Roman Bezrukavnikov, Dennis Gaitsgory, Ivan Mirković, Simon Riche, & Laura Rider An Iwahori-Whittaker model for the Satake category Tome 6 (2019), p. 707-735. <http://jep.centre-mersenne.org/item/JEP_2019__6__707_0> © Les auteurs, 2019. Certains droits réservés. Cet article est mis à disposition selon les termes de la licence L ICENCE INTERNATIONALE D ’ ATTRIBUTION C REATIVE C OMMONS BY 4.0. https://creativecommons.org/licenses/by/4.0/ L’accès aux articles de la revue « Journal de l’École polytechnique — Mathématiques » (http://jep.centre-mersenne.org/), implique l’accord avec les conditions générales d’utilisation (http://jep.centre-mersenne.org/legal/). Publié avec le soutien du Centre National de la Recherche Scientifique Publication membre du Centre Mersenne pour l’édition scientifique ouverte www.centre-mersenne.org Tome 6, 2019, p. 707–735 DOI: 10.5802/jep.104 AN IWAHORI-WHITTAKER MODEL FOR THE SATAKE CATEGORY by Roman Bezrukavnikov, Dennis Gaitsgory, Ivan Mirković, Simon Riche & Laura Rider Abstract. — In this paper we prove, for G a connected reductive algebraic group satisfying a mild technical assumption, that the Satake category of G (with coefficients in a finite field, a finite extension of Qℓ , or the ring of integers of such a field) can be described via Iwahori-Whittaker perverse sheaves on the affine Grassmannian. As applications, we confirm a conjecture of Juteau-Mautner-Williamson describing the tilting objects in the Satake category, and give a new proof of the property that a tensor product of tilting modules is tilting. Résumé (Un modèle d’Iwahori-Whittaker pour la catégorie de Satake) Dans cet article nous montrons, pour G un groupe algébrique réductif connexe satisfaisant à une hypothèse technique mineure, que la catégorie de Satake de G (avec coefficients dans un corps fini, une extension finie des nombres p-adiques, ou l’anneau des entiers d’un tel corps) peut se décrire en termes de faisceaux pervers d’Iwahori-Whittaker sur la grassmannienne affine. Nous en déduisons la démonstration d’une conjecture de Juteau-Mautner-Williamson décrivant les objets basculants dans la catégorie de Satake, et également une nouvelle preuve du fait qu’un produit tensoriel de représentations basculantes est basculant. Contents 1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 2. Constructible sheaves on affine Grassmannians and affine flag varieties . . . . . . . 711 3. Spherical vs. Iwahori-Whittaker. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 4. Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 Mathematics Subject Classification (2010). — 20G05. Keywords. — Affine Grassmannian, perverse sheaves, geometric Satake equivalence, tilting modules, parity sheaves. R.B. was partially supported by NSF grant No. DMS-1601953. D.G. was supported by NSF Grant No. DMS-1063470. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (S.R., grant agreement No. 677147). L.R. was supported by NSF Grant No. DMS-1802378. e-ISSN: 2270-518X http://jep.centre-mersenne.org/ 708 R. Bezrukavnikov, D. Gaitsgory, I. Mirković, S. Riche & L. Rider 1. Introduction 1.1. Another incarnation of the Satake category. — Let G be a connected reductive algebraic group over an algebraically closed field F of positive characteristic, and let k be either a finite field of characteristic ℓ 6= char(F), or a finite extension of Qℓ , or the ring of integers of such an extension. If K := F((z)) and O := F[[z]], the Satake category is the category PervGO (Gr, k) of GO -equivariant (étale) k-perverse sheaves on the affine Grassmannian Gr := GK /GO of G. This category is a fundamental object in Geometric Representation Theory through its appearance in the geometric Satake equivalence, which claims that this category admits a natural convolution product (−) ⋆GO (−), which endows it with a monoidal structure, and that there exists an equivalence of monoidal categories (1.1) ∼ S : (PervGO (Gr, k), ⋆GO ) −→ (Rep(G∨ k ), ⊗). Here the right-hand side is the category of algebraic representations of the split reductive k-group scheme which is Langlands dual to G on finitely generated k-modules; see [MV07] for the original proof of this equivalence in full generality, and [BR18] for a more detailed exposition. (In these references, what is explicitly treated is the analogous equivalence for a complex group G, in which case k can be any Noetherian commutative ring of finite global dimension. The étale setting is similar; see [MV07, §14] and [BR18, §1.1.4] for a few comments.) This category already has another incarnation since (as proved by MirkovićVilonen) the forgetful functor PervGO (Gr, k) −→ Perv(GO ) (Gr, k) from the Satake category to the category of perverse sheaves on Gr which are constructible with respect to the stratification by GO -orbits is an equivalence of categories. The first main result of the present paper provides a third incarnation of this category, as a category PervIW (Gr, k) of Iwahori-Whittaker(1) perverse sheaves on Gr. More precisely we prove that a natural functor (1.2) PervGO (Gr, k) −→ PervIW (Gr, k) is an equivalence of categories, see Theorem 3.9. This result is useful because computations in PervIW (Gr, k) are much easier than in the categories PervGO (Gr, k) or Perv(GO ) (Gr, k), in particular due to the facts that standard/costandard objects have more explicit descriptions and that the “realization functor” b Db PervIW (Gr, k) −→ DIW (Gr, k) is an equivalence of triangulated categories. (1)This terminology is taken from [AB09]. In [ABB+ 05], the term “baby Whittaker” is used for the same construction. J.