.117].) We also have

$$ \begin{align*}\begin{gathered} \mathsf{Shv}(\mathrm{I}_{\mathrm{u}} \backslash \mathrm{Fl}) = \mathrm{Ind} \big( \mathsf{Shv}_{\mathrm{fg}}(\mathrm{I}_{\mathrm{u}} \backslash \mathrm{Fl}) \big), \quad \mathsf{Shv}(\mathrm{I}_{\mathrm{u}} \backslash \mathrm{Gr}) = \mathrm{Ind} \big( \mathsf{Shv}_{\mathrm{fg}}(\mathrm{I}_{\mathrm{u}} \backslash \mathrm{Gr}) \big), \\ \mathsf{Shv}(\mathrm{I} \backslash \widetilde{\mathrm{Fl}}) = \mathrm{Ind} \big( \mathsf{Shv}_{\mathrm{fg}}(\mathrm{I} \backslash \widetilde{\mathrm{Fl}}) \big) \end{gathered}\end{align*} $$

by [Reference Zhu53, Remark 10.162], but the corresponding statement for $\mathrm {I} \backslash \mathrm {Fl}$ does not hold.

2.6 Semiinfinite Iwahori-equivariant sheaves on affine flag varieties

We continue to let X be either $\mathrm {Gr}$ , $\mathrm {Fl}$ , or $\widetilde {\mathrm {Fl}}$ , but let now J be either $\mathrm {I}^{{\frac {\infty }{2}}}$ or $\mathrm {I}_{\mathrm {u}}^{{\frac {\infty }{2}}}$ . Then one can consider the stack quotient $J \backslash X$ . The corresponding $\infty $ -category of sheaves admits the following explicit description, following [Reference Gaitsgory27, §1.2].

Write $J=\operatorname *{{\mathrm {colim}}}_k J_{(k)}$ for the presentation in (2.7). For any k, there exists a presentation (2.9) such that each $X_n$ is stable under the action of $J_{(k)}$ , and an identification

(2.14) $$ \begin{align} J_{(k)} = \lim_{m \geq 0} J_{(k),m} \end{align} $$

where each $J_{(k),m}$ is an affine group scheme of finite type over $\mathbb {F}$ , each transition morphism $J_{(k),m} \to J_{(k),m-1}$ is a quotient morphism with smooth connected unipotent kernel, such that for any $n \geq 0$ there exists $M_n$ such that the action of $J_{(k)}$ on $X_n$ factors through an action of $J_{(k),M_n}$ . Then by the same considerations as in §2.5 the $\infty $ -category $\mathsf {Shv}(J_{(k),m} \backslash X_n)$ does not depend on $m \geq M_n$ up to canonical equivalence, and coincides with $\mathsf {Shv}(J_{(k)} \backslash X_n)$ . For any $n \geq 1$ we have a $!$ -pullback functor $\mathsf {Shv}(J_{(k)} \backslash X_n) \to \mathsf {Shv}(J_{(k)} \backslash X_{n-1})$ corresponding to the $!$ -pullback functor $\mathsf {Shv}(J_{(k),m} \backslash X_n) \to \mathsf {Shv}(J_{(k),m} \backslash X_{n-1})$ for $m \gg 0$ , and we have

$$\begin{align*}\mathsf{Shv}(J_{(k)} \backslash X) = \lim_{n \geq 0} \mathsf{Shv}(J_{(k)} \backslash X_n). \end{align*}$$

In case $J=\mathrm {I}_{\mathrm {u}}^{{\frac {\infty }{2}}}$ we have a fully faithful ‘forgetful’ functor

$$\begin{align*}\mathsf{Shv}(J_{(k)} \backslash X) \to \mathsf{Shv}(X) \end{align*}$$

and in case $J=\mathrm {I}^{{\frac {\infty }{2}}}$ we have a fully faithful ‘forgetful’ functor

$$\begin{align*}\mathsf{Shv}(J_{(k)} \backslash X) \to \mathsf{Shv}(T \backslash X). \end{align*}$$

For any k we have a fully faithful ‘forgetful’ functor

$$\begin{align*}\mathsf{Shv}(J_{(k+1)} \backslash X) \to \mathsf{Shv}(J_{(k)} \backslash X), \end{align*}$$

and we have

$$\begin{align*}\mathsf{Shv}(J \backslash X) = \lim_{k \geq 0} \mathsf{Shv}(J_{(k)} \backslash X). \end{align*}$$

(To justify this, one notes that $J \backslash X = \operatorname *{{\mathrm {colim}}}_k J_{(k)} \backslash X$ since quotient stacks are defined as a colimit over an appropriate Čech nerve and colimits commute with colimits, and then one uses the fact that $\mathsf {Shv}$ transforms colimits into limits.) We also have fully faithful ‘forgetful’ functors

$$\begin{align*}\mathsf{Shv}(\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} \backslash X) \to \mathsf{Shv}(X), \quad \mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}} \backslash X) \to \mathsf{Shv}(T \backslash X). \end{align*}$$

We have canonical exact and continuous ‘forgetful’ functors

$$\begin{align*}\mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}} \backslash X) \xrightarrow{\mathrm{oblv}^{\mathrm{I}^{{\frac{\infty}{2}}} | \mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}}}} \mathsf{Shv}(\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} \backslash X) \xrightarrow{\mathrm{oblv}^{\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}}}} \mathsf{Shv}(X) \end{align*}$$

defined as $!$ -pullback under the maps $X \to \mathrm {I}_{\mathrm {u}}^{{\frac {\infty }{2}}} \backslash X \to \mathrm {I}^{{\frac {\infty }{2}}} \backslash X$ , the second of which is fully faithful. The functor $\mathrm {oblv}^{\mathrm {I}^{{\frac {\infty }{2}}} | \mathrm {I}_{\mathrm {u}}^{{\frac {\infty }{2}}}}$ is the restriction of the forgetful functor

$$\begin{align*}\mathrm{oblv}^T : \mathsf{Shv}(T \backslash X) \to \mathsf{Shv}(X). \end{align*}$$

The latter functor has a left adjoint $\mathrm {Av}^T_!$ and a right adjoint $\mathrm {Av}^T_*$ . These functors restrict to functors

$$\begin{align*}\mathrm{Av}^{\mathrm{I}^{{\frac{\infty}{2}}} | \mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}}}_!, \mathrm{Av}^{\mathrm{I}^{{\frac{\infty}{2}}} | \mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}}}_* : \mathsf{Shv}(\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} \backslash X) \to \mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}} \backslash X) \end{align*}$$

which are left and right adjoint to $\mathrm {oblv}^{\mathrm {I}^{{\frac {\infty }{2}}} | \mathrm {I}_{\mathrm {u}}^{{\frac {\infty }{2}}}}$ respectively.

The functors $p^!$ , $p^*$ , $p^\dagger $ of §2.4 restrict to functors

$$\begin{align*}p^!, p^*, p^\dagger : \mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}} \backslash \mathrm{Gr}) \to \mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}} \backslash \mathrm{Fl}), \quad p^!,p^*,p^\dagger : \mathsf{Shv}(\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} \backslash \mathrm{Gr}) \to \mathsf{Shv}(\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} \backslash \mathrm{Fl}) \end{align*}$$

which are compatible in the obvious way (via the functors $\mathrm {oblv}^{\mathrm {I}^{{\frac {\infty }{2}}} | \mathrm {I}_{\mathrm {u}}^{{\frac {\infty }{2}}}}$ and $\mathrm {oblv}^{\mathrm {I}_{\mathrm {u}}^{{\frac {\infty }{2}}}}$ ) with their non-equivariant counterparts. We also have (isomorphic) pushforward functors

$$\begin{align*}p_*, p_! : \mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}} \backslash \mathrm{Fl}) \to \mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}} \backslash \mathrm{Gr}), \quad p_*, p_! : \mathsf{Shv}(\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} \backslash \mathrm{Fl}) \to \mathsf{Shv}(\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} \backslash \mathrm{Gr}) \end{align*}$$

which are compatible in the obvious way with their non-equivariant counterparts, and the usual adjunctions. Similar comments apply to the morphism q.

It is clear that the full subcategories

$$\begin{align*}\mathsf{Shv}(\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} \backslash \mathrm{Fl}) \subset \mathsf{Shv}(\mathrm{Fl}), \quad \mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}} \backslash \mathrm{Fl}) \subset \mathsf{Shv}(T \backslash \mathrm{Fl}) \end{align*}$$

are stable under the right actions of $\mathsf {Shv}(\mathrm {I} \backslash \mathrm {Fl})$ considered in §2.5, hence are naturally right modules for this algebra object. Similarly, the $\infty $ -categories $\mathsf {Shv}(\mathrm {I}_{\mathrm {u}}^{{\frac {\infty }{2}}} \backslash \mathrm {Gr})$ and $\mathsf {Shv}(\mathrm {I}^{{\frac {\infty }{2}}} \backslash \mathrm {Gr})$ are canonically right modules over the algebra object $\mathsf {Shv}(\mathcal {L}^+ G \backslash \mathrm {Gr})$ of Remark 2.5(1).

If X is as above, for any $\lambda \in \mathbf {Y}$ we have an automorphism of X given by the left action of $z^\lambda \in (\mathcal {L} G)(\mathbb {F})$ , see §2.3. The composition of pushforward under this map with cohomological shift $[-\langle \lambda , 2\rho \rangle ]$ provides an autoequivalence of $\mathsf {Shv}(X)$ , which will be denoted

(2.15) $$ \begin{align} \mathcal{F} \mapsto \mathcal{F} \langle \lambda \rangle, \end{align} $$

and which restricts to an autoequivalence of $\mathsf {Shv}(\mathrm {I}_{\mathrm {u}}^{{\frac {\infty }{2}}} \backslash X)$ . Since $z^\lambda $ commutes with T, we similarly have an autoequivalence of $\mathsf {Shv}(T \backslash X)$ , which restricts to an autoequivalence of $\mathsf {Shv}(\mathrm {I}^{{\frac {\infty }{2}}} \backslash X)$ , and will be denoted by the same symbol. (In case $X=\mathrm {Gr}$ , this construction appears in [Reference Gaitsgory27, §5.1.2].)

2.7 Raskin’s equivalences

Now we recall a construction due to Raskin [Reference Raskin45, §6], as reformulated by Gaitsgory in [Reference Gaitsgory27, §5.2], which will be a fundamental tool for our constructions. (Gaitsgory only considers in detail a special case, corresponding to $X=\mathrm {Gr}$ . He notes however that this construction applies, and has similar properties, for any $\infty $ -category acted on by $\mathcal {L} G$ in an appropriate sense.) For more on the context of this construction, see also [Reference Raskin46].

