The parabolic exotic t-structure

Let G be a connected reductive algebraic group over an algebraically closed field k, with simply connected derived subgroup. The exotic t-structure on the cotangent bundle of its flag variety T^*(G/B), originally introduced by Bezrukavnikov, has been a key tool for a number of major results in geometric representation theory, including the proof of the graded Finkelberg-Mirkovic conjecture. In this paper, we study (under mild technical assumptions) an analogous t-structure on the cotangent bundle of a partial flag variety T^*(G/P). As an application, we prove a parabolic analogue of the Arkhipov-Bezrukavnikov-Ginzburg equivalence. When the characteristic of k is larger than the Coxeter number, we deduce an analogue of the graded Finkelberg-Mirkovic conjecture for some singular blocks.

1. Introduction 1.1. The exotic t-structure. LetĠ be a connected reductive algebraic group over an algebraically closed field k, and letḂ ⊂Ġ be a Borel subgroup. (The undecorated letter G is reserved for another group, to be introduced later.) Let N = T * (Ġ/Ḃ) be the cotangent bundle of its flag variety, and consider the derived category D b CohĠ ×Gm ( N ) of (Ġ × G m )-equivariant coherent sheaves on N . The exotic t-structure is a remarkable t-structure on this category, originally defined in [14]. This t-structure has close connections to derived equivalences coming from the geometric Langlands program [9,8,23], to the cohomology of tilting modules for Lusztig's quantum groups [14] and algebraic groups [2], and other topics in representation theory (see [1] and the references therein). It is defined using a so-called exceptional set of objects in D b CohĠ ×Gm ( N ).
An important and rather nontrivial feature of this t-structure is that the higher t-cohomology of every exceptional object vanishes. This theorem (which was implicit in [14] and proved in different ways in [8,22]) implies that the heart of this tstructure has the familiar structure of a highest weight category. For representationtheoretic applications, this fact plays a similar conceptual role to the Kempf vanishing theorem (for reductive groups) or to the Artin vanishing theorem (for direct images of perverse sheaves under affine maps). In particular, this vanishing theorem plays a crucial role in the proof of the graded Finkelberg-Mirković conjecture [7], which relates the principal block of a reductive group to perverse sheaves on the Langlands dual affine Grassmannian, hence also in the proof of the tilting character formula for reductive groups [3].

Parabolic analogue.
The main result of this paper is a parabolic version of this vanishing theorem. Namely, for any parabolic subgroupṖ ⊂Ġ, the first and third authors have defined in [7] a certain exceptional set in D b CohĠ ×Gm (T * (Ġ/Ṗ )) which generalizes that defined by Bezrukavnikov in the caseṖ =Ḃ. As in this special case, our exceptional set determines a t-structure on D b CohĠ ×Gm (T * (Ġ/Ṗ )), which we again call the exotic t-structure. In Theorem 2.2 we show that, with respect to this t-structure, the higher t-cohomology of every exceptional object vanishes. As a consequence, its heart is a highest weight category; see Corollary 2.3.
Unlike the case of the full flag variety as treated in [22], our proof is indirect, and requires a translation of the problem to the realm of constructible sheaves and "mixed derived categories" in the sense of [6].
1.3. Applications. As an application, we prove a modular parabolic version of a derived equivalence originally due to Arkhipov-Bezrukavnikov-Ginzburg, relating D b CohĠ ×Gm (T * (Ġ/Ṗ )) to a category of constructible sheaves on the affine Grassmannian of the Langlands dual group.
Then, under the additional assumption that the characteristic of k is larger than the Coxeter number ofĠ, we combine this result with [7] to deduce a singular version of the graded Finkelberg-Mirković conjecture, relating a certain block (whose "singularity" is controlled byṖ ) for the reductive group whose Frobenius twist iṡ G to a suitable category of Whittaker perverse sheaves on the dual affine Grassmannian.
1.4. Contents. In Section 2 we state our main result more precisely, and outline our strategy of proof. This result is proved in Section 4, after some preliminaries in Section 3. The applications are deduced in Section 5. Finally, in Appendix A we extend certain results on parity complexes and "mixed derived categories" to the case of Whittaker sheaves. (These results play a technical role in some of our proofs.) 1.5. Acknowledgements. We thank G. Williamson for useful discussions.
2. Statement of the main result 2.1. Notation. LetĠ be a connected reductive algebraic group over a field k of characteristic ℓ, with maximal torus and Borel subgroupṪ ⊂Ḃ ⊂Ġ. (The reason why we decorate our notation with a dot should become clear in §5.5 below. It does not play any role in earlier subsections.) Let alsoḂ + be the Borel subgroup which is opposite toḂ (with respect toṪ ), R be the root system of (Ġ,Ṫ ), and R + ⊂ R be the system of positive roots given by the nonzeroṪ -weights in Lie(Ḃ + ). We will denote by X the character lattice ofṪ , and by S the set of simple reflections of the Weyl group W of (Ġ,Ṫ ) determined by our choice of R + . For s ∈ S, we will denote by α s the corresponding simple root, and by α ∨ s the associated coroot. We will make the following assumptions onĠ and k: (1) ℓ is very good forĠ; (2) the derived subgroup ofĠ is simply connected. By [21,Proposition 2.5.12], our assumption (1) implies the following property: (2.1) Lie(Ġ) admits a nondegenerateĠ-invariant bilinear form.
These assumptions and this property will in particular allow us to use the results of [8]. (Note that, by [23,Corollary 1.6], the condition that ℓ is a "JMW prime" used in [8] is equivalent to the condition that ℓ is good forĠ.) Our assumption (2) allows us to choose weights (ς s ) s∈S such that ς s , α ∨ t = δ s,t for any s, t ∈ S. For any subset I ⊂ S, we then set ς I := s∈I ς s .
The main players of this article will be the "partial Springer resolutions" N I :=Ġ ×Ṗ Iṅ I for I ⊂ S, whereṖ I ⊂Ġ is the standard (with respect toḂ) parabolic subgroup corresponding to I, andṅ I is the Lie algebra of its unipotent radical. Note that N ∅ is the usual Springer resolution ofĠ, and that N I identifies with the cotangent bundle toĠ/Ṗ I (thanks to (2.1)). We let the multiplicative group G m act onṅ I by z · x = z −2 x. This induces an action on N I that commutes with the naturalĠ-action, so we may consideṙ G × G m -equivariant coherent sheaves. We will denote by 1 : CohĠ ×Gm ( N I ) → CohĠ ×Gm ( N I ) the functor of tensoring with the tautological 1-dimensional G m -module, and by n the n-power of this functor (for n ∈ Z).

2.2.
A graded exceptional set. Following the notation and conventions of [7, §9], we fix a subset I ⊂ S, and consider the objects ∆ I (λ) and ∇ I (λ) in D b CohĠ ×Gm ( N I ) characterized in [7,Proposition 9.16]. 1 Here λ ∈ X +,reg I where X +,reg I := {λ ∈ X | ∀s ∈ I, λ, α ∨ s > 0}. In order to define these objects one needs to choose an order ≤ ′ on X. Here we will assume that ≤ ′ is constructed as in [7, §9.4]. Then the objects one obtains are independent of the choices involved in this construction, by [7,Proposition 9.19(1) and Proposition 9.24].
Remark 2.1. In the case I = ∅, the objects ∆ I (λ) and ∇ I (λ) are the same (up to shift) as those introduced by Bezrukavnikov (for characteristic-0 coefficients) in [14]. The general case is similar, replacing characters ofṪ by standard or costandard modules for the Levi factor ofṖ I containingṪ . In particular, when I = S, the object ∆ S (λ) is the Weyl module of highest weight λ−ς S , and ∇ S (λ) is the induced module of highest weight λ − ς S . According to [7,Proposition 9.16], the objects (∇ I (λ) : λ ∈ X +,reg I ) form a graded exceptional set of objects with respect to the order ≤ ′ and the "shift functor" 1 , in the sense of [14, §2.1.5]. That is, we have if µ ≤ ′ λ or if λ = µ and (n, m) = (0, 0), and moreover 1 In [7] we work under the running assumption that ℓ is bigger than the Coxeter number ofĠ.
However, as noticed in [7, Remark 9.1], the results of this particular section hold in the present generality.