É.P. — M., 2019, tome 6 An Iwahori-Whittaker model for the Satake category 709 In the analogous setting of Whittaker D-modules over a field of characteristic 0, this statement already appears in [ABB+ 05]. See Remark 2.1(2) below for a discussion of possible variants for constructible sheaves over C. Let us also mention the conjecture [Bez16, Conj. 59] containing this statement as a special case (see [Bez16, Ex. 60] for more details). 1.2. Relation with the Finkelberg-Mirković conjecture. — One possible justification for the equivalence (1.2) comes from a singular analogue of the FinkelbergMirković conjecture [FM99]. This conjecture states that, if k is a field of positive characteristic ℓ bigger than the Coxeter number of G, if I ⊂ GO is an Iwahori subgroup and Iu ⊂ I is its pro-unipotent radical, there should exist an equivalence of abelian categories ∼ F : PervIu (Gr, k) −→ Rep0 (G∨ k) between the category of Iu -equivariant k-perverse sheaves on Gr and the “extended ∨ principal block” Rep0 (G∨ k ) of Rep(Gk ), i.e., the subcategory consisting of modules over which the Harish-Chandra center of the enveloping algebra of the Lie algebra of G∨ k acts with generalized character 0. This equivalence is expected to be compatible with the geometric Satake equivalence in the sense that for F in PervIu (Gr, k) and G in PervGO (Gr, k) we expect a canonical isomorphism F(F ⋆GO G) ∼ = F(F) ⊗ S(G)(1) . (Here (−) ⋆GO (−) is the natural convolution action of PervGO (Gr, k) on the category PervIu (Gr, k), and (−)(1) is the Frobenius twist.) One might expect similar descriptions for some singular “extended blocks” of Rep(G∨ k ), namely those attached to weights in the closure of the fundamental alcove belonging only to walls parametrized by (non-affine) simple roots, involving some Whittaker-type perverse sheaves.(2) In the “most singular” case, this conjecture postulates the existence of an equivalence ∼ Fsing : PervIW (Gr, k) −→ Rep−ς (G∨ k) between our category of Iwahori-Whittaker perverse sheaves and the extended block of weight −ς, where ς is a weight whose pairing with any simple coroot is 1 (the “Steinberg block”), which should satisfy Fsing (F ⋆GO G) ∼ = Fsing (F) ⊗ S(G)(1) . (Here we assume that ς exists, which holds e.g. if the derived subgroup of G∨ k is simply-connected.) On the representation-theoretic side, it is well known that the assignment V 7→ L((ℓ − 1)ς) ⊗ V (1) induces an equivalence of categories ∼ ∨ Rep(G∨ k ) −→ Rep−ς (Gk ), (2)This extension of the Finkelberg-Mirković conjecture stems from discussions of the fourth author with P. Achar. “Graded versions” of such equivalences are established in [ACR18]. J.É.P. — M., 2019, tome 6 710 R. Bezrukavnikov, D. Gaitsgory, I. Mirković, S. Riche & L. Rider where L((ℓ − 1)ς) is the simple G∨ k -module of highest weight (ℓ − 1)ς; see [Jan03, §II.10.5] or [And18]. Our equivalence (1.2) can be considered a geometric counterpart of this equivalence. 1.3. Relation with results of Lusztig. — Another hint for the equivalence (1.2) is given by some results of Lusztig [Lus83]. Namely, in [Lus83, §6] Lusztig defines some submodules K and J of (a localization of) the affine Hecke algebra H attached to G. By construction K is a (non unital) subalgebra of the localization of H, and J is stable under right multiplication by K. Then [Lus83, Cor. 6.8] states that J is free as a right based K-module (for some natural bases), with a canonical generator denoted Jρ . Now, H (or rather its specialization at q = 1) is categorified by the category of Iwahoriequivariant perverse sheaves on the affine flag variety Fl of G. The subalgebra K (or rather again its specialization) is then categorified by PervGO (Gr, k) (via the pullback functor to Fl), and similarly J is categorified by PervIW (Gr, k). From this perspective, the functor in (1.2) is a categorical incarnation of the map k 7→ Jρ · k considered by Lusztig, and the fact that it is an equivalence can be seen as a categorical upgrade of [Lus83, Cor. 6.8]. 