If X is either $\mathrm {Gr}$ , $\mathrm {Fl}$ , or $\widetilde {\mathrm {Fl}}$ , then one can consider the $\Bbbk $ -linear stable presentable $\infty $ -category $\mathsf {Shv}(\mathcal {L}^+ T \backslash X)$ , which can be described along lines similar to those of §2.5, and we have exact and fully-faithful ‘forgetful’ functors

$$\begin{align*}\mathsf{Shv}(\mathrm{I} \backslash X) \xrightarrow{\mathrm{oblv}^{\mathrm{I} | \mathcal{L}^+ T}} \mathsf{Shv}(\mathcal{L}^+ T \backslash X) \xleftarrow{\mathrm{oblv}^{\mathrm{I}^{{\frac{\infty}{2}}} | \mathcal{L}^+ T}} \mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}} \backslash X). \end{align*}$$

The functor $\mathrm {oblv}^{\mathrm {I} | \mathcal {L}^+ T}$ admits a continuous right adjoint

$$\begin{align*}\mathrm{Av}^{\mathrm{I} | \mathcal{L}^+T}_* : \mathsf{Shv}(\mathcal{L}^+ T \backslash X) \to \mathsf{Shv}(\mathrm{I} \backslash X). \end{align*}$$

(In fact, to justify this claim, one can replace X by an $\mathrm {I}$ -stable subscheme on which the action factors through the quotient by a pro-unipotent normal subgroup of finite codimension. Then any object is equivariant for the subgroup generated by $\mathcal {L}^+T$ and this normal subgroup, and $\mathrm {Av}^{\mathrm {I} | \mathcal {L}^+T}_*$ is realized by the usual $*$ -averaging functor from the latter subgroup to $\mathrm {I}$ .) As explained in [Reference Gaitsgory27, Proposition 5.2.2], the composition

$$\begin{align*}\mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}} \backslash X) \xrightarrow{\mathrm{oblv}^{\mathrm{I}^{{\frac{\infty}{2}}} | \mathcal{L}^+ T}} \mathsf{Shv}(\mathcal{L}^+ T \backslash X) \xrightarrow{\mathrm{Av}^{\mathrm{I} | \mathcal{L}^+T}_*} \mathsf{Shv}(\mathrm{I} \backslash X) \end{align*}$$

is an equivalence of categories, which we will denote by

$$\begin{align*}\mathrm{Ras} : \mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}} \backslash X) \xrightarrow{\sim} \mathsf{Shv}(\mathrm{I} \backslash X); \end{align*}$$

see §2.8 below for details.

The inverse equivalence $\mathrm {Ras}^{-1}$ can be described as follows. As explained in §2.8 below, the functor $\mathrm {oblv}^{\mathrm {I}^{{\frac {\infty }{2}}} | \mathcal {L}^+ T}$ admits a (continuous) left adjoint

$$\begin{align*}\mathrm{Av}_!^{\mathrm{I}^{{\frac{\infty}{2}}} | \mathcal{L}^+ T} : \mathsf{Shv}(\mathcal{L}^+ T \backslash X) \to \mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}} \backslash X). \end{align*}$$

Then it is explained in [Reference Gaitsgory27, §5.2.3] that the composition

$$\begin{align*}\mathrm{Av}_!^{\mathrm{I}^{{\frac{\infty}{2}}} | \mathcal{L}^+ T} \circ \mathrm{oblv}^{\mathrm{I} | \mathcal{L}^+ T} : \mathsf{Shv}(\mathrm{I} \backslash X) \to \mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}} \backslash X) \end{align*}$$

is the inverse to $\mathrm {Ras}$ .

It is clear that these equivalences intertwine the functors $p^*$ , $p^!$ , etc. in the obvious way. Similar considerations replacing $\mathcal {L}^+ T$ by $\mathcal {L}^{++} T$ lead, when X is either $\mathrm {Gr}$ or $\mathrm {Fl}$ , to equivalences of categories

$$\begin{align*}\mathrm{Ras}_{\mathrm{u}} : \mathsf{Shv}(\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} \backslash X) \xrightarrow{\sim} \mathsf{Shv}(\mathrm{I}_{\mathrm{u}} \backslash X). \end{align*}$$

satisfying the same properties, and such that

$$\begin{align*}\mathrm{Ras}_{\mathrm{u}} \circ \mathrm{oblv}^{\mathrm{I}^{{\frac{\infty}{2}}} | \mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}}} \cong \mathrm{oblv}^{\mathrm{I} | \mathrm{I}_{\mathrm{u}}} \circ \mathrm{Ras}. \end{align*}$$

It is clear that in case $X=\mathrm {Fl}$ the equivalences $\mathrm {Ras}$ and $\mathrm {Ras}_{\mathrm {u}}$ are equivalences of right modules over the algebra object $\mathsf {Shv}(\mathrm {I} \backslash \mathrm {Fl})$ , and that in case $X=\mathrm {Gr}$ they are equivalences of right modules over the algebra object $\mathsf {Shv}(\mathcal {L}^+ G \backslash \mathrm {Gr})$ .

2.8 Some details on the proof of Raskin’s equivalences

In this subsection we briefly outline the proof (following [Reference Gaitsgory27, §5.2], see also [Reference Dhillon and Lysenko22, §3.1.9]) of the fact that the functor $\mathrm {Ras}$ of §2.7 is an equivalence, mainly to convince the reader that this statement applies to any kind of coefficients. (No detail of this proof will be required below.) These considerations will use some of the constructions explained in Sections 3–4 below; in fact, the essential ingredient of the proof is the fact that the standard and costandard objects in $\mathsf {Shv}(\mathrm {I} \backslash \mathrm {Fl})$ (whose definition is recalled in §4.1 below) are invertible for the convolution product. The proof that $\mathrm {Ras}_{\mathrm {u}}$ is an equivalence is similar; in this case one needs to work with the ‘free monodromic’ versions of the standard and costandard objects, which will not be explicitly discussed here (see, e.g. [Reference Bezrukavnikov and Riche15, Reference Chen and Dhillon20]).

As in Remark 2.3, we choose a strongly dominant $\lambda \in \mathbf {Y}$ , and take $\mathcal {L}_{(n)} U = z^{-n\lambda } \mathcal {L}^+ U z^{n\lambda }$ in the notation of §2.3. For any $n \geq 0$ we have a forgetful functor

$$\begin{align*}\mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}}_{(n)} \backslash X) \to \mathsf{Shv}(\mathcal{L}^+ T \backslash X), \end{align*}$$

which admits a continuous left adjoint

$$\begin{align*}\mathrm{Av}_!^{\mathrm{I}^{{\frac{\infty}{2}}}_{(n)} | \mathcal{L}^+ T} : \mathsf{Shv}(\mathcal{L}^+ T \backslash X) \to \mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}}_{(n)} \backslash X). \end{align*}$$

(The justification is similar to that for the functor $\mathrm {Av}^{\mathrm {I} | \mathcal {L}^+T}_*$ in §2.7.) These functors form an inductive system parametrized by $\mathbb {Z}_{\geq 0}$ , and the left adjoint $\mathrm {Av}_!^{\mathrm {I}^{{\frac {\infty }{2}}} | \mathcal {L}^+ T}$ to $\mathrm {oblv}^{\mathrm {I}^{{\frac {\infty }{2}}} | \mathcal {L}^+ T}$ is given by

$$\begin{align*}\mathrm{Av}_!^{\mathrm{I}^{{\frac{\infty}{2}}} | \mathcal{L}^+ T} ( \mathcal{F}) = \operatorname*{{\mathrm{colim}}}_{n \geq 0} \mathrm{Av}_!^{\mathrm{I}^{{\frac{\infty}{2}}}_{(n)} | \mathcal{L}^+ T} ( \mathcal{F}) \end{align*}$$

for $\mathcal {F}$ in $\mathsf {Shv}(\mathcal {L}^+ T \backslash X)$ .

These functors have more explicit descriptions when applied to objects in the image of $\mathrm {oblv}^{\mathrm {I} | \mathcal {L}^+ T}$ : in fact, using Lemma 3.2 below one sees that for any $\mathcal {F}$ in $\mathsf {Shv}(\mathrm {I} \backslash X)$ we have

$$\begin{align*}\mathrm{Av}_!^{\mathrm{I}^{{\frac{\infty}{2}}}_{(n)} | \mathcal{L}^+ T} ( \mathrm{oblv}^{\mathrm{I} | \mathcal{L}^+ T}(\mathcal{F})) \cong z^{-n \lambda} \cdot \Delta_{t_{n\lambda}} \star^{\mathrm{I}} \mathcal{F} [n N] \end{align*}$$

where N is the pairing of $\lambda $ with the sum of the positive roots, the notation $t_{n\lambda }$ is explained in §3.1, the corresponding standard object $\Delta _{t_{n\lambda }}$ is defined in §4.1, and we denote by $z^{-n \lambda } \cdot (-)$ the pushforward functor under the automorphism of X given by the left action of $z^{-n \lambda } \in \mathcal {L} G(\mathbb {F})$ . Hence, for any such $\mathcal {F}$ we have

(2.16) $$ \begin{align} \mathrm{Av}_!^{\mathrm{I}^{{\frac{\infty}{2}}} | \mathcal{L}^+ T} ( \mathrm{oblv}^{\mathrm{I} | \mathcal{L}^+ T}(\mathcal{F})) \cong \operatorname*{{\mathrm{colim}}}_{n \geq 0} z^{-n \lambda} \cdot \Delta_{t_{n\lambda}} \star^{\mathrm{I}} \mathcal{F} [nN]. \end{align} $$

Next, one considers the adjunction morphism

(2.17) $$ \begin{align} \mathcal{F} \to \mathrm{Av}^{\mathrm{I} | \mathcal{L}^+T}_* \circ \mathrm{Av}_!^{\mathrm{I}^{{\frac{\infty}{2}}} | \mathcal{L}^+ T}(\mathcal{F}) \end{align} $$

for $\mathcal {F}$ in $\mathsf {Shv}(\mathrm {I} \backslash X)$ , where forgetful functors are omitted. Then using (2.16) one obtains that

$$ \begin{align*} \mathrm{Av}^{\mathrm{I} | \mathcal{L}^+T}_* \circ \mathrm{Av}_!^{\mathrm{I}^{{\frac{\infty}{2}}} | \mathcal{L}^+ T}(\mathcal{F}) &\cong \operatorname*{{\mathrm{colim}}}_{n \geq 0} \mathrm{Av}^{\mathrm{I} | \mathcal{L}^+T}_*(z^{-n \lambda} \cdot \Delta_{t_{n\lambda}} \star^{\mathrm{I}} \mathcal{F})[nN] \\ &\cong \operatorname*{{\mathrm{colim}}}_{n \geq 0} \nabla_{t_{-n \lambda}} \star^{\mathrm{I}} \Delta_{t_{n\lambda}} \star^{\mathrm{I}} \mathcal{F} \end{align*} $$

where the object $\nabla _{t_{-n \lambda }}$ is defined in §4.1. By Lemma 4.1(2) we have a canonical isomorphism $\nabla _{t_{-n \lambda }} \star ^{\mathrm {I}} \Delta _{t_{n\lambda }} \cong \delta _{\mathrm {Fl}}$ ; it follows that we have an identification

$$\begin{align*}\mathrm{Av}^{\mathrm{I} | \mathcal{L}^+T}_* \circ \mathrm{Av}_!^{\mathrm{I}^{{\frac{\infty}{2}}} | \mathcal{L}^+ T}(\mathcal{F}) \cong \mathcal{F}, \end{align*}$$

under which the adjunction morphism (2.17) becomes the identity.