The dual exceptional set is given by (∆ I (λ) : λ ∈ X +,reg I ). In other words, these objects form the unique collection of objects satisfying Here D b CohĠ ×Gm ( N I ) < ′ λ is the full triangulated subcategory of D b CohĠ ×Gm ( N I ) generated by the objects ∇ I (µ) n for µ < ′ λ and n ∈ Z, and the condition (2.4) means that the images of ∆ I (λ) and ∇ I (λ) in the Verdier quotient are isomorphic. These objects in fact satisfy [7,Corollary 9.18]. Moreover, both of the families (∇ I (λ) n : λ ∈ X +,reg I , n ∈ Z) and (∆ I (λ) n : λ ∈ X +,reg I , n ∈ Z) generate D b CohĠ ×Gm ( N I ) as a triangulated category.
2.3. The exotic t-structure. By the general theory of (graded) exceptional sets (see [14,Proposition 4]), the following pair of subcategories defines a bounded tstructure on the triangulated category D b CohĠ ×Gm ( N I ): Here, A ext means the smallest strictly full additive subcategory containing the objects A and closed under extensions. This t-structure is called the exotic tstructure, and its heart will be denoted by It is clear from the definitions that the functor 1 is t-exact for this t-structure.
The main result of this paper is the following.
This result can be rephrased as a cohomology-vanishing statement as follows: since ∇ I (λ) belongs to D b CohĠ ×Gm ( N I ) ≥0 by definition, Theorem 2.2 is equivalent to the statement that along with a similar vanishing statement in negative degrees for ∆ I (λ) (where t H i means the i-th cohomology with respect to the exotic t-structure). Once Theorem 2.2 is established, standard arguments (see e.g. [13], [22, §3.5] or [6,Proposition 3.11]) then imply the following claim, which formed our main motivation for studying this question.
representation-theoretic perverse Table 2.2. t-structures arising in the proof 2.4. Strategy of proof. The proof of Theorem 2.2 will be given in Section 4. In the case I = ∅, this theorem is one of the main results of [23] (see also [8] for a different proof). This special case plays a crucial role in the proof for I = ∅.
Broadly speaking, the strategy of the proof is to carry out a kind of categorical "diagram chase" using the categories and functors on the second, third and fourth columns in the diagram of Figure 2.1. (Precise definitions of all the notation in this diagram will be given in the following sections; we only mention here that the categories in the third and fourth columns are certain "mixed derived categories" in the sense of [6], and that the dashed arrow is not an equivalence but an identification of the right-hand side with a certain summand in the left-hand side; see §4.3 for details.) The leftmost column of this figure is only defined under the stronger assumption that ℓ is larger than the Coxeter number ofĠ, but it motivates our constructions even when it is not available. Each category carries one or two tstructures, which in some cases are already known to satisfy analogues of (2.6). Table 2.2 lists the t-structures that will come up in this paper. In this table, tstructures appearing in the same row correspond to one another under one of the horizontal functors in Figure 2.1.
In more detail, we begin in Section 3 by defining and studying a second tstructure on D b CohĠ ×Gm ( N ∅ ), and relating both t-structures to the affine Grassmannian Gr ′ of the Langlands dual group. In Section 4, we transfer the problem to the rightmost column of Figure 2.1. Specifically, we will reduce the proof of (2.6) to a similar claim for the perverse t-structure on the category D mix IW (Fl I , k). This claim is proved in Appendix A, using variations on some arguments in [27,3,6]. (In a sense, the problem is easier for this category because one can use the categories of Bruhat-constructible sheaves on Fl and Fl I , which have no counterparts in the world of coherent sheaves.) Once the proof of Theorem 2.2 is complete, we will be in a position to obtain an analogue on the bottom line of Figure 2.1 of the equivalence "Ψ" of the upper line. We do this in Section 5, and thereby obtain the parabolic version of the Arkhipov-Bezrukavnikov-Ginzburg equivalence (Theorem 5.5). (The space Gr ′ appearing in Figure 2.1 is the "right coset" version of the affine Grassmannian, but in Section 5 we will switch to the traditional "left coset" version, denoted by Gr.) Finally, when ℓ is larger than the Coxeter number, we can combine this equivalence with the results of [7] to obtain the singular version of the graded Finkelberg-Mirković conjecture (Theorem 5.7).
3. Representation-theoretic t-structure and translation functors 3.1. The representation-theoretic t-structure. In this subsection, we will introduce and study a different t-structure on D b CohĠ ×Gm ( N I ), which is "Koszul dual" (in an appropriate sense) to the exotic t-structure. Proof. The claim follows from the observation that the definition of a graded exceptional set does not depend on the choice of "shift functor." More precisely, in our case the assertion we must prove states that if µ ≤ ′ λ or if λ = µ and (n, m) = (0, 0), and moreover that But these are clearly equivalent to (2.2) and (2.3).
Of course the dual exceptional set is again (∆ I (λ) : λ ∈ X +,reg I ). Using once again [14,Proposition 4], we obtain that the following pair of subcategories forms a bounded t-structure on D b CohĠ ×Gm ( N I ): This t-structure will be called the representation-theoretic t-structure, and its heart will be denoted grRep( N I ). By construction, the functor 1 [1] is t-exact with respect to this t-structure.
Remark 3.2. The motivation for our terminology and notation should become clear in §5.5 below.
3.2. Geometric translation functors. Now we will make use of the "translation functors" There is also a smooth proper map µ I : N ∅,I → N I with fibers isomorphic toṖ I /Ḃ. We define: where ρ I is the halfsum of the positive roots which belong to the sublattice of ZR generated by the simple roots α s with s ∈ I. (Here, as in [7], all the functors are understood to be derived.) Lemma 3.3. The functor Π I is t-exact with respect to both the exotic and representation-theoretic t-structures, and it does not kill any nonzero object.
For the t-exactness statement in this lemma, one should equip both categories D b CohĠ ×Gm ( N ∅ ) and D b CohĠ ×Gm ( N I ) with the exotic t-structure, or both with the representation-theoretic t-structure. In the proof below we denote by W I ⊂ W the subgroup generated by I.
Proof. To show that Π I is left t-exact, by definition it suffices to show that the object Π I (∇ I (µ)) belongs to RT . In view of [14,Proposition 4(c)], we require that for any λ ∈ X and any n, r ∈ Z with r > 0 (for the exotic case) or r − n > 0 (for the representation-theoretic case). By adjunction (see [7,Remark 9.5]), we have if λ / ∈ W I X +,reg .
Using (2.5), we deduce that (3.1) holds under the present assumptions on n and r, so Π I is left t-exact. An analogous argument using the adjunction (Π I −n I [−n I ], Π I ) shows that Π I is also right t-exact. The fact that Π I does not kill any nonzero object follows from similar arguments: if F is a nonzero object in D b CohĠ ×Gm ( N I ), then there exists λ ∈ X +,reg I and r, n ∈ Z such that As above we have Lemma 3.4. The functor Π I is t-exact with respect to the representation-theoretic t-structure.
Proof. We must show that Π I (∆ ∅ (λ)) belongs to D b CohĠ ×Gm ( N I ) ≤0 RT and that Π I (∇ ∅ (λ)) belongs to D b CohĠ ×Gm ( N I ) ≥0 RT . Both of these assertions follow from [7, Proposition 9.24] and the definition of the representation-theoretic t-structure.

3.3.
Mixed derived category of the affine Grassmannian. The proof of Theorem 2.2 will require a "translation" of the problem to a setting involving constructible sheaves on affine flag varieties. This will require in particular an equivalence of categories obtained in [8,23] (which adapts a result obtained by Arkhipov-Bezrukavnikov-Ginzburg [9] for characteristic-0 coefficients), that we explain now.
Let K = C((t)) be the field of formal Laurent series in an indeterminate t, and let O = C[[t]] be the ring of formal power series in t. LetĠ ∨ be the complex reductive algebraic group which is Langlands dual toĠ. By definition this group comes with a maximal torusṪ ∨ whose cocharacter lattice is X, and such that the root system of (Ġ ∨ ,Ṫ ∨ ) identifies with the coroot system of (Ġ,Ṫ ). We will also consider the Borel subgroupḂ ∨ ⊂Ġ ∨ containingṪ ∨ and associated with the negative coroots ofĠ.