1.4. Relation with results of Frenkel-Gaitsgory-Kazhdan-Vilonen. — Finally, a third hint for this equivalence can be found in work of the second author with Frenkel, Kazhdan and Vilonen [FGKV98, FGV01] and more recent work [Gai18]. Working in the context of D-modules over a ground field of characteristic 0 or ℓ-adic sheaves over a ground field of arbitrary characteristic, in [FGV01] the authors defined a candidate for the role of a Whittaker category on Gr using geometry of a complete curve and moduli stacks of bundles over it. A more direct, local definition of such a category is proposed in [Gai18], where it is also shown that the two constructions produce equivalent categories; the methods of [Gai18] rely on recently developed techniques of ∞-categories. Notice also that [Ras16, Th. 2.7.1(2)] implies an equivalence between the above categories and the Iwahori-Whittaker category. It was shown in [FGV01] (see in particular [FGV01, §§1.2.4–1.2.5]) that their Whittaker category is a free right module over the monoidal category PervGO (Gr, k). Thus, combining these works, we obtain another proof of the equivalence between PervIW (Gr, k) and PervGO (Gr, k) ∼ = Rep(G∨ k ), valid when we work with characteristic-0 coefficients. The above results cannot be automatically carried over to our present context, which is that of sheaves with coefficients of positive characteristic. However, the latter equivalence does generalize to our context, and amounts to our equivalence (1.2). Of course, we use different methods to prove it. As explained in [FGV01, §1.1], in the case of characteristic-0 coefficients these properties are closely related to the Casselmann-Shalika formula, and in fact our proof uses the geometric counterpart to this formula known as the geometric CasselmannShalika formula. (See also [AB09, §1.1.1] for the relation between the “Whittaker” and “Iwahori-Whittaker” conditions in the classical setting of modules over the affine Hecke algebra.) J.É.P. — M., 2019, tome 6 An Iwahori-Whittaker model for the Satake category 711 1.5. Application to tilting objects. — In Section 4 we provide a number of applications of this statement. An important one is concerned with the description of the tilting objects in the Satake category. Namely, in the case when k is a field of characteristic ℓ, the tilting modules (see e.g. [Jan03, Chap. E]) form an interesting family of objects in the category Rep(G∨ k ). It is a natural question to try to characterize topologically the GO -equivariant perverse sheaves on Gr corresponding to these objects. A first answer to this question was obtained by Juteau-Mautner-Williamson [JMW16]: they showed that, under some explicit conditions on ℓ, the parity sheaves on Gr for the stratification by GO -orbits are perverse, and that their images under (1.1) are the indecomposable tilting objects in Rep(G∨ k ). This result was later extended by Mautner and the fourth author [MR18] to the case when ℓ is good for G, and it played a crucial role in the proof (by Achar and the fifth author) of the Mirković-Vilonen conjecture (or more precisely the corrected version of this conjecture suggested by Juteau [Jut08]) on stalks of standard objects in the Satake category [AR15]. It is known (see [JMW16]) that if ℓ is bad then the GO -constructible parity sheaves on Gr are not necessarily perverse; so the answer to our question must be different in general. A conjecture was proposed by Juteau-Mautner-Williamson to cover this case, namely that the perverse cohomology objects of the parity complexes are tilting in PervGO (Gr, k) (so that all tilting objects are obtained by taking direct sums of direct summands of the objects obtained in this way). In our main application we confirm this conjecture, see Theorem 4.10, hence obtain an answer to our question in full generality. Using this description we prove a geometric analogue of a fundamental result for tilting modules, namely that these objects are preserved by tensor product and by restriction to a Levi subgroup. (On the representation-theoretic side, these results are due to Mathieu [Mat90] in full generality; see [JMW16, §1.1] for more references.) In fact, combined with the Satake equivalence, our proof can also be considered as providing a new complete proof of these properties of tilting modules. In [BR18], Baumann and the fourth author also use these facts to obtain a slight simplification of the proof of the geometric Satake equivalence. (Note that the proofs in the present paper do not rely on the latter result.) Acknowledgements. — The final stages of this work were accomplished while the fourth author was a fellow of the Freiburg Institute for Advanced Studies, as part of the Research Focus “Cohomology in Algebraic Geometry and Representation Theory” led by A. Huber-Klawitter, S. Kebekus and W. Soergel. We thank P. Achar and G. Williamson for useful discussions on the subject of this paper, and the referees for their helpful comments. 2. Constructible sheaves on affine Grassmannians and affine flag varieties 2.1. Notation. — Let F be an algebraically closed field of characteristic p > 0. Let G be a connected reductive algebraic group over F, let B − ⊂ G be a Borel subgroup, J.