At this point, to conclude it suffices to show that the functor $\mathrm {Ras}$ is conservative, or equivalently that it does not kill any object. Let $U^-$ be the unipotent radical of the Borel subgroup of G opposite to B with respect to T; then the multiplication morphism

$$\begin{align*}\mathcal{L}^{++} (U^-) \times \mathcal{L}^+ T \times \mathcal{L}^+ U \to \mathrm{I} \end{align*}$$

is an isomorphism, hence if $\mathcal {G}$ is in $\mathsf {Shv}(\mathrm {I}^{{\frac {\infty }{2}}} \backslash X)$ we have

(2.18) $$ \begin{align} \mathrm{Ras}(\mathcal{G}) = \mathrm{Av}^{\mathrm{I} | \mathcal{L}^+T}_*(\mathcal{G}) \cong \mathrm{Av}^{\mathcal{L}^{++}(U^-)}_*(\mathcal{G}). \end{align} $$

The basic observation here is that there exists a nonzero morphism $\mathcal {G}' \to \mathcal {G}$ where $\mathcal {G}' \in \mathsf {Shv}(X)$ is an object which is equivariant with respect to the action of the group $z^{-n\lambda } \mathcal {L}^{++} (U^-) z^{n\lambda }$ for some $n \gg 0$ , so that $\mathrm {Av}_*^{z^{-n\lambda } \mathcal {L}^{++} (U^-) z^{n\lambda }}(\mathcal {G}) \neq 0$ . The latter object naturally belongs to

$$\begin{align*}\mathsf{Shv}((z^{-n\lambda} \mathrm{I} z^{n\lambda}) \backslash X), \end{align*}$$

hence $z^{n\lambda } \cdot \mathrm {Av}_*^{z^{-n\lambda } \mathcal {L}^{++} (U^-) z^{n\lambda }}(\mathcal {G})$ is a nonzero object in $\mathsf {Shv}(\mathrm {I} \backslash X)$ , and one checks as above that

$$ \begin{align*} \mathrm{Av}^{\mathcal{L}^{++}(U^-)}_*(\mathcal{G})& = \mathrm{Av}^{\mathcal{L}^{++}(U^-)}_* \circ \mathrm{Av}_*^{z^{-n\lambda} \mathcal{L}^{++} (U^-) z^{n\lambda}}(\mathcal{G}) \\ &\cong \nabla_{t_{-n \lambda}} \star^{\mathrm{I}} (z^{n\lambda} \cdot \mathrm{Av}_*^{z^{-n\lambda} \mathcal{L}^{++} (U^-) z^{n\lambda}}(\mathcal{G})). \end{align*} $$

Using once again the invertibility of $\nabla _{t_{-n \lambda }}$ , this object is nonzero. This concludes the proof, in view of (2.18).

3.1 Weyl groups

Recall the notation of §2.3. Let $\mathbf {Y}_+ \subset \mathbf {Y}$ be the subset of dominant coweights determined by $\mathfrak {R}_+$ , and let $\mathbf {Y}_{++}$ be the subset of strongly dominant coweights, i.e. coweights $\lambda $ such that $\langle \lambda ,\alpha \rangle> 0$ for all $\alpha \in \mathfrak {R}^+$ . Let also $\mathfrak {R}^\vee \subset \mathbf {Y}$ be the system of coroots, let $\mathfrak {R}^\vee _+ \subset \mathfrak {R}^\vee $ be the subset of positive coroots, and let $\mathfrak {R}_{\mathrm {s}}^\vee \subset \mathfrak {R}^\vee _+$ be the associated basis. We will denote by $\preceq $ the order on $\mathbf {Y}$ such that $\lambda \preceq \mu $ iff $\mu - \lambda $ is a sum of positive coroots; then $\mathbf {Y}_{\succeq 0} = \{\mu \in \mathbf {Y} \mid \mu \succeq 0\}$ is the free commutative monoid generated by $\mathfrak {R}_{\mathrm {s}}^\vee $ .

Let

$$\begin{align*}W_{\mathrm{aff}} = W_{\mathrm{f}} \ltimes \mathbb{Z}\mathfrak{R}^\vee \quad \text{and} \quad W_{\mathrm{ext}} = W_{\mathrm{f}} \ltimes \mathbf{Y} \end{align*}$$

denote the affine and extended affine Weyl groups, respectively. We will need to distinguish the elements of $\mathbf {Y}$ from their images in $W_{\mathrm {ext}}$ ; for that, for $\lambda \in \mathbf {Y}$ we will write $t_\lambda $ for the corresponding element of $W_{\mathrm {ext}}$ . It is a standard fact that there exists a unique subset $S_{\mathrm {aff}} \subset W_{\mathrm {aff}}$ such that $(W_{\mathrm {aff}}, S_{\mathrm {aff}})$ is a Coxeter system with associated length function given by

$$\begin{align*}\ell(t_\lambda w) = \sum_{\substack{\alpha \in \mathfrak{R}_+ \\ w^{-1}(\alpha) \in \mathfrak{R}_+}} |\langle \lambda, \alpha \rangle| + \sum_{\substack{\alpha \in \mathfrak{R}_+ \\ w^{-1}(\alpha) \in -\mathfrak{R}_+}} |1 + \langle \lambda,\alpha\rangle| \qquad\text{for }w \in W_{\mathrm{f}}, \lambda \in \mathbb{Z}\mathfrak{R}^\vee. \end{align*}$$

This formula makes sense more generally for $\lambda \in \mathbf {Y}$ , which allows us to extend $\ell $ to $W_{\mathrm {ext}}$ . (This extension still satisfies $\ell (w)=\ell (w^{-1})$ for any $w \in W_{\mathrm {ext}}$ .) Note that for $\lambda \in \mathbf {Y}$ and $w \in W_{\mathrm {f}}$ we have

(3.1) $$ \begin{align} \ell(t_\lambda w) = \begin{cases} \ell(t_\lambda) + \ell(w) & \text{if }\lambda \in \mathbf{Y}_+; \\ \ell(t_\lambda) - \ell(w) & \text{if }\lambda \in -\mathbf{Y}_{++}. \end{cases} \end{align} $$

The subgroup $W_{\mathrm {aff}} \subset W_{\mathrm {ext}}$ is normal and, setting $\Omega = \{w \in W_{\mathrm {ext}} \mid \ell (w)=0\}$ , multiplication induces an isomorphism

$$\begin{align*}\Omega \ltimes W_{\mathrm{aff}} \xrightarrow{\sim} W_{\mathrm{ext}}, \end{align*}$$

where $\Omega $ acts on $W_{\mathrm {aff}}$ by Coxeter group automorphisms. We extend the Bruhat order on $W_{\mathrm {aff}}$ to $W_{\mathrm {ext}}$ by declaring that, for $\omega ,\omega ' \in \Omega $ and $w,w' \in W_{\mathrm {aff}}$ ,

$$\begin{align*}\omega w \leq \omega' w' \quad \Leftrightarrow \quad \omega=\omega' \text{ and } w \leq w'. \end{align*}$$

For $\lambda \in \mathbf {Y}$ we will denote by $w_\lambda $ the unique element of minimal length in $t_\lambda W_{\mathrm {f}}$ . If $v_\lambda $ is the unique element of $W_{\mathrm {f}}$ of minimal length such that $v_\lambda (\lambda ) \in \mathbf {Y}_+$ , then we have

$$\begin{align*}w_\lambda = t_\lambda v_\lambda^{-1} \qquad\text{and}\qquad \ell(w_\lambda) = \ell(t_\lambda) - \ell(v_\lambda), \end{align*}$$

see, e.g. [Reference Mautner and Riche42, Lemma 2.4]. In particular, we have

(3.2) $$ \begin{align} w_\lambda = \begin{cases} t_\lambda & \text{if }\lambda \in \mathbf{Y}_+, \\ t_\lambda w_\circ & \text{if }\lambda \in - \mathbf{Y}_{++} \end{cases} \end{align} $$

(which of course is consistent with (3.1).)

Set $W_{\mathrm {ext}}^{+} := \{w \in W_{\mathrm {ext}} \mid \forall \mu \in \mathbf {Y}_+, \, \ell (t_\mu w) = \ell (t_\mu ) + \ell (w)\}$ . (Note that, in view of (3.1), all the elements of the form $t_\lambda w_{\mathrm {f}}$ with $\lambda \in \mathbf {Y}_+$ and $w_{\mathrm {f}} \in W_{\mathrm {f}}$ belong to $W_{\mathrm {ext}}^+$ .) If $w,y \in W_{\mathrm {ext}}$ , then as in [Reference Achar and Riche4, Lemma 2.4] the following conditions are equivalent:

1. (1) there exists $\mu \in \mathbf {Y}$ such that $t_\mu w$ and $t_\mu y$ belong to $W_{\mathrm {ext}}^+$ and $t_\mu w \leq t_\mu y$ ;

2. (2) for any $\mu \in \mathbf {Y}$ such that $t_\mu w$ and $t_\mu y$ belong to $W_{\mathrm {ext}}^+$ , we have $t_\mu w \leq t_\mu y$ .