Let W aff := W ⋉ X be the extended affine Weyl group and let W Cox aff ⊂ W aff be the "true" affine Weyl group, i.e. the subgroup W ⋉ ZR. (The image of λ ∈ X in W aff will be denoted t λ .) Then W Cox aff has a natural structure of Coxeter group (such that S is a subset of the set of simple reflections), and whose length function extends to W aff . We will denote by 0 W aff ⊂ W aff the subset of elements w which are minimal in the coset W w. This subset is in bijection with X, via the map sending λ ∈ X to the minimal element w λ in W t λ . (See e.g. [22, §2.2] for details and references on this subject.) Consider the (left version of the) affine Grassmannian We denote by Iw ⊂Ġ ∨ (O) the Iwahori subgroup associated withḂ ∨ , i.e. the inverse image ofḂ ∨ under the "evaluation at t = 0" morphismĠ ∨ (O) →Ġ ∨ . We consider the action of Iw on Gr ′ induced by right multiplication inĠ ∨ (K ). The orbits for this action are parametrized in a natural way by the subset 0 W aff ⊂ W aff , and we will denote by Gr ′ w the orbit associated with w. Since each of these orbits is isomorphic to an affine space, following [6] we can consider the mixed derived category D mix (Iw) (Gr ′ , k) := K b Parity (Iw) (Gr ′ , k) and its perverse t-structure, where Parity (Iw) (Gr ′ , k) is the category of parity complexes on Gr ′ with respect to the stratification by Iw-orbits, in the sense of [19]. In particular, this theory provides standard and costandard (mixed) perverse sheaves 2 ∆ mix w and ∇ mix w for all w ∈ 0 W aff , and indecomposable tilting perverse sheaves T mix w . We will denote by {1} the autoequivalence of D mix (Iw) (Gr ′ , k) induced by the cohomological shift in Parity (Iw) (Gr ′ , k), and by [1] the usual shift of complexes; then the "Tate twist" 1 : The results of [8,23] provide an equivalence of triangulated categories From these properties we see that Ψ is t-exact if D b CohĠ ×Gm ( N ∅ ) is equipped with the representation-theoretic t-structure and D mix (Iw) (Gr ′ , k) is equipped with the perverse t-structure.
Remark 3.5. See [7,Remark 11.3] for a comparison of the conventions used in [8] and in [23]. The assumptions in §2.1 come from [8]; in [23], there are slightly more restrictive conditions on the group. Note that [8] and parts of [23] work instead with the "left coset" affine GrassmannianĠ ∨ (K )/Ġ ∨ (O), but as explained in [23, §3.2] it is straightforward to pass back and forth between this variety and Gr ′ . We will use Gr ′ for now because it is more convenient for the arguments in Section 4, but in Section 5, we will switch to (a positive characteristic analogue of) The triangulated category D mix (Iw) (Gr ′ , k) also admits a second interesting t-structure, called the adverse t-structure, defined in [8, §A.2]. This t-structure consists of the subcategories From this definition we see that the functor −1 [1] is t-exact with respect to the adverse t-structure and (using also (3.2 3.4. Highest weight structure. The following analogue of Theorem 2.2 for the representation-theoretic t-structure turns out to be much easier to prove. Lemma 3.6. The objects ∆ I (λ) and ∇ I (λ) (λ ∈ X +,reg I ) belong to grRep( N I ).
Proof. Let us first treat the case where I = ∅. In this case, we can use the equivalence of categories Ψ introduced in §3.3. Since this equivalence takes the representation-theoretic t-structure on D b CohĠ ×Gm ( N ∅ ) to the adverse t-structure on D mix (Iw) (Gr ′ , k), it suffices to prove that the objects ∆ mix w and ∇ mix w (w ∈ 0 W aff ) belong to the heart of the adverse t-structure. In turn, this claim follows from [8, Lemma A.5 and Proposition A. 16]. Now suppose that I = ∅, and let λ ∈ X +,reg I . By [7, Proposition 9.24], we have ∆ I (λ) ∼ = Π I (∆ ∅ (λ)) n I [n I ]. By Lemma 3.4 and the previous paragraph, we conclude that ∆ I (λ) ∈ grRep( N I ). Similar reasoning applies to ∇ I (λ).
Remark 3.7. Under the assumption that ℓ is bigger than the Coxeter number oḟ G, one can alternatively prove Lemma 3.6 by using the fact that the functor Φ I considered in §5.5 is t-exact if D b CohĠ ×Gm ( N I ) is endowed with the representationtheoretic t-structure and D b Rep I (G) with its tautological t-structure, and sends standard, resp. costandard, objects to standard, resp. costandard, objects.
As in §2.3, this lemma implies that the category grRep( N I ) is a graded highest weight category in the sense of [26, §7], with weight poset (X +,reg I , ≤ ′ ), standard objects (∆ I (λ) : λ ∈ X +,reg I ), costandard objects (∇ I (λ) : λ ∈ X +,reg I ), and shift functor 1 [1]. In particular, we can consider the tilting objects in grRep( N I ). The indecomposable tilting object associated with λ will be denoted T RT I (λ), and the multiplicity of a standard, resp. costandard, object ∆ I (λ) m [m], resp. ∇ I (λ) m [m], in a tilting object T will be denoted Note that, in the case I = ∅, the isomorphisms in (3.2) imply that for λ ∈ X we have Remark 3.8. In the case I = ∅, the object T RT ∅ (λ) coincides with the indecomposable exotic parity object in D b CohĠ ×Gm ( N ∅ ) associated with λ studied in [2]. Once Corollary 2.3 is established, this claim can be generalized to any subset I ⊂ S.

Compatibility with translation functors.
We are now in a position to refine some claims from Lemma 3.3. Lemma 3.9. For any λ ∈ X +,reg I , the object Π I (∆ I (λ)), resp. Π I (∇ I (λ)), belongs to grRep( N ∅ ), and it admits a filtration whose subquotients are the objects of the form and m is the length of the unique element w ∈ W I such that µ = w(λ) (each appearing once).
Proof. The first assertion follows from Lemma 3.3 and Lemma 3.6. We will prove the second one for the object ∆ I (λ); the case of ∇ I (λ) is similar.
By general properties of graded highest weight categories (see e.g. [26, §7.4]), we know that an object F ∈ grRep( N ∅ ) admits a filtration whose subquotients are standard objects if and only if for all µ ∈ X and all n ∈ Z. Moreover, if F admits such a filtration, then the number of occurences of a specific standard object ∆ ∅ (µ) n [n] as a subquotient in such a filtration is the dimension of (3.5) We apply this criterion to F = Π I (∆ I (λ)). As in the proof of Lemma 3.3 we have Using (2.5), we deduce that Hom(Π I (∆ I (λ)), ∇ ∅ (µ) m [n]) vanishes unless m = n = ℓ(w) and λ = wµ for some w ∈ W I , and is 1-dimensional otherwise. In particular, we have confirmed (3.4) for F = Π I (∆ I (λ)), and we have shown that the space (3.5) is 1-dimensional if µ ∈ W I (λ) and m is the length of the unique element w ∈ W I such that µ = w(λ), and 0-dimensional otherwise. Proof. It follows from Lemma 3.9 that the functor Π I sends tilting objects in grRep( N I ) to tilting objects in grRep( N ∅ ). Hence indeed Π I (T RT I (λ)) is tilting, and Lemma 3.9 gives us information on the multiplicities (Π I (T RT ). More precisely, using also the construction of the order ≤ ′ (see in particular [7,Equation (9.9)]), this information shows that , with multiplicity 1, and that all the other indecomposable direct summands are of the form T RT {λ} was also a direct summand, then we would have The first condition would imply that m > 0, and the second one that m < 0; a contradiction.
Remark 3.11. We expect (but do not prove in general) that in fact Π I (T RT By definition of the order ≤ ′ , X λ is a closed subvariety of Gr ′ , and U λ is open in X λ . We will consider the open and closed embeddings The definition of the mixed derived category in [6] applies to locally closed unions of Iw-orbits in Gr ′ also; in particular we can consider the categories D mix (Iw) (X λ , k) and D mix (Iw) (U λ , k), these categories possess perverse t-structures, and they are related by functors (i λ ) * , (i λ ) * , (i λ ) ! , (j λ ) * , (j λ ) ! , (j λ ) * which satisfy the usual adjunction properties. Moreover, we have a fully-faithful and t-exact pushforward morphism associated with the closed embedding X λ → Gr ′ , whose essential image contains T mix w λ . Therefore we may (and will) consider this object as belonging to D mix (Iw) (X λ , k). In §4.3 below we will prove the following claim.
belong to the heart of the adverse t-structure.