É.P. — M., 2019, tome 6 712 R. Bezrukavnikov, D. Gaitsgory, I. Mirković, S. Riche & L. Rider and let T ⊂ B − be a maximal torus. Let also B + ⊂ G be the Borel subgroup opposite to B − (with respect to T ), and let U + be its unipotent radical. We denote by X := X ∗ (T ) the character lattice of T , by X ∨ := X∗ (T ) its cocharacter lattice, by ∆ ⊂ X the root system of (G, T ), and by ∆∨ ⊂ X ∨ the corresponding coroots. We choose the system of positive roots ∆+ ⊂ ∆ consisting of the T -weights ∨ ∨ ∨ in Lie(U + ), and denote by X ∨ the corresponding subset + ⊂ X , resp. X ++ ⊂ X of dominant cocharacters, resp. of strictly dominant cocharacters. We also denote by ∆s ⊂ ∆ the corresponding subset of simple roots, and set 1 X ρ= α ∈ Q ⊗Z X. 2 + α∈∆ For any α ∈ ∆s we choose an isomorphism between the additive group Ga and the root subgroup Uα of G associated with α, and denote it uα . We will assume(3) that there exists ς ∈ X ∨ such that hς, αi = 1 for any α ∈ ∆s ; ∨ then we have X ∨ ++ = X + + ς. (Such a cocharacter might not be unique; we fix a choice once and for all.) Let Wf be the Weyl group of (G, T ), and let W := Wf ⋉ X ∨ be the corresponding (extended) affine Weyl group. For λ ∈ X ∨ we will denote by tλ the associated element of W . If w ∈ W and w = tλ v with λ ∈ X ∨ and v ∈ Wf , we set X X |1 + hλ, αi|. |hλ, αi| + ℓ(w) = α∈∆+ v(α)∈∆+ α∈∆+ v(α)∈−∆+ The restriction of ℓ to the semi-direct product W Cox of Wf with the coroot lattice is the length function for a natural Coxeter group structure, and if we set Ω := {w ∈ W | ℓ(w) = 0} then multiplication induces a group isomorphism ∼ W Cox ⋊ Ω −→ W. 2.2. The affine Grassmannian and the affine flag variety. — For the facts we state here, we refer to [Fal03]. We set K := F((z)), O := F[[z]], and consider the group ind-scheme GK (denoted LG in [Fal03]) and its group subscheme GO (denoted L+ G in [Fal03]). We denote by I − ⊂ GO the Iwahori subgroup associated with B − , i.e., the inverse image of B − under the morphism GO → G sending z to 0. We consider the affine Grassmannian Gr and the affine flag variety Fl defined by Gr := GK /GO , Fl := GK /I − . We denote by π : Fl → Gr the projection morphism. Any λ ∈ X ∨ defines a point z λ ∈ TK ⊂ GK , hence a point Lλ := z λ GO ∈ Gr. We set Grλ := GO · Lλ . (3)This assumption holds in particular if G is semisimple of adjoint type. J.É.P. — M., 2019, tome 6 An Iwahori-Whittaker model for the Satake category 713 Then Grλ only depends on the Wf -orbit of λ. Moreover, the Bruhat decomposition implies that F Gr = Grλ . λ∈X ∨ + λ We will denote by jλ : Gr → Gr the embedding. − For λ ∈ X ∨ + , we will denote by Pλ ⊂ G the parabolic subgroup of G containing B associated with the subset of ∆s consisting of those simple roots which are orthogonal to λ. Then Pλ is the stabilizer of Lλ in G, so that we have a canonical isomorphism ∼ G/Pλ −→ G·Lλ . Under this identification, it is known that the map pλ : Grλ → G/Pλ sending x to limt→0 t · x (where we consider the Gm -action on Gr via loop rotation) is a morphism of algebraic varieties, and realizes Grλ as an affine bundle over G/Pλ (see e.g. [NP01, Lem. 2.3]). It is well known (see e.g. [Lus83] or [NP01, §2]) that if λ ∈ X ∨ + , then we have X dim(Grλ ) = hλ, 2ρi = hλ, αi. α∈∆+ We denote by 4 the order on X ∨ + determined by λ4µ Then for λ, µ ∈ X∨ + iff µ − λ is a sum of positive coroots. we have Grλ ⊂ Grµ iff λ 4 µ. 2.3. Some categories of sheaves on Gr and Fl. — We let ℓ be a prime number which is different from p, and let k be either a finite extension of Qℓ , or the ring of integers in such an extension, or a finite field of characteristic ℓ. In this paper we will be concerned with the constructible derived categories Dcb (Gr, k) and Dcb (Fl, k) of étale k-sheaves on Gr and Fl, respectively. If K ⊂ GO is a subgroup, we will also denote b b by DK (Gr, k) and DK (Fl, k) the (constructible) K-equivariant derived category of ksheaves on Gr and Fl, in the sense of Bernstein-Lunts [BL94]. Each of these categories is endowed with the perverse t-structure, whose heart will be denoted Perv(Gr, k), Perv(Fl, k), PervK (Gr, k) and PervK (Fl, k) respectively. Remark 2.1 (1) Since Gr and Fl are ind-varieties and not varieties, the definition of the categories considered above requires some care; see e.g. [Nad05, §2.2] or [Gai01, App.] for details. We will not mention this point in the body of the paper, and simply refer to objects in these categories as complexes of sheaves. (2) Recall that by [MV07] the category Rep(G∨ R ) of algebraic representations of the ∨ group scheme GR over any Noetherian commutative base ring R of finite global dimension is equivalent to the corresponding category of spherical perverse sheaves on GrC in its analytic topology. More restrictive assumptions on the base ring in the present paper come from our need to use the Artin-Schreier sheaf (see Section 3.2), which is only defined in the context of étale sheaves over a variety in positive characteristic; this setting yields categories of sheaves with coefficients as above. Notice however J.