Since our present setting and claim are slightly different from those in [Reference Achar and Riche4], we briefly recall the argument. It is clear that the second condition implies the first one, since for any $w,y \in W_{\mathrm {ext}}$ there exists $\mu \in \mathbf {Y}$ such that $t_\mu w$ and $t_\mu y$ belong to $W_{\mathrm {ext}}^+$ . Conversely, assume that the first condition is satisfied, and fix $\mu \in \mathbf {Y}$ which satisfies this condition. If $\mu ' \in \mathbf {Y}$ is any element such that $t_\mu w$ and $t_\mu y$ belong to $W_{\mathrm {ext}}^+$ , and if $\nu \in \mathbf {Y}_+$ is such that $(\eta + \mu )-\mu ' \in \mathbf {Y}_+$ , then using [Reference Achar and Riche3, Lemma 2.1] and the definition of $W_{\mathrm {ext}}^+$ we see that $t_{\eta +\mu } w \leq t_{\eta +\mu } y$ , and then that $t_{\mu '} w \leq t_{\mu '} y$ .

We will write

$$\begin{align*}w \leq^{\frac{\infty}{2}} y \end{align*}$$

if the conditions (1)–(2) above are satisfied. It is clear that this defines an order on $W_{\mathrm {ext}}$ , which is called the semiinfinite order.

3.2 Iwahori orbits on $\mathrm {Fl}$ and $\widetilde {\mathrm {Fl}}$

The orbits of $\mathrm {I}$ on $\mathrm {Fl}$ are naturally labelled by $W_{\mathrm {ext}}$ , in the following way. If $w \in W_{\mathrm {ext}}$ and $w=t_\mu w_{\mathrm {f}}$ for some $w_{\mathrm {f}} \in W_{\mathrm {f}}$ , then we can consider the $\mathbb {F}$ -point $w_{\mathrm {f}} \mathrm {I} \in \mathrm {Fl}$ , its image $z^\mu w_{\mathrm {f}} \mathrm {I}$ under multiplication by $z^\mu $ , and the $\mathrm {I}$ -orbit $\mathrm {Fl}_w$ of this point. Then we have

$$\begin{align*}\mathrm{Fl} = \bigsqcup_{w \in W_{\mathrm{ext}}} \mathrm{Fl}_w. \end{align*}$$

Moreover, it is a standard fact that each $\mathrm {Fl}_w$ is also an $\mathrm {I}_{\mathrm {u}}$ -orbit, isomorphic to an affine space of dimension $\ell (w)$ , and that for any $w \in W_{\mathrm {ext}}$ we have

$$\begin{align*}\overline{\mathrm{Fl}_w} = \bigsqcup_{y \leq w} \mathrm{Fl}_y. \end{align*}$$

We will denote by

$$\begin{align*}j_w : \mathrm{Fl}_w \to \overline{\mathrm{Fl}_w}, \quad \overline{i}_w : \overline{\mathrm{Fl}_w} \to \mathrm{Fl} \end{align*}$$

the embeddings, and set $i_w = \overline {i}_w \circ j_w$ .

For $\alpha \in \mathfrak {R}$ and $n \in \mathbb {Z}$ , we will denote by $U_{\alpha + n \hbar }$ the $1$ -parameter subgroup in $\mathcal {L} G$ which identifies with the image of the morphism $x \mapsto u_\alpha (x z^n)$ for any choice of an isomorphism $u_\alpha $ between the additive group $\mathbb {G}_{\mathrm {a}}$ and the root subgroup of G associated with $\alpha $ . We have $U_{\alpha + n\hbar } \subset \mathrm {I}$ iff either $\alpha \in \mathfrak {R}_+$ and $n \geq 0$ , or $\alpha \in -\mathfrak {R}_+$ and $n>0$ . We fix an arbitrary order on $\mathfrak {R}_+$ . The following statement is standard.

Lemma 3.2. Let $\mu \in \mathbf {Y}_{+}$ and $w_{\mathrm {f}} \in W_{\mathrm {f}}$ , and set $w=t_\mu w_{\mathrm {f}}$ . Then multiplication and action on $z^\mu w_{\mathrm {f}} \mathrm {I}$ induce an isomorphism of $\mathbb {F}$ -schemes

$$\begin{align*}\left( \prod_{\substack{\alpha \in \mathfrak{R}_+ \\ w_{{\mathrm{f}}}^{-1}(\alpha) \in \mathfrak{R}_+}} \prod_{n=0}^{\langle \alpha, \mu \rangle-1} U_{\alpha + n\hbar} \right) \times \left( \prod_{\substack{\alpha \in \mathfrak{R}_+ \\ w_{{\mathrm{f}}}^{-1}(\alpha) \in -\mathfrak{R}_+}} \prod_{n=0}^{\langle \alpha, \mu \rangle} U_{\alpha + n\hbar} \right) \xrightarrow{\sim} \mathrm{Fl}_{w}. \end{align*}$$

In addition to the $\infty $ -categories considered in §2.5, we will also consider for $w \in W_{\mathrm {ext}}$ the $\infty $ -categories

$$\begin{align*}\mathsf{Shv}(\mathrm{I} \backslash \mathrm{Fl}_w), \quad \mathsf{Shv}(\mathrm{I} \backslash \overline{\mathrm{Fl}_w}), \quad \mathsf{Shv}(\mathrm{I}_{\mathrm{u}} \backslash \mathrm{Fl}_w), \quad \mathsf{Shv}(\mathrm{I}_{\mathrm{u}} \backslash \overline{\mathrm{Fl}_w}), \end{align*}$$

and the push/pull functors associated with the maps $i_w$ , $j_w$ , $\overline {i}_w$ .

The $\mathrm {I}$ -orbits on $\widetilde {\mathrm {Fl}}$ are also naturally labelled by $W_{\mathrm {ext}}$ ; in fact, setting $\widetilde {\mathrm {Fl}}_w = q^{-1}(\mathrm {Fl}_w)$ for $w \in W_{\mathrm {ext}}$ , each $\widetilde {\mathrm {Fl}}_w$ is an $\mathrm {I}$ -orbit, and we have

$$\begin{align*}\widetilde{\mathrm{Fl}} = \bigsqcup_{w \in W_{\mathrm{ext}}} \widetilde{\mathrm{Fl}}_w. \end{align*}$$

3.3 Orbits on $\mathrm {Gr}$

The $\mathrm {I}$ -orbits on $\mathrm {Gr}$ are naturally labelled by $\mathbf {Y}$ , in the following way. For $\lambda \in \mathbf {Y}$ , we will denote by $\mathrm {Gr}_\lambda $ the orbit of $z^\lambda \mathcal {L}^+ G \in \mathrm {Gr}$ . Each $\mathrm {Gr}_\lambda $ is also an $\mathrm {I}_{\mathrm {u}}$ -orbit, is isomorphic to an affine space of dimension $\ell (w_\lambda )$ , and for any $\lambda \in \mathbf {Y}$ we have

$$\begin{align*}\overline{\mathrm{Gr}_\lambda} = \bigsqcup_{\substack{\mu \in \mathbf{Y} \\ w_\mu \leq w_\lambda}} \mathrm{Gr}_\mu. \end{align*}$$

We will denote by

$$\begin{align*}j_\lambda : \mathrm{Gr}_\lambda \to \overline{\mathrm{Gr}_\lambda}, \quad \overline{i}_\lambda : \overline{\mathrm{Gr}_\lambda} \to \mathrm{Gr} \end{align*}$$

the embeddings, and set $i_\lambda = \overline {i}_\lambda \circ j_\lambda $ .

The following statement is the analogue for $\mathrm {Gr}$ of Lemma 3.2.

Lemma 3.3. Let $\mu \in \mathbf {Y}_{+}$ . Then multiplication and action on $z^\mu \mathcal {L}^+G$ induce an isomorphism of $\mathbb {F}$ -schemes

$$\begin{align*}\prod_{\alpha \in \mathfrak{R}_+} \prod_{n=0}^{\langle \alpha, \mu \rangle-1} U_{\alpha + n\hbar} \xrightarrow{\sim} \mathrm{Gr}_{\mu}. \end{align*}$$

If $w=t_\lambda w_{\mathrm {f}}$ with $\lambda \in \mathbf {Y}$ and $w_{\mathrm {f}} \in W_{\mathrm {f}}$ , then we have $p(\mathrm {Fl}_w) = \mathrm {Gr}_\lambda $ ; more precisely,

(3.3) $$ \begin{align} p_{|\mathrm{Fl}_w}: \mathrm{Fl}_w \to \mathrm{Gr}_\lambda \qquad \begin{array}{@{}c@{}} \text{is a trivial affine space bundle of} \\ \text{relative dimension }\ell(w) - \ell(w_\lambda). \end{array} \end{align} $$

The $\mathcal {L}^+G$ -orbits on $\mathrm {Gr}$ are naturally labelled by $\mathbf {Y}_+$ , and we will denote by $\mathrm {Gr}^{\lambda }$ the orbit attached to $\lambda $ . For any $\lambda \in \mathbf {Y}_+$ the $\mathrm {I}$ -orbit $\mathrm {Gr}_\lambda $ is open and dense in $\mathrm {Gr}^{\lambda }$ , and we have

$$\begin{align*}\dim(\mathrm{Gr}^\lambda) = \langle \lambda, 2\rho \rangle. \end{align*}$$

In this setting also one can consider the categories of sheaves on $\mathrm {I} \backslash \mathrm {Gr}_\lambda $ and its variants involving $\overline {\mathrm {Gr}_\lambda }$ and $\mathrm {I}_{\mathrm {u}}$ , and we have push/pull functors relating these categories.

3.4 Semiinfinite orbits

The orbits of $\mathrm {I}^{{\frac {\infty }{2}}}$ on $\widetilde {\mathrm {Fl}}$ , $\mathrm {Fl}$ or $\mathrm {Gr}$ are called semiinfinite orbits. The semiinfinite orbits on $\mathrm {Fl}$ , resp. on $\widetilde {\mathrm {Fl}}$ , resp. on $\mathrm {Gr}$ , are again labelled naturally by $W_{\mathrm {ext}}$ , resp. $W_{\mathrm {ext}}$ , resp. $\mathbf {Y}$ . For $w \in W_{\mathrm {ext}}$ , resp. $w \in W_{\mathrm {ext}}$ , resp. $\lambda \in \mathbf {Y}$ , the corresponding semiinfinite orbit is denoted by

$$\begin{align*}\mathfrak{S}_w \subset \mathrm{Fl}, \quad \text{resp.} \quad \widetilde{\mathfrak{S}}_w \subset \widetilde{\mathrm{Fl}}, \quad \text{resp.} \quad \mathrm{S}_\lambda \subset \mathrm{Gr}. \end{align*}$$

Here we have $\widetilde {\mathfrak {S}}_w = q^{-1}(\mathfrak {S}_w)$ .