In the rest of this subsection we show that Lemma 4.1 implies Theorem 2.2. We fix λ ∈ X +,reg I , and will prove that ∆ I (λ) belongs to ExCoh( N I ). The case of ∇ I (λ) can be treated similarly.
By general properties of highest weight categories (see e.g. [26,Theorem 7.14]), we have an exact sequence where coker is an extension of objects of the form ∆ I (µ) m [m] with m ∈ Z and µ ∈ X +,reg I such that µ < ′ λ. Applying the exact functor Π I (see Lemma 3.3) we deduce an exact sequence . In view of Lemma 3.10, we can choose an isomorphism with ν < ′ λ and ν / ∈ W I (λ). Using Lemma 3.9 we see that Hom(Π I (∆ I (λ)), T ) = 0, so that the first arrow in (4.1) factors through an embedding Π I (∆ I (λ)) ֒→ T RT ∅ (λ) whose cokernel is a direct summand of the third term in (4.1). In conclusion, we have constructed a distinguished triangle , whose first term belongs to the triangulated subcategory generated by the objects of the form ∆ ∅ (µ) n with µ ∈ W I (λ), and whose third term belongs to the triangulated subcategory generated by the objects of the form ∆ ∅ (ν) n with ν < ′ λ and ν / ∈ W I (λ). Applying Ψ to (4.2) and using (3.3) we obtain a distinguished triangle The comments above and (3.2) show that all three objects in this triangle are supported on X λ , and that Now, consider the natural distinguished triangle where the first two arrows are the adjunction maps. Adjunction and the first equality in (4.4) show that the composition of the first arrow in (4.3) with the second arrow in (4.5) vanishes. In view of [12, Proposition 1.1.9], this implies that there exists a unique morphism of triangles whose middle morphism is the identity. (Here, the upper line is (4.3), and the lower line is (4.5).) Similar considerations using the second property in (4.4) produces a morphism of triangles in the reverse direction, and applying the uniqueness claim in [12, Proposition 1.1.9] to both compositions of these morphisms we see that they are isomorphisms, inverse to each other. In particular,we have shown that there exists an isomorphism By Lemma 4.1 the right-hand side belongs to the heart of the adverse t-structure. By t-exactness of Ψ (see §3.3), it follows that Π I (∆ I (λ)) belongs to the heart of the exotic t-structure. In view of Lemma 3.3, this shows that ∆ I (λ) belongs to the heart of the exotic t-structure, and finishes the proof of Theorem 2.2.

4.2.
Iwahori-Whittaker mixed derived categories. The proof of Lemma 4.1 will require a new "translation," via the Koszul duality from [3] (which adapts a characteristic-0 construction due to Bezrukavnikov-Yun [16]), to a setting involving Whittaker parity complexes on affine flag varieties. This will require in particular a passage from the classical topology to theétale topology. Sinceétale sheaves are defined only for certain choices of coefficients, in the next two subsections we replace k by a large enough finite field of characteristic ℓ (or by a finite extension of Q ℓ ′ for some prime ℓ ′ if ℓ = 0), that we will still denote by k for simplicity. Lemma 4.1 of course makes sense for such coefficients, and this variant will imply the lemma as stated in §4.1.
The general theory of Whittaker parity complexes on partial flag varieties of Kac-Moody groups is developed in Appendix A (after partial results obtained in [27,3]). In this subsection we explain how to apply these general results in the present case of affine flag varieties.
Let F be an algebraically closed field such that char(F) = p > 0 and p = ℓ (or p = ℓ ′ if ℓ = 0). We now consider the connected reductive F-groupĠ ∨ F which is Langlands dual toĠ, and its maximal torusṪ ∨ F and Borel subgroupḂ ∨ F , defined as in §3.3. We will also denote by H the simply-connected cover of the derived subgroup ofĠ ∨ F . We can then consider the groups H(F((t))) and (By our conventions, this ind-variety is connected.) As in [27, §11.7] and [3, §7.2] we can consider the "Iwahori-Whittaker" derived category D b IW (Fl, k) and its "mixed" variant D mix IW (Fl, k) (defined again as an appropriate bounded homotopy category of parity complexes). These categories are defined using the action of an Iwahori subgroup Iw + H associated with a positive Borel subgroup in H. The orbits of Iw + H on Fl are parametrized in a natural way by W Cox aff and we denote the orbit associated with w by Fl w . If we set 0 W Cox aff := 0 W aff ∩ W Cox aff , then the indecomposable parity complexes in D b IW (Fl, k) are naturally parametrized by 0 W Cox aff × Z, and we will denote by E IW w the object corresponding to (w, 0). The complex concentrated in degree 0, and with degree-0 term E IW w , is an object of D mix IW (Fl, k), which will be denoted E IW,mix w . The cohomological shift in the triangulated category D b IW (Fl, k) restricts to an autoequivalence of the subcategory of parity complexes, and thus induces an autoequivalence of D mix IW (Fl, k). This autoequivalence will be denoted by {1}. To I ⊂ S we can also associate a parabolic affine flag variety is the parahoric subgroup corresponding to the parabolic subgroup of H containing the negative Borel subgroup and associated which I. The natural projection q I : Fl → Fl I is a smooth, proper morphism. In the same way as for Fl, we can consider the Iwahori-Whittaker derived category D b IW (Fl I , k) and its "mixed" variant D mix IW (Fl I , k). The latter category admits a natural perverse t-structure, defined by the same procedure as in [6]; see Appendix A for details.
Remark 4.2. In Appendix A, for completeness we work in the setting of (partial) flag varieties of Kac-Moody groups. It seems very likely that the partial affine flag varieties considered above are special cases of (products of) flag varieties of (untwisted affine) Kac-Moody groups, but we do not know of a reference for this claim. (See, however, [20, Chap. XIII] for similar claims when the base field is C, and [25, §9.f] for the case I = ∅.) In any case, all the properties of flag varieties of Kac-Moody groups that we use in Appendix A have well-known analogues for affine flag varieties, so that all of our results apply in this setting also.   In fact we will treat the case of (j λ ) ! (j λ ) * T mix w λ ; the case of (j λ ) * (j λ ) * T mix w λ is similar and left to the reader.
We first remark that all the connected components of Gr ′ are isomorphic as indvarieties stratified by the Iw-orbits. Hence we can assume that λ belongs to ZR, or equivalently that w λ ∈ W Cox aff , or in other words that T mix w λ is supported on the connected component (Gr ′ ) 0 of the base point. Now, recall the "Koszul duality" equivalence [3,Theorem 7.4]. This equivalence satisfies κ • 1 = 1 [1] • κ, and sends standard, resp. costandard, perverse sheaves to standard, resp. costandard, perverse sheaves (in a way compatible with labellings). Hence it is t-exact if D mix (Iw) ((Gr ′ ) 0 , k) is endowed with the adverse t-structure and D mix IW (Fl, k) is endowed with the perverse t-structure. By [3,Theorem 7.4], it also satisfies Fix λ ∈ X +,reg I , and recall the distinguished triangle (4.5) in D mix (Iw) (Gr ′ ) 0 , k . Applying κ and using (4.6) we deduce a distinguished triangle On the partial affine flag variety Fl I , let (Fl I ) w λ be the Iw + H -orbit corresponding to w λ , and consider the embeddings As in §4.1 we have pullback and pushforward functors between mixed derived categories associated with these maps, and using these functors we obtain a canonical distinguished triangle We now want to identify the triangles (4.7) and (4.9). Applying κ −1 to the composition of the first arrow in (4.7) and the second arrow in (4.9) gives a map . This map is 0 by adjunction, since the second term is supported on X λ U λ . Hence our original composition also vanishes, and as in §4.1 we obtain a unique morphism of triangles [1] whose second vertical arrow is the identity. Consider now the space of maps between the first object in (4.9) and the third object in (4.7): we have . In view of Lemma A.8, the object (q I ) * κ((i λ ) * (i λ ) * T mix w λ ) belongs to the triangulated subcategory of D mix IW (Fl I , k) generated the objects ∇ IW,I,mix v with v = w λ ; hence this Hom-space vanishes. There is thus a unique morphism of triangles from (4.9) to (4.7) whose middle arrow is the identity and, as in §4.1, applying [12, Proposition 1.1.9] to both compositions of these morphisms gives that they are inverse isomorphisms. We have finally identified (4.7) and (4.9), hence proved in particular that there exists an isomorphism (4. 10) κ((j λ ) ! (j λ ) * T mix w λ ) ∼ = (q I ) * ∆ IW,I,mix w λ {ℓ(w I 0 )}. We can finally conclude: Lemma 4.4 guarantees that the right-hand side of (4.10) is perverse; hence (j λ ) ! (j λ ) * T mix w λ is adverse, which finishes the proof.