É.P. — M., 2019, tome 6 714 R. Bezrukavnikov, D. Gaitsgory, I. Mirković, S. Riche & L. Rider that some constructions involving the Artin-Schreier sheaf do have an analogue for constructible sheaves in the classical topology (see e.g. [Wan15] for the example of Fourier-Deligne transform). We expect that such a counterpart of the Whittaker category can also be defined (see [AG, Rem. 10.3.6] for a possible approach); this would allow one to extend our main result to more general coefficient rings. If K ′ ⊂ K ⊂ GO are subgroups, we will denote by b b ForK K ′ : DK (Gr, k) −→ DK ′ (Gr, k), b b ForK K ′ : DK (Fl, k) −→ DK ′ (Fl, k) the natural forgetful functors. If K/K ′ is of finite type, these functors have both a K ! right and a left adjoint, which will be denoted ∗IndK K ′ and IndK ′ respectively. If we write X for Gr or Fl, these functors can be described explicitly by

∗ ′ e e IndK and !IndK K ′ (F) = a∗ k ⊠ F K ′ (F) = a! k ⊠ F [2(dim K/K )], e (−) is the functor sending an object F to the only object in Db (K ×K ′ X, k) where k ⊠ K ′ b whose pullback to K × X (an object of DK×K ′ (K × X, k), where K acts on K × X ′ via h · (g, x) = (gh−1 , h · x)) is isomorphic to kK ⊠L k F. When K = {1} we will write K ForK for For{1} . 2.4. Convolution. — We will make extensive use of the convolution construction, b defined as follows. Consider F, G in DG (Gr, k), and the diagram O pGr q Gr mGr Gr × Gr ←−−−− GK × Gr −−−−→ GK ×GO Gr −−−−−→ Gr, where pGr and q Gr are the quotient morphisms, and mGr is induced by the GK -action on Gr. Consider the action of GO × GO on GK × Gr defined by (g1 , g2 ) · (h1 , h2 GO ) = (g1 h1 (g2 )−1 , g2 h2 GO ). Then the functor (q Gr )∗ induces an equivalence of categories ∼ b b DG (GK ×GO Gr, k) −→ DG (GK × Gr, k). O O ×GO e G such that Hence there exists a unique object F ⊠ L

e G = (pGr )∗ F ⊠k G . (q Gr )∗ F ⊠ The convolution product of F and G is defined by (2.1)
eG . F ⋆GO G := (mGr )∗ F ⊠ b This construction endows the category DG (Gr, k) with the structure of a monoidal O category. A similar formula defines a right action of this monoidal category on b DK (Gr, k), for any K ⊂ GO . (This action will again be denoted ⋆GO .) Remark 2.2. — Note that if k is not a field, the convolution product considered above is not the same as the one considered (when F and G are perverse sheaves) in [MV07]: the product considered in [MV07] is rather defined as pH0 (F ⋆GO G) in our notation. Lemma 2.3. — Assume that k is a field. If F belongs to Perv(Gr, k) and G belongs to PervGO (Gr, k), then F ⋆GO G belongs to Perv(Gr, k). J.É.P. — M., 2019, tome 6 An Iwahori-Whittaker model for the Satake category 715 Proof. — This claim follows from the description of convolution in terms of nearby cycles obtained in [Gai01, Prop. 6]. (In [Gai01], only the case of characteristic-0 coefficients is treated. However the same proof applies in general, simply replacing [Gai01, Prop. 1] by [MV07, Prop. 2.2].)  Remark 2.4. — The description of convolution in terms of nearby cycles as in [Gai01] works for general coefficients (if convolution is defined as in (2.1)). The nearby cycles functor is t-exact in this generality, but this description also involves a (derived) external tensor product. If k is not a field this tensor product operation is not t-exact, which explains the failure of Lemma 2.3 in this setting. A very similar construction as the one considered above, based on the diagram − q Fl pFl mFl Fl × Fl ←−−−− GK × Fl −−−−→ GK ×I Fl −−−−→ Fl, − provides a convolution product ⋆I on DIb− (Fl, k), which endows this category with the structure of a monoidal category, and defines a right action of this monoidal b category on DK (Fl, k), for any K ⊂ GO . Again the same formulas, using the diagram Fl mFl qGr pFl − Gr Gr −→ Gr, −→ GK ×I Gr −−−− −− GK × Gr −−− Fl × Gr ←−− allows to define a bifunctor b b (Gr, k), DK (Fl, k) × DIb− (Gr, k) −→ DK − which will once again be denoted ⋆I . The following lemma is standard; its proof is left to interested readers. b Lemma 2.5. — For any subgroup K ⊂ GO , any F in DK (Fl, k) and any G in b DGO (Gr, k), there exists a canonical isomorphism − GO O ∼ F ⋆I ForG G I − (G) = π∗ (F) ⋆ b in DK (Gr, k). In the following lemma we consider the convolution bifunctor b b b (Fl, k) −→ DK (Fl, k) (−) ⋆GO (−) : DK (Gr, k) × DG O (constructed once again using formulas similar to those above). Its proof is easy, and left to the reader. b Lemma 2.6. — Let F in DK (Gr, k) and G in DIb− (Fl, k). Then there exists a canonical isomorphism − G π ∗ (F) ⋆I G ∼ = F ⋆GO ∗Ind −O (G) I in b DK (Fl, k). J.É.P. — M., 2019, tome 6 716 R. Bezrukavnikov, D. Gaitsgory, I. Mirković, S. Riche & L. Rider 3. Spherical vs. Iwahori-Whittaker 3.1. Equivariant perverse sheaves on Gr. — For λ ∈ X ∨ + , we will denote by