These orbits do not have well-defined dimensions or codimensions. Nevertheless, we assign to each $\mathrm {I}^{{\frac {\infty }{2}}}$ -orbit on $\mathrm {Fl}$ or $\mathrm {Gr}$ an integer that will be called its pseudodimension, by the following formulas:

$$ \begin{align*} \operatorname{\mathrm{psdim}} (\mathfrak{S}_w) &= \langle \lambda, 2\rho\rangle + \ell(w_{\mathrm{f}}) &&\text{if }w = t_\lambda w_{\mathrm{f}}\text{ with }w_{\mathrm{f}} \in W_{\mathrm{f}}\text{ and }\lambda \in \mathbf{Y}, \\ \operatorname{\mathrm{psdim}} (\mathrm{S}_\lambda) &= \langle \lambda, 2\rho\rangle && \text{for }\lambda \in \mathbf{Y}. \end{align*} $$

If $w=t_\lambda w_{\mathrm {f}}$ with $\lambda \in \mathbf {Y}$ and $w_{\mathrm {f}} \in W_{\mathrm {f}}$ , then we have $p(\mathfrak {S}_w)=\mathrm {S}_\lambda $ ; more precisely,

(3.4) $$ \begin{align} p_{|\mathfrak{S}_w}: \mathfrak{S}_w \to \mathrm{S}_\lambda \qquad \begin{array}{@{}c@{}} \text{is a trivial affine space bundle of} \\ \text{relative dimension }\ell(w_{\mathrm{f}}). \end{array} \end{align} $$

We will also consider the ind-closures

$$\begin{align*}\overline{\mathfrak{S}_w} \subset \mathrm{Fl}, \qquad \overline{\mathrm{S}_\lambda} \subset \mathrm{Gr}. \end{align*}$$

Here it is a standard fact (see [Reference Mirković and Vilonen44, Proposition 3.1]) that

(3.5) $$ \begin{align} \overline{\mathrm{S}_\lambda} = \bigsqcup_{\mu \preceq \lambda} \mathrm{S}_\mu. \end{align} $$

We will denote by

$$\begin{align*}\mathbf{j}_w : \mathfrak{S}_w \to \overline{\mathfrak{S}_w}, \quad \overline{\mathbf{i}}_w : \overline{\mathfrak{S}_w} \to \mathrm{Fl}, \quad \mathbf{j}_\lambda : \mathrm{S}_\lambda \to \overline{\mathrm{S}_\lambda}, \quad \overline{\mathbf{i}}_\lambda : \overline{\mathrm{S}_\lambda} \to \mathrm{Gr} \end{align*}$$

the embeddings, and set $\mathbf {i}_w = \overline {\mathbf {i}}_w \circ \mathbf {j}_w$ , $\mathbf {i}_\lambda = \overline {\mathbf {i}}_\lambda \circ \mathbf {j}_\lambda $ .

Following the same procedure as in §2.6, one can define the categories

$$\begin{align*}\mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}} \backslash \mathfrak{S}_w), \quad \mathsf{Shv}(\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} \backslash \mathfrak{S}_w), \quad \mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}} \backslash \overline{\mathfrak{S}_w}), \quad \mathsf{Shv}(\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} \backslash \overline{\mathfrak{S}_w}) \end{align*}$$

for $w \in W_{\mathrm {ext}}$ , and the analogous categories for $\mathrm {S}_\lambda $ and $\overline {\mathrm {S}_\lambda }$ for $\lambda \in \mathbf {Y}$ . Denoting by

$$\begin{align*}\mathbf{k}_w : \mathrm{Spec}(\mathbb{F}) \to \mathfrak{S}_w, \quad \mathbf{k}_\lambda : \mathrm{Spec}(\mathbb{F}) \to \mathrm{S}_\lambda \end{align*}$$

the inclusions of the base points (for $w \in W_{\mathrm {ext}}$ and $\lambda \in \mathbf {Y}$ respectively) and by $\mathsf {Vect}_\Bbbk $ the $\infty $ -category of (complexes of) $\Bbbk $ -vector spaces, as in [Reference Gaitsgory27, Proposition 1.3.5] one sees that the functors $(\mathbf {k}_w)^!$ and $(\mathbf {k}_\lambda )^!$ induce equivalences

(3.6) $$ \begin{align} \mathsf{Shv}(\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} \backslash \mathfrak{S}_w) \cong \mathsf{Vect}_\Bbbk, \quad \mathsf{Shv}(\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} \backslash \mathrm{S}_\lambda) \cong \mathsf{Vect}_\Bbbk \end{align} $$

for any $w \in W_{\mathrm {ext}}$ and $\lambda \in \mathbf {Y}$ ; under these equivalences, $\omega _{\mathfrak {S}_w}$ and $\omega _{\mathrm {S}_\lambda }$ correspond to $\Bbbk \in \mathsf {Vect}_\Bbbk $ . We have push/pull functors between these categories associated with the maps $\mathbf {i}_w$ , $\overline {\mathbf {i}}_w$ , $\mathbf {j}_w$ , $\mathbf {i}_\lambda $ , $\overline {\mathbf {i}}_\lambda $ , $\mathbf {j}_\lambda $ .

We define the perverse t-structure on $\mathsf {Shv}(\mathrm {I}_{\mathrm {u}}^{{\frac {\infty }{2}}} \backslash \mathfrak {S}_w)$ to be the cohomological shift of the transport of the tautological t-structure on $\mathsf {Vect}_\Bbbk $ along the left-hand equivalence in (3.6) such that the object

$$\begin{align*}\omega_{\mathfrak{S}_w}[-\operatorname{\mathrm{psdim}} (\mathfrak{S}_w)] \end{align*}$$

belongs to the heart. (Here, as in §2.2, to simplify notation we write $\omega _{\mathfrak {S}_w}$ for $\omega _{\mathrm {I}_{\mathrm {u}} \backslash \mathfrak {S}_w}$ . Similar conventions will be used for variants of this construction below.) This t-structure will be denoted

$$\begin{align*}(\mathsf{Shv}(\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} \backslash \mathfrak{S}_w)^{\leq 0}, \mathsf{Shv}(\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} \backslash \mathfrak{S}_w)^{\geq 0}), \end{align*}$$

and its heart will be denoted $\mathsf {Shv}(\mathrm {I}_{\mathrm {u}}^{{\frac {\infty }{2}}} \backslash \mathfrak {S}_w)^\heartsuit $ . We similarly define the perverse t-structure on $\mathsf {Shv}(\mathrm {I}_{\mathrm {u}}^{{\frac {\infty }{2}}} \backslash \mathrm {S}_\lambda )$ in such a way that

$$\begin{align*}\omega_{\mathrm{S}_\lambda}[-\operatorname{\mathrm{psdim}} (\mathrm{S}_\lambda)] \end{align*}$$

belongs to the heart, and will use similar notation. In both cases, the corresponding cohomology functors will be denoted ${}^{\mathrm {p}} \hspace {-0.5pt} \mathcal {H}^n(-)$ ( $n \in \mathbb {Z}$ ). Any object $\mathcal {F}$ in the heart of one of these t-structures is a direct sum of copies of the appropriate shift of the dualizing sheaf; this multiplicity of the dualizing sheaf will be called the rank of $\mathcal {F}$ , and denoted $\operatorname {\mathrm {rank}} (\mathcal {F})$ . (For most objects of interest this quantity will be finite, but in general it can be infinite.)

Similar considerations hold for the $\mathrm {I}^{{\frac {\infty }{2}}}$ -equivariant versions of these $\infty $ -categories and functors; in this case we have

(3.7) $$ \begin{align} \mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}} \backslash \mathfrak{S}_w) \cong \mathsf{Shv}(T \backslash \mathrm{pt}), \quad \mathsf{Shv}(\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} \backslash \mathrm{S}_\lambda) \cong \mathsf{Shv}(T \backslash \mathrm{pt}). \end{align} $$

We normalize these equivalences in such a way that the forgetful functors

(3.8) $$ \begin{align} \mathrm{oblv}^{\mathrm{I}^{{\frac{\infty}{2}}} | \mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}}} : \mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}} \backslash \mathfrak{S}_w) \to \mathsf{Shv}(\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} \backslash \mathfrak{S}_w) \end{align} $$

(3.9) $$ \begin{align} \mathrm{oblv}^{\mathrm{I}^{{\frac{\infty}{2}}} | \mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}}} : \mathsf{Shv}(\mathrm{I}^{{\frac{\infty}{2}}} \backslash \mathrm{S}_\lambda) \to \mathsf{Shv}(\mathrm{I}_{\mathrm{u}}^{{\frac{\infty}{2}}} \backslash \mathrm{S}_\lambda) \end{align} $$

correspond to the forgetful functor

$$\begin{align*}\mathrm{oblv}^T : \mathsf{Shv}(T \backslash \mathrm{pt}) \to \mathsf{Shv}(\mathrm{pt})=\mathsf{Vect}_\Bbbk \end{align*}$$

under these equivalences and those of (3.6). We will also use perverse t-structure to mean the unique t-structure on $\mathsf {Shv}(\mathrm {I}^{{\frac {\infty }{2}}} \backslash \mathfrak {S}_w)$ , resp. $\mathsf {Shv}(\mathrm {I}^{{\frac {\infty }{2}}} \backslash \mathrm {S}_\lambda )$ , such that (3.8), resp. (3.9), is t-exact with respect to the perverse t-structures. We will use notation similar to that above for these t-structures. The definition of the rank of an object in the heart of the perverse t-structure translates verbatim to this setting.