Application: a singular version of the mixed Finkelberg-Mirković conjecture
In this section we assume that k is an algebraic closure of F ℓ .

5.1.
Whittaker perverse sheaves on Gr. We continue with the notation of §4.2.
We also denote byḂ ∨,+ F the Borel subgroup ofĠ ∨ F which is opposite toḂ ∨ F with respect toṪ ∨ F . (In other words, as the notation suggests,Ḃ ∨,+ F is the positive Borel subgroup ofĠ ∨ F .) We will now work with the F-version of the affine Grassmannian (in its usual incarnation in terms of cosets for right multiplication), defined by . We have a natural embedding ] → Gr, and we will denote by L λ the image of λ ∈ X.
Let again I ⊂ S be a subset, and consider the associated parabolic subgrouṗ P ∨,+ F,I ⊂Ġ ∨ containingḂ ∨,+ F . Let alsoL ∨ F,I be the Levi factor ofṖ ∨,+ F,I containinġ T ∨ F , and setU ∨ F,I :=U ∨ F ∩L ∨ F,I , whereU ∨ F is the unipotent radical of the (negative) Borel subgroupḂ ∨ F ⊂Ġ ∨ F . We set ] →Ġ ∨ k is the natural morphism, and (Ṗ ∨,+ F,I ) u is the unipotent radical ofṖ ∨,+ F,I . Fixing an Artin-Schreier local system onU ∨ F,I and pulling it back to Q I , we obtain a notion of "Whittaker" complexes on Gr (see e.g. §A.2); 3 we denote the corresponding category of parity complexes by Parity Wh,I (Gr, k), and the associated mixed derived category by D mix Wh,I (Gr, k). (In case I = ∅, this construction recovers the familiar Iwahori-constructible categories.) As in §4.2, this category is endowed with a natural "perverse" t-structure, whose heart will be denoted Perv mix Wh,I (Gr, k), and the corresponding standard and costandard objects are perverse (by an appropriate analogue of Proposition A.9). By standard arguments (see e.g. [6]), this implies that the realization functor (see [12, §3.1] or [11,Appendix]) induces an equivalence of categories Wh,I (Gr, k). The orbits of Q I on Gr are parametrized in a natural way by X, where λ corresponds to the orbit of L λ . To understand the combinatorics of orbits on (partial) affine flag varieties, it is usually simpler to work with the negative Iwahori subgroup Iw F (defined as ev −1 (Ḃ ∨ F )). But here, because we will eventually want to combine our constructions with those of [8] and [7, Section 11], we instead work with the positive Iwahori subgroup Iw + F , defined as ev −1 (Ḃ ∨,+ F ). We now explain how to compare the resulting combinatorics of orbits.
We set . Then the Iw F -orbits on Fl ′ F (for the action induced by right multiplication) are parametrized in a natural way by W aff , and those on Gr ′ F by 0 W aff . Consider a 3 Here we understand theétale derived category of k-sheaves on Gr as the direct limit of the categories of k 0 -sheaves, where k 0 runs over finite subfields of k.
"Cartan" anti-automorphism ofĠ ∨ F which acts as the identity onṪ ∨ F and sendsḂ ∨ F toḂ ∨,+ F . (With the notation of [7, Remark 11.3 (2)], this antiautomorphism can be chosen as the composition of the F-version of the automorphism ϕ with the map g → g −1 .) This map induces an isomorphism Gr ′ F → Gr, which sends the Iw F -orbit corresponding to w λ to the Iw + F -orbit of L λ . Lemma 5.1. For λ ∈ X, the Q I -orbit on Gr labelled by λ supports a nonzero local system which is Q I -equivariant against the pullback of the Artin-Schreier local system iff λ ∈ −X +,reg I . Proof. We translate our problem in terms of Gr ′ F following the principles presented above. We will denote by Q − I the analogue of Q I where the roles of positive and negative roots are switched (so that Q − I is the appropriate analogue in the present setting of the subgroup U K ·U # K of §A.2.) Then we have to show that the Q − I -orbit labelled with w λ supports a (nonzero) Whittaker local system iff λ ∈ −X +,reg I . By the general considerations in §A.2, the latter condition holds iff vw λ is minimal in vw λ W I for any v ∈ W .
First, we assume that w λ satisfies this property. Then ℓ(vw λ w) = ℓ(vw λ ) + ℓ(w) = ℓ(v) + ℓ(w λ ) + ℓ(w) for any v ∈ W and w ∈ W I . In particular, w λ w I 0 belongs to 0 W aff ; it must then coincide with w w I 0 (λ) . In view of [7, Lemma 10.2], this implies that w I 0 (λ) belongs to X +,reg I , so that λ ∈ −X +,reg I . Conversely, assume that λ ∈ −X +,reg I . Then, by the converse implication in [7,Lemma 10.2], w w I 0 (λ) w I 0 belongs to 0 W aff , hence coincides with w λ ; moreover we have ℓ(vw λ w) = ℓ(v) + ℓ(w λ w) = ℓ(v) + ℓ(w λ ) + ℓ(w) for any v ∈ W and w ∈ W I . This implies that vw λ is minimal in vw λ W I for any v ∈ W , and finishes the proof. (λ), T Wh,I (λ) the corresponding normalized indecomposable parity complex, standard mixed perverse sheaf, costandard mixed perverse sheaf, simple mixed perverse sheaf and indecomposable mixed tilting perverse sheaf associated with λ respectively; see [6] for details on these notions. We will also denote by E Wh,I,mix (λ) the object of D mix Wh,I (Gr, k) consisting of the complex with E Wh,I (λ) in degree 0, and 0 in other degrees. (When I = ∅, we will sometimes omit the superscripts.) 5.2. Averaging functor. We have a natural "averaging" functor Wh,I (Gr, k), defined by the same procedure as in §A.3.
In the following lemma we use the notion of the "naive" quotient of an additive category by a full additive subcategory as in §A.3. Proof. These claims are special cases of the analogue in the present context of Lemma A.11. In particular, from the proof of Lemma 5.1, for λ ∈ −X +,reg I , w λ is minimal in w λ W I and so the shifts appearing in the second statement of Lemma A.11 are 0.

5.3.
Study of tilting objects in ExCoh( N I ). We denote by Tilt(ExCoh( N I )) the full additive subcategory of ExCoh( N I ) whose objects are the tilting objects. (This notion does make sense now that Corollary 2.3 is proved.) By the general theory of (graded) highest weight categories, we know that the isomorphism classes of indecomposable objects in this category are in a natural bijection with X +,reg For any λ ∈ X +,reg I , we will denote by T exo I (λ) the object associated with the pair (λ, 0); then the object associated with (λ, n) is T exo I (λ) n . It is also known that the "realization" functor (1) For any T in Tilt(ExCoh( N ∅ )), the object Π I (T ) belongs to ExCoh( N I ), and is tilting therein.
Proof. (1) Let T ∈ Tilt(ExCoh( N ∅ )). Then T is an extension of objects of the form ∆ ∅ (λ) n for λ ∈ X and n ∈ Z. By [7, Corollary 9.21 and Proposition 9.24(2)], we deduce that Π I (T ) is an extension (in the sense of triangulated categories) of objects of the form ∆ I (µ) n [m] for µ ∈ X +,reg , n ∈ Z and m ∈ Z ≤0 , hence that Now, T is also an extension of objects ∇ ∅ (λ) n for λ ∈ X and n ∈ Z. Using now [7, Corollary 9.21 and Proposition 9.24(1)], we deduce similarly that By [14,Lemma 4], the properties (5.3) and (5.4) imply that Π I (T ) belongs to ExCoh( N I ), and is tilting therein.
(2) As observed in the proof of Lemma 5.1, if λ / ∈ −X +,reg I then there exists v ∈ W such that vw λ is not minimal in vw λ W I , or in other words such that vw λ admits a reduced expression ending with a simple reflection s ∈ I. In view of [22, Proof of Corollary 4.2] this shows that T exo ∅ (λ) is then a direct summand of an object which is killed by Π I .

Exotic sheaves and mixed perverse sheaves.