J! (λ, k) := pH0 (jλ )! kGrλ [hλ, 2ρi] , resp. J∗ (λ, k) := pH0 (jλ )∗ kGrλ [hλ, 2ρi] , the standard, resp. costandard, GO -equivariant perverse sheaf on Gr associated with λ. We will also denote by J!∗ (λ, k) the image of any generator of the free rank-1 k-module HomPervGO (Gr,k) (J! (λ, k), J∗ (λ, k)). If k is a field then J!∗ (λ, k) is a simple perverse sheaf, which is both the head of J! (λ, k) and the socle of J∗ (λ, k). Recall the notion of highest weight category, whose definition is spelled out e.g. in [Ric16, Def. 7.1]. (These conditions are obvious extensions of those considered in [BGS96, §3.2], which were preceded by a related study in [CPS88].) Lemma 3.1. — Assume that k is a field. The category PervGO (Gr, k) is a highest ∨ weight category with weight poset (X ∨ + , 4), standard objects {J! (λ, k) : λ ∈ X + }, and costandard objects {J∗ (λ, k) : λ ∈ X ∨ + }. Moreover, if char(k) = 0 then this category is semisimple. Proof. — The first claim is an easy consequence of [MV07, Prop. 10.1(b)]; see [BR18, Prop. 1.12.4] for details. If char(k) = 0, the semisimplicity of the category PervGO(Gr,k) is well known: see [Gai01, Prop. 1] (or [BR18, §1.4] for an expanded version).  Remark 3.2 (1) If k is a field of characteristic 0, the semisimplicity of the category PervGO(Gr,k) implies in particular that the natural maps J! (λ, k) → J!∗ (λ, k) → J∗ (λ, k) are isomorphisms. (2) For any coefficients k, we have (
k if n = 0 and λ = µ; HomDb PervGO (Gr,k) J! (λ, k), J∗ (µ, k)[n] = 0 otherwise. In fact, to prove this it suffices to prove the similar claim for perverse sheaves on Z, where Z ⊂ Gr is any closed finite union of GO -orbits containing Grλ and Grµ . In the case k is a field, this claim is a consequence of Lemma 3.1 (or rather its version for Z). The case when k is the ring of integers in a finite extension of Qℓ follows. Indeed, since the category PervGO (Z, k) has enough projectives we can consider the complex of k-modules M = R HomPervGO (Z,k) (J! (λ, k), J∗ (µ, k)). If k0 is the residue field of k, it is not difficult (using the results of [MV07, §8 & §10], ∼ and in particular the fact that k0 ⊗L k J? (λ, k) = J? (λ, k0 ) for ? ∈ {!, ∗}, see [MV07, Prop. 8.1]) to check that L k0 ⊗ k M ∼ = R HomPervGO (Z,k0 ) (J! (λ, k0 ), J∗ (µ, k0 )). J.É.P. — M., 2019, tome 6 An Iwahori-Whittaker model for the Satake category 717 We deduce that the left-hand side is isomorphic to k0 in the derived category of k0 -vector spaces; this implies that M is isomorphic to k in the derived category of k-modules. In Section 4 we will also encounter some Iu− -equivariant perverse sheaves on Gr, where Iu− is the pro-unipotent radical of I − . In particular, we have F − Gr = Iu · Lµ , µ∈X ∨ Gr and we will denote by ∆Gr µ (k), resp. ∇µ (k), the standard, resp. costandard, perverse sheaf associated with µ, i.e., the !-direct image (resp. ∗-direct image) of the constant perverse sheaf of rank 1 on Iu− · Lµ . (These objects are perverse sheaves thanks to [BBDG82, Cor. 4.1.3], because the orbits Iu− · Lµ are affine spaces.) 3.2. Category of Iwahori-Whittaker perverse sheaves. — We now denote by I + ⊂ GO the Iwahori subgroup associated with B + . We also denote by Iu+ the pro-unipotent radical of I + , i.e., the inverse image of U + under the map I + → B + . We assume that there exists a primitive p-th root of unity in k, and fix one. This choice determines a character ψ of the prime subfield of F (with values in k× ), and we denote by Lkψ the corresponding Artin-Schreier local system on Ga . (Below, some arguments using Verdier duality will also involve the Artin-Schreier local system Lk−ψ associated with the character ψ −1 ; clearly these two versions play similar roles.) We also consider the “generic” character χ : U + → Ga defined as the composition Q Y + α uα −−− U + −→ −→ U + /[U + , U + ] ←−− Ga −−−→ Ga , ∼ α∈∆s and denote by χI + its composition with the projection Iu+ ։ U + . We can then define the “Iwahori-Whittaker” derived category b DIW (Gr, k) as the (Iu+ , χ∗I + (Lkψ ))-equivariant constructible derived category of k-sheaves on Gr (see e.g. [AR16, App. A] for a review of the construction of this category). This category admits a perverse t-structure, whose heart will be denoted PervIW (Gr, k), and moreover the “realization functor” b Db PervIW (Gr, k) −→ DIW (Gr, k) is an equivalence of triangulated categories. Note that any ring of coefficients considered above appears in an ℓ-modular triple (K, O, L) where K is a finite extension of Qℓ , O is its ring of integers, and L is the residue field of O. In this setting the embedding O ֒→ K and the projection O → L induce bijections between the p-th roots of unity in O, K and L. Therefore, choosing a primitive root in any of these rings provides primitive roots in all three rings, and we can then consider extension of scalars functors L b b K ⊗O (−) : DIW (Gr, O) −→ DIW (Gr, K), L b b L ⊗O (−) : DIW (Gr, O) −→ DIW (Gr, L). J.