Consider a finite union Z of closures of orbits $\mathfrak {S}_w$ ( $w \in W_{\mathrm {ext}}$ ), and denote by $i : Z \to \mathrm {Fl}$ the (closed) embedding. If $w \in W_{\mathrm {ext}}$ is such that $\mathfrak {S}_w \subset Z$ , for simplicity we will still denote by $\mathbf {i}_w : \mathfrak {S}_w \to Z$ the embedding. Then one can consider as above the $\infty $ -categories $\mathsf {Shv}(\mathrm {I}^{{\frac {\infty }{2}}} \backslash Z)$ and $\mathsf {Shv}(\mathrm {I}_{\mathrm {u}}^{{\frac {\infty }{2}}} \backslash Z)$ . Below we will require the following statement.

4.1 Standard and costandard complexes on $\mathrm {Fl}$

For $w \in W_{\mathrm {ext}}$ we set

$$\begin{align*}\nabla_w := (i_w)_* \omega_{\mathrm{Fl}_w}[-\ell(w)], \quad \Delta_w := (i_w)_! \omega_{\mathrm{Fl}_w}[-\ell(w)]. \end{align*}$$

When $w=e$ the objects $\Delta _e$ and $\nabla _e$ are isomorphic, and coincide with the unit object $\delta _{\mathrm {Fl}}$ for the monoidal product $\star ^{\mathrm {I}}$ . For any $w,y \in W_{\mathrm {ext}}$ and $n \in \mathbb {Z}$ we have

(4.1) $$ \begin{align} \mathrm{Hom}_{\mathsf{Shv}(\mathrm{I} \backslash \mathrm{Fl})}(\Delta_w, \nabla_y [n]) \cong \begin{cases} \mathsf{H}^n_T(\mathrm{pt}; \Bbbk) & \text{if }w=y\text{ and }n \in 2\mathbb{Z}_{\geq 0}; \\ 0 & \text{otherwise,} \end{cases} \end{align} $$

where $\mathsf {H}^n_T(\mathrm {pt}; \Bbbk )$ identifies with the $n/2$ -th symmetric power of the $\Bbbk $ -vector space $\Bbbk \otimes _{\mathbb {Z}} \mathbf {X}$ for $n \in 2\mathbb {Z}_{\geq 0}$ . In the $\mathrm {I}_{\mathrm {u}}$ -equivariant context we have

(4.2) $$ \begin{align} \mathrm{Hom}_{\mathsf{Shv}(\mathrm{I}_{\mathrm{u}} \backslash \mathrm{Fl})}(\mathrm{oblv}^{\mathrm{I} | \mathrm{I}_{\mathrm{u}}}(\Delta_w), \mathrm{oblv}^{\mathrm{I} | \mathrm{I}_{\mathrm{u}}}(\nabla_y) [n]) \cong \begin{cases} \Bbbk & \text{if }w=y\text{ and }n = 0; \\ 0 & \text{otherwise.} \end{cases} \end{align} $$

Recall that we have the perverse t-structures on the stable $\infty $ -categories $\mathsf {Shv}(\mathrm {I} \backslash \mathrm {Fl})$ and $\mathsf {Shv}(\mathrm {I}_{\mathrm {u}} \backslash \mathrm {Fl})$ from §2.5, which restrict to t-structures on the full subcategories $\mathsf {Shv}_{\mathrm {fg}}(\mathrm {I} \backslash \mathrm {Fl})$ and $\mathsf {Shv}_{\mathrm {fg}}(\mathrm {I}_{\mathrm {u}} \backslash \mathrm {Fl})$ respectively. Since each immersion $\mathrm {Fl}_w \to \overline {\mathrm {Fl}_w}$ is affine, the complexes $\Delta _w$ and $\nabla _w$ are perverse sheaves. For any $w \in W_{\mathrm {ext}}$ there exists (up to scalar) a unique nonzero morphism $\Delta _w \to \nabla _w$ , whose image will be denoted $\mathrm {IC}_w$ . It is a simple perverse sheaf, namely the intersection cohomology complex associated with the stratum $\mathrm {Fl}_w$ (and the trivial local system). It is also the unique simple quotient of $\Delta _w$ , and the unique simple subobject of $\nabla _w$ .

The following lemma gathers standard properties of standard and costandard complexes. For proofs, see, e.g. [Reference Achar and Riche5, Lemmas 4.1.4, 4.1.7 and 4.1.9].

Below we will also use the following properties (which hold both in $\mathrm {I}$ -equivariant and in $\mathrm {I}_{\mathrm {u}}$ -equivariant perverse sheaves).

4.2 Standard and costandard complexes on $\mathrm {Gr}$

Similarly to the setting considered in §4.1, in the $\infty $ -category $\mathsf {Shv}(\mathrm {I} \backslash \mathrm {Gr})$ , for any $\lambda \in \mathbf {Y}$ we have objects

$$\begin{align*}\overline{\Delta}_\lambda \quad \text{and} \quad \overline{\nabla}_\lambda, \end{align*}$$

supported on $\overline {\mathrm {Gr}_\lambda }$ , which are obtained by $!$ - and $*$ -pushforward of $\omega _{\mathrm {Gr}_\lambda }[-\ell (w_\lambda )]$ under the map $i_\lambda $ respectively. By (3.3), we have

$$\begin{align*}\begin{aligned} p_*\Delta_w &= \overline{\Delta}_\lambda[\ell(w_\lambda) - \ell(w)], \\ p_*\nabla_w &= \overline{\nabla}_\lambda[\ell(w) - \ell(w_\lambda)], \end{aligned} \qquad \text{for }\lambda \in \mathbf{Y}\text{ and }w \in t_\lambda W_{\mathrm{f}}. \end{align*}$$

In particular, for $w = t_\lambda w_{\mathrm {f}}$ with $\lambda \in \mathbf {Y}$ and $w_{\mathrm {f}} \in W_{\mathrm {f}}$ , in view of (3.2) we have

(4.3) $$ \begin{align} p_*\Delta_w &\cong \overline{\Delta}_\lambda[-\ell(w_{\mathrm{f}})], \ \ p_*\nabla_w \cong \overline{\nabla}_\lambda[\ell(w_{\mathrm{f}})] \quad \text{if } \lambda \in \mathbf{Y}_+, \end{align} $$

(4.4) $$ \begin{align} p_*\Delta_w &\cong \overline{\Delta}_\lambda[\ell(w_{\mathrm{f}})-\ell(w_\circ)], \ \ p_*\nabla_w \cong \overline{\nabla}_\lambda[\ell(w_\circ) - \ell(w_{\mathrm{f}})] \quad \text{if }\lambda \in -\mathbf{Y}_{++}. \end{align} $$

For $\lambda , \mu \in \mathbf {Y}$ and $n \in \mathbb {Z}$ , as in (4.1) we have

(4.5) $$ \begin{align} \mathrm{Hom}_{\mathsf{Shv}(\mathrm{I} \backslash \mathrm{Gr})}(\overline{\Delta}_\lambda, \overline{\nabla}_\mu [n]) \cong \begin{cases} \mathsf{H}^n_T(\mathrm{pt}; \Bbbk) & \text{if }\lambda=\mu\text{ and }n \in 2\mathbb{Z}_{\geq 0}; \\ 0 & \text{otherwise,} \end{cases} \end{align} $$

and as in (4.2) we have

(4.6) $$ \begin{align} \mathrm{Hom}_{\mathsf{Shv}(\mathrm{I}_{\mathrm{u}} \backslash \mathrm{Gr})}(\mathrm{oblv}^{\mathrm{I} | \mathrm{I}_{\mathrm{u}}}(\overline{\Delta}_\lambda), \mathrm{oblv}^{\mathrm{I} | \mathrm{I}_{\mathrm{u}}}(\overline{\nabla}_\mu) [n]) \cong \begin{cases} \Bbbk & \text{if }\lambda=\mu\text{ and }n=0; \\ 0 & \text{otherwise.} \end{cases} \end{align} $$

With respect to the perverse t-structures on $\mathsf {Shv}(\mathrm {I} \backslash \mathrm {Gr})$ and $\mathsf {Shv}(\mathrm {I}_{\mathrm {u}} \backslash \mathrm {Gr})$ , the objects $\overline {\Delta }_\lambda $ and $\overline {\nabla }_\lambda $ are perverse. For any $\lambda \in \mathbf {Y}$ , the image of the unique (up to scalar) nonzero morphism $\overline {\Delta }_\lambda \to \overline {\nabla }_\lambda $ will be denoted $\overline {\mathrm {IC}}_\lambda $ ; once again, this is a simple perverse sheaf, namely the intersection cohomology complex associated with the stratum $\mathrm {Gr}_\lambda $ (and the constant local system). For $\lambda =0$ we have $\overline {\Delta }_0=\overline {\nabla }_0=\overline {\mathrm {IC}}_0$ , and this object will be denoted $\delta _{\mathrm {Gr}}$ ; it is the image of the unit object in the monoidal $\infty $ -category $(\mathsf {Shv}(\mathcal {L}^+ G \backslash \mathrm {Gr}), \star ^{\mathcal {L}^+ G})$ .

It is a standard fact that the functor $p^\dagger $ (see §2.5) is t-exact for the perverse t-structures, that its restriction to perverse sheaves is fully faithful, and that for any $\lambda \in \mathbf {Y}$ we have

$$\begin{align*}p^\dagger \overline{\mathrm{IC}}_\lambda \cong \mathrm{IC}_{w_\lambda w_\circ}. \end{align*}$$

4.3 Spherical complexes on $\mathrm {Gr}$

Recall the $\infty $ -category $\mathsf {Shv}(\mathcal {L}^+ G \backslash \mathrm {Gr})$ considered in Remark 2.5(1). On this $\infty $ -category too we have a perverse t-structure, which restricts to a t-structure on the full subcategory $\mathsf {Shv}_{\mathrm {fg}}(\mathcal {L}^+ G \backslash \mathrm {Gr})$ , and such that the functor $\mathrm {oblv}^{\mathcal {L}^+G | \mathrm {I}}$ is t-exact. Its heart will be denoted $\mathsf {Shv}(\mathcal {L}^+ G \backslash \mathrm {Gr})^\heartsuit $ . By [Reference Mirković and Vilonen44, Proposition 4.2] this heart is stable under the convolution bifunctor $\star ^{\mathcal {L}^+ G}$ . For $\lambda \in \mathbf {Y}_+$ we will denote by $\mathcal {I}_*(\lambda )$ , resp. $\mathcal {I}_!(\lambda )$ , the $0$ -th perverse cohomology of the $*$ -pushforward, resp. $!$ -pushforward, associated with the embedding $\mathrm {Gr}^\lambda \to \mathrm {Gr}$ of the constant local system on $\mathrm {Gr}^\lambda $ placed in degree $-\dim (\mathrm {Gr}^\lambda )$ . Then the image of the unique (up to scalar) nonzero morphism $\mathcal {I}_!(\lambda ) \to \mathcal {I}_*(\lambda )$ is $\overline {\mathrm {IC}}_\lambda $ .