Recall now the equivalence Ψ from §3.3. Here we will rather consider the variant of this equivalence considered in [7, §11.3], which will be denoted Note that in [7] the affine Grassmannian is defined over the complex numbers, while here we work withétale sheaves on the F-version of this variety. The fact that these two constructions give rise to equivalent categories follows from the general principles from [12, §6.1]. We now denote by Perv sph (Gr, k) the abelian category ofĠ ∨ F F[[t]] -equivariant k-perverse sheaf on Gr. This category is equipped with a symmetric monoidal structure given by the convolution product ⋆; moreover if we denote by Rep f (Ġ) the category of finite-dimensional algebraicĠmodules, then the geometric Satake equivalence provides an equivalence of abelian monoidal categories see [24] for the original source and [10] for a more detailed exposition of the proof. The same considerations as in [7, §11.2] (see also [15, §4.4]) show that the convolution construction also provides an action of the monoidal category (Perv sph (Gr, k), ⋆) on D mix Wh,I (Gr, k) on the right, which will also be denoted ⋆. The following theorem is a "parabolic version" of the main results of [8] and [23]. (2) for any λ ∈ −X +,reg I there exist isomorphisms P I (J Wh,I ! (λ)) ∼ = ∆ I (w I 0 (λ)), P I (J Wh,I * (λ)) ∼ = ∇ I (w I 0 (λ)), P I (E Wh,I,mix (λ)) ∼ = T exo I (w I 0 (λ)); (3) for any F in D mix Wh,I (Gr, k) and G ∈ Perv sph (Gr, k), there exists a bifunctorial isomorphism P I (F ⋆ G) ∼ = P I (F ) ⊗ S(G); (4) the following diagram commutes up to isomorphism: Proof. The equivalence P restricts to an equivalence of additive categories sending E(λ) to T exo ∅ (λ), see [8,Proposition 8.4]. Lemma 5.2 and Lemma 5.4 imply that Π I • P ′ factors through a functor P ′ I : Parity Wh,I (Gr, k) → Tilt(ExCoh( N I )).
Passing to bounded homotopy categories and composing with the equivalence (5.2) we deduce a functor P I : D mix Wh,I (Gr, k) → D b CohĠ ×Gm ( N I ) which satisfies (1).
Consider now the diagram where the maps labelled "real" are the functors (5.2). The leftmost square in this diagram commutes by definition, and the middle square commutes by construction of P ′ I . The rightmost square commutes by [3, Proposition 2.3]. (To be able to apply this result we need to check that the functor Π I admits a "lift" to the filtered versions. This however follows from [11,Example A.2].) The bottom part of the diagram commutes by definition of P I .
We claim that the top part of the diagram also commutes (up to isomorphism). This will again follow from [3, Proposition 2.3] once we justify that the functor P lifts to filtered versions. (Here, the filtered version of D mix Wh,∅ (Gr, k) that we consider is the same as in [3, Comments preceding Lemma 2.4]; the corresponding realization functor is then the identity.) In fact, P is defined (see [8, §7]) by applying an additive functor P 0 : Parity Wh,∅ (Gr, k) → CohĠ ×Gm ( N ∅ ) termwise to complexes of parity complexes. Using the filtered version of the category D mix Wh,∅ (Gr, k) constructed in [4, §2.5] and that of D b CohĠ ×Gm ( N ∅ ) constructed in [12, §3.1], we can again apply P 0 termwise to filtered objects formed from Parity Wh,∅ (Gr, k) to obtain filtered objects formed from CohĠ ×Gm ( N ∅ ), and hence obtain the required lift. This finishes the justification of the commutativity of the diagram in (4).
To conclude, it only remains to prove (3). By construction of the convolution action of Perv sph (Gr, k) on D mix Wh,I (Gr, k), it suffices to construct such an isomorphism when G is parity (in addition to being perverse), i.e. when S(G) is a tiltinġ G-module. In this case the functor (−) ⊗ S(G) stabilizes Tilt(ExCoh( N ∅ )) by [22,Proposition 4.10]. The equivalence P intertwines the functors (−)⋆G and (−)⊗S(G) (see [8,Proposition 7.2]); therefore the same property holds for its restriction P ′ .
The functor Π I clearly commutes with the functors (−) ⊗ S(G); from this we deduce that (−) ⊗ S(G) also preserves the subcategory Tilt (ExCoh( N I )). Now the functor Av I commutes with (−) ⋆ G (see e.g. [27, (11.1.1)] for a similar statement); hence the latter functor preserves the kernel of the former, namely the subcategory E(λ) : λ / ∈ −X +,reg I ⊕,Z of Parity Wh,∅ (Gr, k). By construction of the functor P ′ I out of P ′ , Av I and Π I , we finally deduce that this functor also intertwines the functors (−) ⋆ G and (−) ⊗ S(G). And using [3, Proposition 2.3] once again we deduce (3).

5.5.
Relation with representations of reductive groups. From now on we fix a connected reductive group G over k with simply-connected derived subgroup, and assume that ℓ > h, where h is the Coxeter number of G. We also fix a maximal torus and a Borel subgroup T ⊂ B ⊂ G. We then assume thatĠ, resp.Ḃ, resp.Ṫ , is the Frobenius twist of G, resp. B, resp. T . (Of course,Ġ is a reductive group that is isomorphic to G, but it plays a different conceptual role.) Note that ℓ is automatically very good forĠ, so that the assumptions of §2.1 hold. We will identify the lattice of characters of T with X, in such a way that the composition of the Frobenius morphism T →Ṫ with the character λ ∈ X = X * (Ṫ ) is the character ℓλ of T . We will consider the "dilated and shifted" action of W aff on X defined by w · ℓ µ = w(µ + ρ) − ρ, t λ · ℓ µ = µ + ℓλ for w ∈ W and λ, µ ∈ X. (Here, ρ ∈ 1 2 X is as usual the halfsum of the positive roots.) Denote by Rep ∅ (G) the "extended principal block" of the category Rep f (G) of finite-dimensional algebraic G-modules, that is, the Serre subcategory generated by the simple modules whose highest weight has the form w · ℓ 0 with w ∈ W aff . Here the weight w · ℓ 0 is dominant iff w belongs to the subset 0 W aff ⊂ W aff . In particular, Rep ∅ (G) contains the Weyl and induced modules of highest weight w · ℓ 0 for w ∈ 0 W aff , denoted M(w· ℓ 0) and N(w· ℓ 0) respectively. The corresponding simple object will be denoted L(w · ℓ 0).
Given a subset I ⊂ S, we let Rep I (G) be the Serre subcategory of Rep f (G) generated by the simple modules whose highest weight has the form w · ℓ (−ς I ) for w ∈ W aff . This category is a direct summand of Rep f (G) and is "singular at I" in that the stabilizer of −ς I under the dot action of W aff is the subgroup W I of W generated by I. Here, the weight w· ℓ (−ς I ) is dominant iff w belongs to 0 W I aff ⊂ W aff (see [7, §10.1]). In particular, Rep I (G) contains the Weyl and induced modules of highest weight w · ℓ (−ς I ) for w ∈ 0 W I aff , denoted M(w · ℓ (−ς I )) and N(w · ℓ (−ς I )) respectively, and the corresponding simple module L(w · ℓ (−ς I )).
Recall the bijection X ∼ − → 0 W aff considered in §3.3. As explained in [7, Lemma 10.2], this bijection restricts to a bijection X +,reg Recall also the functor constructed in [7, §10.3]. (In the notation of [7], we have Φ I := Ω I • κ I .) According to [7,Proposition 10.6], this is a degrading functor with respect to 1 [1]: that is, there exists a natural isomorphism Φ I • 1 [1] ∼ − → Φ I , such that Φ I induces an isomorphism (5.5) n∈Z for all F , G ∈ D b CohĠ ×Gm ( N I ). Moreover, by [7,Proposition 10.3] we have It is clear from these properties that Φ I is t-exact if D b CohĠ ×Gm ( N I ) is endowed with the representation-theoretic t-structure, and D b Rep I (G) with its tautological t-structure. In particular, this provides a grading on the category Rep I (G) in the sense of [13,Definition 4.3.1] (see also [7,Definition 11.5]); in other words, under our present assumptions grRep( N I ) is a "graded version" of Rep I (G).