É.P. — M., 2019, tome 6 718 R. Bezrukavnikov, D. Gaitsgory, I. Mirković, S. Riche & L. Rider These functors will play a crucial role in our arguments below. (Here the first functor is t-exact, but the second one is only right t-exact.) For λ ∈ X ∨ we set Xλ := I + · Lλ . Then again we have F Gr = Xλ . λ∈X ∨ Lemma 3.3. — The orbit Xλ supports an (Iu+ , χ∗I + (Lkψ ))-equivariant local system iff λ ∈ X∨ ++ . λ Sketch of proof. — Let λ ∈ X ∨ + , and consider the affine bundle Gr → G/Pλ (see Secλ tion 2.2). The decomposition of Gr in Iu+ -orbits is obtained by pullback from the decomposition of G/Pλ into U + -orbits; in particular, for µ ∈ Wf · λ, Xµ supports an (Iu+ , χ∗I + (Lkψ ))-equivariant local system iff its image in G/Pλ is a free U + -orbit. ∨ If λ ∈ / X∨ ++ there is no such orbit in G/Pλ , and if λ ∈ X ++ there is exactly one, corresponding to Xλ .  λ Note that if λ ∈ X ∨ ++ , since Xλ is open dense in Gr we have (3.1) dim(Xλ ) = hλ, 2ρi and Xλ ⊂ Xµ For λ ∈ X∨ ++ iff λ 4 µ. we will denote by ∆IW λ (k), resp. ∇IW λ (k), the standard, resp. costandard, (Iu+ , χ∗I + (Lkψ ))-equivariant perverse sheaf on Gr associated with λ, i.e., the !-extension, resp. ∗-extension, to Gr of the free rank-1 (Iu+ , χ∗I + (Lkψ ))-equivariant perverse sheaf on Xλ . (Once again, these objects are perverse sheaves thanks to [BBDG82, Cor. 4.1.3].) We will also denote by ICIW λ (k) the image of any generator of the rank-1 free k-module IW HomPervIW (Gr,k) (∆IW λ (k), ∇λ (k)). If k is a field then ICIW λ (k) is a simple perverse sheaf. Note that since ς is minimal in X ∨ ++ for 4, we have (3.2) IW IW ∆IW ς (k) = ∇ς (k) = ICς (k). Lemma 3.4. — Assume that k is a field of characteristic 0. Then the i-th cohomology of the stalks of ICIW λ (k) vanish unless i ≡ dim Xλ (mod 2). Sketch of proof. — Since the morphism π is smooth, by standard properties of perverse sheaves (see e.g. [BBDG82, §4.2.6]) it suffices to prove a similar statement on Fl instead of Gr. Now the Decomposition Theorem implies that all simple (Iu+ , χ∗I + (Lkψ ))equivariant perverse sheaves on Fl can be obtained from the one corresponding to the J.É.P. — M., 2019, tome 6 An Iwahori-Whittaker model for the Satake category 719 orbit of the base point by convolving on the right with I − -equivariant simple perverse sheaves on Fl corresponding to orbits of dimension either 0 or 1. Standard arguments (going back at least to [Spr82]) show that these operations preserve the parity-vanishing property of stalks, and the claim follows.  Remark 3.5. — Assume that k is a field. Following [JMW14](4) we will say that an b object of DIW (Gr, k) is even, resp. odd, if its restriction and corestriction to each stratum is concentrated in even, resp. odd, degrees, and that it is parity if it is isomorphic to a direct sum F ⊕ F′ with F even and F′ odd. Using this language, Lemma 3.4 states that if char(k) = 0 then the objects ICIW λ (k) are parity, of the same parity as dim(Xλ ). Corollary 3.6. — Assume that k is a field. The category PervIW (Gr, k) is a highest ∨ IW weight category with weight poset (X ∨ ++ , 4), standard objects {∆λ (k) : λ ∈ X ++ }, ∨ and costandard objects {∇IW λ (k) : λ ∈ X ++ }. Moreover, if char(k) = 0 then this category is semisimple. Proof. — The first claim is standard, as e.g. in [BGS96, §3.3]. For the second claim, we observe that the orbits Xλ (for λ ∈ X ∨ ++ ) have dimensions of constant parity on each connected component of Gr, see (3.1). Using this and Lemma 3.4, the semisimplicity can be proved exactly as in the case of the category PervGO (Gr, k). Namely, we have to prove that IW IW Ext1PervIW (Gr,k) (ICIW (k), ICIW b (Gr,k) (ICλ λ (k), ICµ (k)) = HomDIW µ (k)[1]) vanishes for any λ, µ. If Xλ and Xµ belong to different connected components of Gr IW then this claim is obvious; otherwise ICIW λ (k) and ICµ (k) are either both even or both odd (see Remark 3.5), so that the desired vanishing follows from [JMW14, Cor. 2.8].  Remark 3.7 (1) Once Corollary 3.6 is known, one can refine Lemma 3.4 drastically: if k is a field of characteristic 0, then the simple perverse sheaves ICIW λ (k) are clean, in the sense that if iµ : Xµ → Gr is the embedding, for any µ 6= λ we have

IW ! (3.3) i∗µ ICIW λ (k) = iµ ICλ (k) = 0. In fact, as in Remark 3.2(1), the semisimplicity claim in Corollary 3.6 implies that the natural maps IW IW ∆IW λ (k) −→ ICλ (k) −→ ∇λ (k) are isomorphisms, which is equivalent to (3.3). (See also [ABB+ 05, Cor. 2.2.3] for a different proof of (3.3).) This observation can be used to give a new proof of the main result of [FGV01], hence of the geometric Casselman-Shalika formula. (4)In [JMW14] the authors consider the setting of “ordinary” constructible complexes. However, as observed already in [RW18, §11.1] or [AMRW19, §6.2], their considerations apply verbatim in our Iwahori-Whittaker setting. J.É.P. — M., 2019, tome 6 720 R. Bezrukavnikov, D. Gaitsgory, I. Mirković, S. Riche & L. Rider (2) The same arguments as in Remark 3.2(2) show that for any coefficients k, any λ, µ ∈ X ∨ ++ and any n ∈ Z we have (