Let us denote by $G^\vee _\Bbbk $ the split reductive group scheme over $\Bbbk $ which is Langlands dual to G, and by $\mathsf {Rep}(G^\vee _\Bbbk )^\heartsuit $ the abelian category of finite-dimensional algebraic $G^\vee _\Bbbk $ -modules. Then the geometric Satake equivalence of [Reference Mirković and Vilonen44] provides an equivalence of monoidal abelian categories

$$\begin{align*}\mathrm{Sat} : \left( \mathsf{Shv}_{\mathrm{fg}}(\mathcal{L}^+ G \backslash \mathrm{Gr})^\heartsuit, \star^{\mathcal{L}^+ G} \right) \xrightarrow{\sim} \left( \mathsf{Rep}(G^\vee_\Bbbk)^\heartsuit, \otimes \right). \end{align*}$$

The construction of this equivalence determines a canonical maximal torus $T^\vee _\Bbbk \subset G^\vee _\Bbbk $ , whose lattice of weights is $\mathbf {Y}$ .

Recall that, for $\lambda \in \mathbf {Y}_+$ , the $G^\vee _\Bbbk $ -module

$$\begin{align*}\mathrm{Sat}(\mathcal{I}_*(\lambda)), \quad \text{resp.} \quad \mathrm{Sat}(\mathcal{I}_!(\lambda)), \quad \text{resp.} \quad \mathrm{Sat}(\overline{\mathrm{IC}}_\lambda), \end{align*}$$

is the dual Weyl (i.e. induced) module $\mathsf {N}^{\vee }(\lambda )$ , resp. Weyl module $\mathsf {M}^{\vee }(\lambda )$ , resp. simple module, of highest weight $\lambda $ , see [Reference Mirković and Vilonen44, §13] or [Reference Achar and Riche5, Lemma 1.5.2 and Remark 1.5.3(1)].

4.4 Wakimoto sheaves on $\mathrm {Fl}$

In the abelian category $\mathsf {Shv}(\mathrm {I} \backslash \mathrm {Fl})^\heartsuit $ we have a collection of objects

$$\begin{align*}(\mathcal{W}_{t_\lambda} : \lambda \in \mathbf{Y}) \end{align*}$$

called the Wakimoto sheaves, whose definition is due to Mirković. These objects satisfy (and are determined by) the following properties:

• • if $\lambda \in \mathbf {Y}_+$ , resp. $\lambda \in -\mathbf {Y}_+$ , then we have $\mathcal {W}_{t_\lambda } = \Delta _{t_\lambda }$ , resp. $\mathcal {W}_{t_\lambda } = \nabla _{t_\lambda }$ (in particular we have $\mathcal {W}_{t_0} = \delta _{\mathrm {Fl}}$ );

• • if $\lambda , \mu \in \mathbf {Y}$ , we have a canonical isomorphism

  (4.7) $$ \begin{align} \mathcal{W}_{t_\lambda} \star^{\mathrm{I}} \mathcal{W}_{t_\mu} \cong \mathcal{W}_{t_{\lambda+\mu}}. \end{align} $$

For details about this construction, see [Reference Achar and Riche5, §4.2]. (Note however that in [Reference Achar and Riche5] the fixed Iwahori subgroup is associated with the negative Borel subgroup, while here it is attached to the positive Borel subgroup.)

We extend this definition to general elements of $W_{\mathrm {ext}}$ as follows: if $w = t_\lambda w_{\mathrm {f}} \in W_{\mathrm {ext}}$ with $\lambda \in \mathbf {Y}$ and $w_{\mathrm {f}} \in W_{\mathrm {f}}$ , we set

$$\begin{align*}\mathcal{W}_w = \mathcal{W}_{t_\lambda} \star^{\mathrm{I}} \Delta_{w_{\mathrm{f}}}. \end{align*}$$

Here, using (3.1) and Lemma 4.1 we see that if $\mu , \lambda +\mu \in \mathbf {Y}_+$ we have

(4.8) $$ \begin{align} \mathcal{W}_w \cong \nabla_{t_{-\mu}} \star^{\mathrm{I}} \Delta_{t_{\lambda+\mu} w_{\mathrm{f}}}, \end{align} $$

and if $\mu $ is furthermore strongly dominant we have

(4.9) $$ \begin{align} \mathcal{W}_w \cong \Delta_{t_{\lambda+\mu}} \star^{\mathrm{I}} \nabla_{t_{-\mu} w_{\mathrm{f}}}. \end{align} $$

In particular, this object is perverse and supported on $\overline {\mathrm {Fl}_w}$ by Lemma 4.1(3).

Some immediate properties of these objects are gathered in the following lemma. (Here, (4) involves the semiinfinite order on $W_{\mathrm {ext}}$ , whose definition is recalled in §3.1.)

Proof. (1) is a direct consequence of (4.7). (2) and (3) are special cases of (4.8) and (4.9) respectively.

For (4), we write $w=t_\lambda w_{\mathrm {f}}$ and $y=t_{\mu } y_{\mathrm {f}}$ with $\lambda ,\mu \in \mathbf {Y}$ and $w_{\mathrm {f}}, y_{\mathrm {f}} \in W_{\mathrm {f}}$ , and choose $\nu \in \mathbf {Y}$ such that $\lambda +\nu $ and $\mu + \nu $ are dominant. Then we have

$$\begin{align*}\mathcal{W}_w \cong \nabla_{t_{-\nu}} \star^{\mathrm{I}} \Delta_{t_\nu w}, \quad \mathcal{W}_y \cong \nabla_{t_{-\nu}} \star^{\mathrm{I}} \Delta_{t_\nu y}, \end{align*}$$

so that by the invertibility of $\nabla _{t_{-\nu }}$ (see Lemma 4.1(2)) and Lemma 4.2(2) we have

$$\begin{align*}\dim \mathrm{Hom}_{\mathsf{Shv}(\mathrm{I} \backslash \mathrm{Fl})}(\mathcal{W}_w, \mathcal{W}_y) = \begin{cases} 1 & \text{if }t_\nu w \leq t_\nu y; \\ 0 & \text{otherwise.} \end{cases} \end{align*}$$

This is equivalent to the condition in the lemma by definition of $\leq ^{\frac {\infty }{2}}$ .

4.5 A subcategory of constructible sheaves

For later use, in this subsection we study certain full subcategories in $\mathsf {Shv}_{\mathrm {fg}}(\mathrm {I} \backslash \widetilde {\mathrm {Fl}})$ and $\mathsf {Shv}_{\mathrm {fg}}(\mathrm {I} \backslash \mathrm {Fl})$ characterized in terms of Wakimoto objects. Here we insist that, contrary to the situation in most subsections above, we work in the $\infty $ -categories $\mathsf {Shv}_{\mathrm {fg}}$ and not $\mathsf {Shv}$ . We will use the standard fact (already recalled above for $\mathrm {Fl}$ ) that the perverse t-structures on the $\infty $ -categories $\mathsf {Shv}(\mathrm {I} \backslash \widetilde {\mathrm {Fl}})$ and $\mathsf {Shv}(\mathrm {I} \backslash \mathrm {Fl})$ restrict to t-structures on the full subcategories $\mathsf {Shv}_{\mathrm {fg}}(\mathrm {I} \backslash \widetilde {\mathrm {Fl}})$ and $\mathsf {Shv}_{\mathrm {fg}}(\mathrm {I} \backslash \mathrm {Fl})$ .

Proof. We will explain the proof of (1); the statement in (2) can be justified similarly, identifying $\mathsf {Shv}_{\mathrm {fg}}(\mathrm {I} \backslash \widetilde {\mathrm {Fl}})$ with $\mathsf {Shv}_{\mathrm {fg}}(\mathrm {Fl}' / \mathrm {I}_{\mathrm {u}})$ , see Remark 2.5(2).

So, we fix $\mathcal {F}$ in $\mathsf {Shv}_{\mathrm {fg}}(\mathrm {I} \backslash \mathrm {Fl})$ . It is clear that if $\mathcal {F}$ satisfies (1a), then it satisfies (1b). Since Wakimoto sheaves are perverse (see §4.4) and satisfy the formula in Lemma 4.3(1), it is clear also that if $\mathcal {F}$ satisfies (1c) then it satisfies (1a).

To conclude, we assume that $\mathcal {F}$ satisfies (1b), and fix data as in this condition. By [Reference Achar and Riche5, Proposition 4.4.4] (a statement due to Arkhipov–Bezrukavnikov) there exists a finite subset $A \subset W_{\mathrm {ext}}$ such that for all $x,y \in W_{\mathrm {ext}}$ we have

$$\begin{align*}(i_x)^*(\Delta_y \star^{\mathrm{I}} \mathcal{F}) \neq 0 \quad \Rightarrow \quad x \in y \cdot A. \end{align*}$$

Choose $n \geq N$ such that $t_{n \lambda } \cdot A \subset \{t_\mu w_{\mathrm {f}} : w_{\mathrm {f}} \in W_{\mathrm {f}}, \, \mu \in \mathbf {Y}_+\}$ . Then

$$\begin{align*}\Delta_{t_\lambda} \star^{\mathrm{I}} \mathcal{F} = \mathcal{W}_{t_\lambda} \star^{\mathrm{I}} \mathcal{F} \end{align*}$$

is concentrated in nonpositive perverse degrees by assumption. By standard arguments (see, e.g. [Reference Achar and Riche5, Lemma 4.4.3]) it follows that this complex belongs to the full subcategory of $\mathsf {Shv}_{\mathrm {fg}}(\mathrm {I} \backslash \mathrm {Fl})$ generated under extensions by the objects $\Delta _y[n]$ with $y \in \{t_\mu w_{\mathrm {f}} : w_{\mathrm {f}} \in W_{\mathrm {f}}, \, \mu \in \mathbf {Y}_+\}$ and $n \in \mathbb {Z}_{\geq 0}$ . For such y, by Lemma 4.3(2) we have $\Delta _y = \mathcal {W}_y$ . Convolving on the left with $\mathcal {W}_{t_{-\lambda }}$ and using Lemma 4.3(1) once again, we deduce that $\mathcal {F}$ satisfies (1c), which concludes the proof.