Remark 5.6. In view of [7,Theorem 8.16, Remark 8.17 and Proposition 9.25], the functors Φ ∅ and Φ I intertwine the "geometric translation functors" Π I and Π I and the usual translation functors for G-modules, denoted T I ∅ and T ∅ I in [7]. Using [18,Proposition E.11], it follows that in the setting of Lemma 3.10 we in fact have Π I (T RT I (λ)) ∼ = T RT ∅ (λ) under the present assumptions. 5.6. The singular Mirković-Vilonen conjecture. The following theorem is a "singular analogue" of [7, Proposition 11.6 and Theorem 11.7]. Here we denote by ForĠ G : Rep f (Ġ) → Rep f (G) the restriction functor associated with the Frobenius morphism G →Ġ.
Proof. The proof is the same as for [7,Theorem 11.7] but for completeness we repeat it. We first construct a functor Q I : D mix Wh,I (Gr, k) → D b Rep I (G) as the composition Φ I • P I . Using the isomorphism from Theorem 5.5(1) and the natural isomorphism Φ I • 1 [1] ∼ − → Φ I we obtain the isomorphism ε I : Q I ∼ − → Q I • 1 . By the first two isomorphisms in Theorem 5.5(2) and (5.6), we also obtain isomorphisms (5.7) Q I (J Wh,I ! (λ)) ∼ = M(w w I 0 (λ) · ℓ (−ς I )), Q I (J Wh,I * (λ)) ∼ = N(w w I 0 (λ) · ℓ (−ς I )) for any λ ∈ −X +,reg I . In particular, this shows that the complexes Q I (J Wh,I ! (λ)) and Q I (J Wh,I * (λ)) belong to Rep I (G), which implies that Q I is t-exact if the category D mix Wh,I (Gr, k) is endowed with the perverse t-structure and D b Rep I (G) with its tautological t-structure. We will still denote by Q I the restriction of this functor to the hearts of these t-structures, which provides the wished-for exact functor Perv mix Wh,I (Gr, k) → Rep I (G). By (5.1), Theorem 5.5, and (5.5), this functor induces isomorphisms for any F and G in Perv mix Wh,I (Gr, k) and any k ∈ Z. In particular, this implies that Q I is faithful. Now J Wh,I ! * (λ) is the image of any nonzero morphism J Wh,I ! (λ) → J Wh,I * (λ) and L(w w I 0 (λ) · ℓ (−ς I )) is the image any nonzero morphism M(w w I 0 (λ) · ℓ (−ς I )) → N(w w I 0 (λ) · ℓ (−ς I )). Combining these facts with (5.7) gives that Q I (J Wh,I ! * (λ)) ∼ = L(w w I 0 (λ) · ℓ (−ς I )), hence finally that (Perv mix (Wh,I) (Gr, k), Q I , ε I ) gives a grading on Rep I (G).
As Q I is exact and given the isomorphisms (5.7), one sees that, for any λ ∈ −X +,reg I , Q I (T Wh,I (λ)) is a tilting G-module, and that it admits T(w w I 0 (λ) · ℓ (−ς I )) as a direct summand. The isomorphism (5.8) provides a ring isomorphism Since the left-hand side is local by [17,Theorem 3.1], this shows that Q I (T Wh,I (λ)) is indecomposable, and so isomorphic to T(w w I 0 (λ) · ℓ (−ς I )). By Theorem 5.5(3), for F in D mix Wh,I (Gr, k) and G in Perv sph (Gr, k) we have Q I (F ⋆ G) ∼ = Φ I (P I (F ) ⊗ S(G)). Combining this with the isomorphisms in [7, Theorem 1.1 and Theorem 1.2], we deduce a bifunctorial isomorphism Wh,I (Gr, k) then this implies that Q I (F ⋆ G) belongs to Rep I (G). Since Q I is t-exact and does not kill any nonzero object, this in turn implies that F ⋆ G is perverse, which proves Points (1) and (2b), and finishes the proof.
Appendix A. Whittaker mixed perverse sheaves on partial flag varieties In this appendix we assume that the reader is familiar (to a certain extent at least) with the theory of parity complexes (from [19]) and of mixed derived categories (from [6]). Our aim is to study such objects and categories in the case of Whittaker sheaves on partial flag varieties of Kac-Moody groups. The case of Bruhat-constructible sheaves on partial flag varieties (corresponding, in the notation used below, to the case when K = ∅) is known, mainly from [19,6], as is that of Whittaker sheaves on the full flag variety (corresponding to the case J = ∅), mainly from [27,3]. The general case will usually be deduced from one of these special cases.
A.1. Notation. In this section we consider the setting of [27,Part III] or [3, § §6.1-6.2]. In particular, we consider an algebraically closed field F of characteristic p > 0 and a Kac-Moody root datum (I, X, {α i } i∈I , {α ∨ i } i∈I ). (Note that the symbol "I" used here is unrelated to the set I considered in the body of the paper.) Let G be the associated Kac-Moody group over F in the sense of Mathieu. We also denote by B ⊂ G the Borel subgroup, and by X := G /B the associated flag variety. (See [27, §9.1] for a reminder on this construction, and for references to the original sources.) If W is the Weyl group of G , and if S ⊂ W are the simple reflections (in canonical bijection with I), then we have a decomposition into B-orbits where X w is a locally closed subvariety isomorphic to an affine space of dimension ℓ(w).
For any subset J ⊂ I of finite type we also have a partial flag variety X J . We will denote by W J ⊂ W the (finite) subgroup generated by the simple reflections corresponding to elements in J, and by W J ⊂ W the subset of elements w such that w is minimal in wW J . Then we have a stratification . We also have a natural proper morphism of ind-schemes q J : X → X J . For any w ∈ W J and v ∈ W J , we have q J (X wv ) = X J w , and the morphism X wv → X J w induced by q J identifies with the natural projection from A ℓ(w)+ℓ(v) to A ℓ(w) . (Note however that it is not known-at least to us-if q J is a smooth morphism in general; see [27, Remark 9.2.1] for details on this question.) A.2. Whittaker derived categories. We let ℓ be a prime number different from p, and k be either a finite field of characteristic ℓ, or a finite extension of Q ℓ . Then it makes sense to considerétale k-sheaves on X J (for J ⊂ I of finite type). We will assume that k contains a primitive p-th root of unity; after fixing a choice of such a root of unity we can consider the associated Artin-Schreier local system L AS on G a,F .
Let now K ⊂ I be another subset of finite type, and consider the associated parabolic subgroup P K of G , and its pro-unipotent radical U K . Let also L K ⊂ P K be the Levi subgroup, and U # K ⊂ L K be the unipotent radical of the Borel subgroup of L K which is opposite to B ∩L K with respect to the canonical maximal torus. Then the orbits of U K · U # K on X J are also in a natural bijection with W J . We will denote by K X J w the orbit associated with w. (When J or K is ∅, we will usually omit the corresponding superscript. This convention will be applied more generally to any notation used in this appendix and involving J or K.) After choosing an identification of each simple root subgroup of U # K with the additive group G a,F , the quotient identifies with a product of #K copies of G a,F . Composing with the addition map to G a,F , we obtain a group homomorphism U # K /[U # K , U # K ] → G a,F . The composition of this morphism with the projection We will denote by the (étale) (U K ·U # K , (χ K ) * L AS )-equivariant derived category of k-sheaves on X J . (See [5, Appendix A] for a brief review of the construction of this category. When K = ∅, one recovers the usual Bruhat-constructible derived category.) In the case J = ∅, as explained in [27, §11.1] (see also [16,Lemma 4.2.1] for more details), the orbit K X w supports a nonzero (U K ·U # K , (χ K ) * L AS )-equivariant local system iff w is minimal in W K w. The subset of W consisting of elements satisfying this condition will be denoted K W . For the orbits on X J , we observe that, for w ∈ W J , the orbit K X J w supports a nonzero (U K ·U # K , (χ K ) * L AS )-equivariant local system iff each orbit in (q J ) −1 ( K X J w ) supports a (U K ·U # K , (χ K ) * L AS )-equivariant local system, i.e. iff w belongs to In fact one, using standard Coxeter groups combinatorics one can check that 0 is the longest element in W J . For any element w ∈ K W J , we have standard and costandard perverse sheaves in D b Wh,K (X J , k), denoted K ∆ J w and K ∇ J w respectively, and obtained as !-and *extensions of the perversely shifted rank-1 (U K · U # K , (χ K ) * L AS )-equivariant local system on K X J w . (The fact that these objects are perverse sheaves is guaranteed by [12,Corollaire 4.1.3].) The following lemma is an extension of [27, Lemma 11.1.1], with an essentially identical proof.
Parity Wh,∅ (X J , k) → Parity Wh,K (X J , k), which will also be denoted Av J K . For w ∈ W J , we will denote by E J w the indecomposable object of Parity Wh,∅ (X J , k) parametrized by w (see the comments following Proposition A.2). We will also denote by the full subcategory of Parity Wh,∅ (X J , k) whose objects are the direct sums of cohomological shifts of objects of the form E J w with w ∈ W J K W J . Then we consider the "naive" quotient i.e. the additive category whose objects are those of Parity Wh,∅ (X J , k), and whose morphisms are obtained from those in Parity Wh,∅ (X J , k) by quotienting by the morphisms which factor through an object of E J w : w ∈ W J K W J ⊕,Z . Lemma A.5. For any w ∈ W J we have is open in the support of this object, and its restriction to this stratum is Wh,∅ (X , k). This isomorphism also shows that the restriction of E ww J 0 to any stratum X x with x ∈ wW J is k[ℓ(ww J 0 )]; it follows (using distinguished triangles associated with open/closed decompositions) that the restriction of (q J ) * E ww J 0 to X J w is z∈WJ k[ℓ(ww J 0 ) − 2ℓ(z)]. Since this stratum is open in the support of this object, we deduce that Wh,∅ (X J , k), and then that On the other hand, by the projection formula we have proving that G ′ = (q J ) * G = 0. From this one can deduce that G = 0, which completes the proof.
Proposition A.6. Assume that char(ℓ) = 2. 4 The functor Av J K vanishes on E J w : w ∈ W J K W J ⊕,Z , and induces an equivalence of categories . 4 As should be clear from the proof, this assumption can be refined to the one considered in [27, Theorem 11.5.1].
Proof. The case J = ∅ is equivalent to [27,Theorem 11.5.1]. Let us now explain how the general case can be deduced from this one. Let w ∈ W J K W J . By Lemma A.5 we have (q J ) * E J w ∼ = E ww J 0 . Here ww J 0 / ∈ K W by (A.1), so that Av K (E ww J 0 ) = 0. We deduce that Av K ((q J ) * E J w ) ∼ = (q J ) * Av J K (E J w ) = 0, which implies that Av J K (E J w ) vanishes. This proves the first claim of the statement, and hence also that Av J K factors through a functor Parity Wh,∅ (X J , k) E J w : w ∈ W J K W J ⊕,Z → Parity Wh,K (X J , k). We will now argue that this functor is fully faithful; essential surjectivity is then easy to see.
We need to show that for any F , G in Parity Wh,∅ (X J , k), the functor Av J K induces an isomorphism between the quotient of Hom(F , G) by the morphisms factoring through an object of E J w : w ∈ W J K W J ⊕,Z and Hom(Av J K (F ), Av J K (G)). The comments after Proposition A.2 (in the special case K = ∅) show that we can assume that G = (q J ) * H for some H in Parity Wh,∅ (X , k); then we have Av J K (G) = (q J ) * Av K (H), and using adjunction and the obvious isomorphism (q J ) * • Av J K ∼ = Av K • (q J ) * we obtain isomorphisms Hom(F , G) ∼ = Hom((q J ) * F , H), Hom(Av J K (F ), Av J K (G)) ∼ = Hom(Av K ((q J ) * F ), Av K (H)). Moreover, under these identifications our morphism is induced by Av K . Taking into account the known case J = ∅, it therefore suffices to prove that the isomorphism identifies the subspace V 1 of the left-hand side consisting of morphisms factoring through an object of E J w : w ∈ W J K W J ⊕,Z with the subspace V 2 of the right-hand side consisting of morphisms factoring through an object of E w : w ∈ W K W ⊕,Z .
From Lemma A.5 and (A.1) it is clear that (A.2) maps V 1 into V 2 . On the other hand, if w / ∈ K W then E w is annihilated by Av K , so the same is true for (q J ) * (q J ) * E w . Hence (q J ) * E w cannot admit as direct summands objects of the form E J x [n] with x ∈ K W J ; in other words this object belongs to E J w : w ∈ W J K W J ⊕,Z . It follows that the inverse map of (A.2) sends V 2 into V 1 , as desired.
Remark A.7. It follows in particular from Proposition A.6 that the functor Av J K sends indecomposable parity complexes to indecomposable parity complexes.
A.4. Mixed derived category. Following [6], we define the "mixed derived category" D mix Wh,K (X J , k) := K b Parity Wh,K (X J , k). As in [6] the autoequivalence induced by the cohomological shift in the category Parity Wh,K (X J , k) will be denoted by {1}, and the cohomological shift (of complexes of objects of Parity Wh,K (X J , k)) will be denoted by [1]. This category also admits a "Tate twist" autoequivalence 1 defined as {−1} [1].
The "recollement" formalism constructed in [6] applies in this setting (see also [3, §6.2]), and we have "mixed" standard and costandard objects K ∆ J,mix w and K ∇ J,mix w . ∼ = K ∇ J,mix w J {ℓ(w J )}. Proof. The proof is identical to that of Lemma A.1, once we have proved the appropriate compatibility statement, in mixed derived categories, for the functor (q J ) * and the *or !-pushforward functor under the embedding of a statum in X and X J . However, by adjunction it suffices to prove a similar statement for pullback functors. In turn, this property is clear from the Whittaker analogue of [6, Remark 2.7].
A.5. Standard and costandard objects. The goal of this subsection is to prove the following claim.
Proposition A.9. For any w ∈ K W J , the objects K ∆ J,mix w and K ∇ J,mix w belong to the heart of the perverse t-structure.
For the proof of this claim we need some preliminary lemmas.
Lemma A.10. Let w ∈ W , and write w = w K w K with w K ∈ W K and w K ∈ K W . Then we have . Proof. In the case w K = id, these isomorphisms are proved in [3, Lemma 6.1]. We deduce the general case as follows. We will only give the details for the first isomorphism; the second one can be treated similarly. Recall from [6,Lemma 4.9] that there exists a morphism ∆ mix id −ℓ(w K ) → ∆ mix wK whose cone is an extension (in the sense of triangulated categories) of objects which belong to the essential image of the forgetful functors from some equivariant mixed derived categories for some parabolic subgroups of the form P L with ∅ = L ⊂ K. Convolving with ∆ mix w K on the right and using [6, Proposition 4.4(1)] we deduce a morphism ∆ mix w K −ℓ(w K ) → ∆ mix w whose cone satisfies a similar property. It is easily seen that this cone is killed by Av K ; see e.g. [16,Lemma 4.4.6] for similar considerations. Therefore we obtain an isomorphism Av K (∆ mix w K ) −ℓ(w K ) ∼ − → Av K (∆ mix w ), which concludes the proof.
Lemma A.11. Let w ∈ W J . If W K w ∩ K W J = ∅, then Otherwise, write w = w K w K with w K ∈ W K and w K ∈ K W (so that w K belongs to K W J ). Then we have we deduce that Av J K (∆ J,mix w ) ∼ = K ∆ J,mix w and Av J K (∇ J,mix w ) ∼ = K ∇ J,mix w are perverse too (where the isomorphisms follow from Lemma A.11 again).
Let us also note the following property.
Lemma A.12. For any w ∈ K W J we have isomorphisms Moreover, these objects are perverse.
Proof. As usual we only treat the case of standard objects; the case of costandard objects is similar.
We begin with the case K = ∅. Here we have ∆ J,mix w = (q J ) * ∆ mix w by Lemma A.8, so the isomorphism between our two objects follows from the comparison of the two isomorphisms in [27, Lemma 9.4.2(1)]. We have already observed in the course of the proof of Proposition A.9 that this objects belongs to p D mix Wh,∅ (X , k) ≥0 . Now it is clear that the * -pullback of (q J ) * ∆ J,mix w {ℓ(w J 0 )} to a stratum X x vanishes unless x ∈ wW J and that in this cases it is isomorphic to k{ℓ(w)+ℓ(w J 0 )}. Since ℓ(w)+ℓ(w J 0 ) ≥ ℓ(x), we deduce that this object also belongs to p D mix Wh,∅ (X , k) ≤0 . The case of a general subset K follows from the case K = ∅ by applying the functor Av J K , as in the proofs of Lemma A.11 and Proposition A.9.