4.6 Dual Wakimoto sheaves

In this subsection we introduce a family of objects in $\mathsf {Shv}(\mathrm {I} \backslash \mathrm {Fl})$ which is ‘homologically dual’ to the family of Wakimoto sheaves, and whose construction is due (in the setting of representations of affine Kac–Moody algebras) to Gaitsgory in [Reference Gaitsgory28]. These objects do not belong to the subcategory $\mathsf {Shv}_{\mathrm {fg}}(\mathrm {I} \backslash \mathrm {Fl})$ , so that they cannot be seen in the usual framework of bounded derived categories of constructible sheaves.

Let $w \in W_{\mathrm {ext}}$ , and write $w = t_\lambda w_{\mathrm {f}}$ with $\lambda \in \mathbf {Y}$ and $w_{\mathrm {f}} \in W_{\mathrm {f}}$ . Then we set $\mathbf {Y}_{+,w} = \{\mu \in \mathbf {Y}_+ \mid \lambda +\mu \in \mathbf {Y}_+\}$ , and for any $\mu \in \mathbf {Y}_{+,w}$ we set

$$\begin{align*}\mathcal{W}_w^{\vee,\mu} = \nabla_{t_{-\mu}} \star^{\mathrm{I}} \nabla_{t_{\mu} w}. \end{align*}$$

For any $\mu \in \mathbf {Y}_{+,w}$ and $\eta \in \mathbf {Y}_+$ we have $\mu +\eta \in \mathbf {Y}_{+,w}$ , and there exists a canonical morphism

(4.10) $$ \begin{align} \mathcal{W}_w^{\vee,\mu} \to \mathcal{W}_w^{\vee,\mu+\eta} \end{align} $$

which is defined as follows. Recall from (4.1) that there exists a canonical morphism $\Delta _{t_\eta } \to \nabla _{t_\eta }$ . Convolving on the left with $\nabla _{t_{-\eta }}$ and using Lemma 4.1(2) we deduce a canonical morphism

$$\begin{align*}\delta_{\mathrm{Fl}} \to \nabla_{t_{-\eta}} \star^{\mathrm{I}} \nabla_{t_\eta}, \end{align*}$$

and then a canonical morphism

$$\begin{align*}\mathcal{W}_w^{\vee,\mu} = \nabla_{t_{-\mu}} \star^{\mathrm{I}} \delta_{\mathrm{Fl}} \star^{\mathrm{I}} \nabla_{t_{\mu} w} \to \nabla_{t_{-\mu}} \star^{\mathrm{I}} \nabla_{t_{-\eta}} \star^{\mathrm{I}} \nabla_{t_\eta} \star^{\mathrm{I}} \nabla_{t_{\mu} w}. \end{align*}$$

Here we have $\ell (t_{-\mu -\nu }) = \ell (t_{-\mu }) + \ell (t_{-\nu })$ and $\ell (t_{\eta +\mu } w) = \ell (t_\eta ) + \ell (t_{\mu } w)$ (see (3.1)), hence using Lemma 4.1(1) we have

$$\begin{align*}\nabla_{t_{-\mu}} \star^{\mathrm{I}} \nabla_{t_{-\eta}} \star^{\mathrm{I}} \nabla_{t_\eta} \star^{\mathrm{I}} \nabla_{t_{\mu} w} \cong \nabla_{t_{-\mu-\eta}} \star^{\mathrm{I}} \nabla_{t_{\mu+\eta} w}, \end{align*}$$

which provides the desired morphism (4.10).

Let us now denote by $\unlhd $ the preorder on $\mathbf {Y}$ defined by

$$\begin{align*}\lambda \unlhd \mu \quad \text{iff }\mu-\lambda \in \mathbf{Y}_+. \end{align*}$$

It is easy to see that the construction explained above defines a functor from (the ordinary category associated with) the preordered set $(\mathbf {Y}_{+,w}, \unlhd )$ to the homotopy category of the $\infty $ -category $\mathsf {Shv}(\mathrm {I} \backslash \mathrm {Fl})$ .

Lemma 4.5. For any $w \in W_{\mathrm {ext}}$ , there exists a functor (in the sense of $\infty $ -categories)

$$\begin{align*}(\mathbf{Y}_{+,w}, \unlhd) \to \mathsf{Shv}(\mathrm{I} \backslash \mathrm{Fl}) \end{align*}$$

which ‘lifts’ the construction of objects and morphisms above.

Proof. The justification is similar to that given in [Reference Gaitsgory27, §2.7]: the claim follows from the fact that the mapping spaces $\mathrm {Maps}(\mathcal {W}_w^{\vee ,\mu }, \mathcal {W}_w^{\vee ,\mu +\eta })$ are discrete, because they identify with the spaces

$$\begin{align*}\mathrm{Maps}(\delta_{\mathrm{Fl}}, \nabla_{t_{-\eta}} \star^{\mathrm{I}} \nabla_{t_\eta}) \cong \mathrm{Maps}(\Delta_{t_{\eta}}, \nabla_{t_\eta}), \end{align*}$$

which are discrete by (4.1), so that the functor defined at the level of homotopy categories automatically lifts to the $\infty $ -categorical level.

With Lemma 4.5 at hand, using the notation above one can set

$$\begin{align*}\mathcal{W}^\vee_w = \operatorname*{{\mathrm{colim}}}_{\mu \in \mathbf{Y}_{+,w}} \mathcal{W}_{w}^{\vee,\mu}. \end{align*}$$

Proposition 4.7. For any $w,y \in W_{\mathrm {ext}}$ and $n \in \mathbb {Z}$ we have canonical isomorphisms

$$\begin{align*}\mathrm{Hom}_{\mathsf{Shv}(\mathrm{I}_{\mathrm{u}} \backslash \mathrm{Fl})}(\mathrm{oblv}^{\mathrm{I} | \mathrm{I}_{\mathrm{u}}}(\mathcal{W}_w), \mathrm{oblv}^{\mathrm{I} | \mathrm{I}_{\mathrm{u}}}(\mathcal{W}^\vee_y) [n]) \cong \begin{cases} \Bbbk & \text{if }w=y\text{ and }n = 0; \\ 0 & \text{otherwise.} \end{cases} \end{align*}$$

Proof. The Wakimoto objects $\mathrm {oblv}^{\mathrm {I} | \mathrm {I}_{\mathrm {u}}}(\mathcal {W}_w)$ are compact. As a consequence, to prove the proposition it suffices to prove that if $\mu \in \mathbf {Y}_{+,w}$ is sufficiently large (with respect to the order $\unlhd $ ) we have

$$\begin{align*}\mathrm{Hom}_{\mathsf{Shv}(\mathrm{I}_{\mathrm{u}} \backslash \mathrm{Fl})}(\mathrm{oblv}^{\mathrm{I} | \mathrm{I}_{\mathrm{u}}}(\mathcal{W}_w), \mathrm{oblv}^{\mathrm{I} | \mathrm{I}_{\mathrm{u}}}(\mathcal{W}^{\vee,\mu}_{y}) [n]) \cong \begin{cases} \Bbbk & \text{if }w=y\text{ and }n = 0; \\ 0 & \text{otherwise,} \end{cases} \end{align*}$$

However, if $\mu \in \mathbf {Y}_{+,y} \cap \mathbf {Y}_{+,w}$ we have $\mathcal {W}_w \cong \nabla _{t_{-\mu }} \star ^{\mathrm {I}} \Delta _{t_{\mu } w}$ . It is a standard fact that for any $\nu \in \mathbf {Y}$ there exists an auto-equivalence of $\mathsf {Shv}(\mathrm {I}_{\mathrm {u}} \backslash \mathrm {Fl})$ which makes the following diagram commutative:

(This equivalence can, e.g. be constructed using convolution with a cofree standard object as considered in [Reference Chen and Dhillon20, §5.3] or [Reference Dhillon, Li, Yun and Zhu21] and an appropriate analogue of Lemma 4.1(2). Details will be discussed elsewhere.) Using this, the claims follow from (4.2).

Remark 4.8. Using later results of this paper one can show the analogue of Proposition 4.7 in the $\mathrm {I}$ -equivariant setting, i.e. that we have

$$\begin{align*}\mathrm{Hom}_{\mathsf{Shv}(\mathrm{I} \backslash \mathrm{Fl})}(\mathcal{W}_w, \mathcal{W}^\vee_y [n]) \cong \begin{cases} \mathsf{H}^n_T(\mathrm{pt}; \Bbbk) & \text{if }w=y\text{ and }n \in 2\mathbb{Z}_{\geq 0}; \\ 0 & \text{otherwise.} \end{cases} \end{align*}$$

(More precisely, the formula reduces to (5.1) using Proposition 5.6 and Corollary 6.14.) The proof above does not immediately apply because Wakimoto objects are not compact in this setting; see Remark 5.2(2) below for more details.

4.7 Wakimoto sheaves on $\mathrm {Gr}$

For $\lambda \in \mathbf {Y}$ we also define

$$\begin{align*}\overline{\mathcal{W}}_\lambda = p_*\mathcal{W}_{t_\lambda}. \end{align*}$$

By Lemma 4.3 and (4.3)–(4.4), we have

(4.11) $$ \begin{align} \overline{\mathcal{W}}_\lambda \cong \begin{cases} \overline{\Delta}_\lambda & \text{if }\lambda \in \mathbf{Y}_+, \\ \overline{\nabla}_\lambda[\ell(w_\circ)] & \text{if }\lambda \in -\mathbf{Y}_{++}. \end{cases} \end{align} $$

More generally, for any $\lambda \in \mathbf {Y}$ and $w_{\mathrm {f}} \in W_{\mathrm {f}}$ we have

(4.12) $$ \begin{align} p_*\mathcal{W}_{t_\lambda w_{\mathrm{f}}} \cong \mathcal{W}_{t_\lambda} \star^{\mathrm{I}} p_*\Delta_{w_{\mathrm{f}}} \cong \mathcal{W}_{t_\lambda} \star^{\mathrm{I}} \delta_{\mathrm{Gr}}[-\ell(w_{\mathrm{f}})] \cong \overline{\mathcal{W}}_\lambda[-\ell(w_{\mathrm{f}})]. \end{align} $$

For any $\lambda \in \mathbf {Y}$ , the complex $\overline {\mathcal {W}}_\lambda $ is supported on $\overline {\mathrm {Gr}_\lambda }$ . Finally, it follows from (4.7) that for $\lambda ,\mu \in \mathbf {Y}$ we have