k if λ = µ and n = 0; IW IW HomDb PervIW (Gr,k) ∆λ (k), ∇µ (k)[n] = 0 otherwise. (In this case, the existence of enough projectives in PervIW (Z, k) can be checked using the techniques of [RSW14, §2].) 3.3. Statement. — We consider the functor b b Φ : DG (Gr, k) −→ DIW (Gr, k) O defined by GO Φ(F) = ∆IW F. ς (k) ⋆ In view of (3.2) (or, alternatively, arguing as in [BBM04]), in this definition ∆IW ς (k) IW IW can be replaced by ∇ς (k) or ICς (k); in particular this shows that the conjugate of Φ by Verdier duality is the similar functor using the character ψ −1 instead of ψ. Lemma 3.8. — The functor Φ is t-exact for the perverse t-structures. Proof. — In the case where k is a field, the claim follows from Lemma 2.3. The general case follows using an ℓ-modular triple (K, O, L) as in Section 3.2 with k = O, and the associated extension of scalars functors. Namely, if F is in PervGO (Gr, O), then L L K ⊗O Φ(F) ∼ = Φ(K ⊗O F) is perverse; hence any perverse cohomology object pHi (Φ(F)) with i 6= 0 is torsion. On the other hand, L L L ⊗O Φ(F) ∼ = Φ(L ⊗O F) p i lives in perverse degrees −1 and 0 since L ⊗L O F lives in these degrees. If H (Φ(F)) were nonzero for some i > 0, then taking i maximal with this property we would p i obtain that pHi (L ⊗L O Φ(F)) 6= 0, a contradiction. On the other hand, if H (Φ(F)) was nonzero for some i < 0, then taking i minimal with this property we would obtain that pHi−1 (L ⊗L  O Φ(F)) 6= 0, a contradiction again. We will denote by Φ0 : PervGO (Gr, k) −→ PervIW (Gr, k) the restriction of Φ to the hearts of the perverse t-structures, so that Φ0 is an exact functor between abelian categories. The main result of this section is the following theorem, whose proof will be given in the next subsection. Theorem 3.9. — The functor Φ0 is an equivalence of categories. J.É.P. — M., 2019, tome 6 An Iwahori-Whittaker model for the Satake category 721 3.4. Proof of Theorem 3.9. — As explained in Section 3.2, any ring of coefficients considered above appears in an ℓ-modular triple (K, O, L) where K is a finite extension of Qℓ , O is its ring of integers, and L is the residue field of O. Therefore we fix such a triple, and will treat the three cases in parallel. The starting point of our proof will be the geometric Casselman-Shalika formula, first conjectured in [FGKV98] and then proved independently in [FGV01] and [NP01] (see also Remark 3.7(1)). We consider the composition χK + χU + : UK −−−−→ (Ga )K −→ Ga , K where the second map is the “residue” morphism defined by X fi z i 7−→ f−1 . i∈Z + For µ ∈ X ∨ + we set Sµ := UK · Lµ ; then there exists a unique function χµ : Sµ → Ga + such that χµ (u · Lµ ) = χU + (u) for any u ∈ UK . The geometric Casselman-Shalika K ∨ formula states that for λ, µ ∈ X + we have (
K if λ = µ and i = h2ρ, λi; (3.4) Hic Sµ , J!∗ (λ, K)|Sµ ⊗K χ∗µ (LK ψ) = 0 otherwise. In the following lemma, we denote by χ′µ : z −ς Xµ+ς → Ga the unique function such that χ′µ (z −ς · u · Lµ+ς ) = χI + (u) for u ∈ Iu+ . Lemma 3.10. — For k ∈ {K, O, L}, for any λ, µ ∈ X ∨ + with λ 6= µ we have Hchλ+µ,2ρi Grλ ∩ (z −ς Xµ+ς ), (χ′µ )∗ (Lkψ )|Grλ ∩(z−ς Xµ+ς ) ) = 0. Proof. — For α ∈ ∆ and n ∈ Z>0 we denote by Uα,n ⊂ GO the image of the morphism x 7→ uα (xz n ). As explained e.g. in [NP01, Lem. 2.2], the action on Lµ+ς induces an isomorphism Y α∈∆+ hµ+ς,αi−1 Y ∼ Uα,j −→ Xµ+ς . j=0 Multiplying by z −ς we deduce that z −ς Xµ+ς ⊂ Sµ , and moreover that χ′µ is the restriction of χµ to z −ς Xµ+ς . By [MV07, Th. 3.2], we have dim(Grλ ∩Sµ ) = hλ+µ, ρi; it follows that dim(Grλ ∩ (z −ς Xµ+ς )) 6 hλ + µ, ρi. If this inequality is strict, then our vanishing claim is obvious (see e.g. [FK88, Th. I.8.8]). Otherwise, each irreducible component of Grλ ∩ (z −ς Xµ+ς ) of dimension hλ + µ, ρi is dense (hence open) in an irreducible component of Grλ ∩ Sµ ; therefore to prove the lemma it suffices to prove that (3.5) Hchλ+µ,2ρi Grλ ∩ Sµ , χ∗µ (Lkψ )|Grλ ∩Sµ ) = 0. J.É.P. — M., 2019, tome 6 722 R. Bezrukavnikov, D. Gaitsgory, I. Mirković, S. Riche & L. Rider Finally, we note that since we are considering the top cohomology, the O-module hλ+µ,2ρi Grλ ∩ Sµ , χ∗µ (LO Hc ψ )|Grλ ∩Sµ ) is free, and the natural morphisms hλ+µ,2ρi K ⊗O Hchλ+µ,2ρi Grλ ∩ Sµ , χ∗µ (LO Grλ ∩ Sµ , χ∗µ (LK ψ )|Grλ ∩Sµ ), ψ )|Grλ ∩Sµ ) −→ Hc hλ+µ,2ρi L ⊗O Hchλ+µ,2ρi Grλ ∩ Sµ , χ∗µ (LO Grλ ∩ Sµ , χ∗µ (LLψ )|Grλ ∩Sµ ) ψ )|Grλ ∩Sµ ) −→ Hc are isomorphisms; hence it suffices to prove (3.5) in case k = K. So, from now on we assume that k = K. The geometric Casselman-Shalika formula (3.4) implies that for any F in PervGO (Gr, K) we have Hic (Sµ , F|Sµ ⊗K χ∗µ (LK ψ )) = 0 for i 6= h2ρ, µi; therefore the morphism (jλ )! KGrλ [hλ, 2ρi] → J! (λ, K) induces an isomorphism  

∼ Hchλ+µ,2ρi Sµ , (jλ )! KGrλ |S ⊗K χ∗µ (LK ) −→ Hchµ,2ρi Sµ , J! (λ, K) ⊗K χ∗µ (LK ψ ψ) . µ Now we have J! (λ, K) ∼ = J!∗ (λ, K) by Remark 3.2(1); hence the right-hand side vanishes if λ 6= µ by (3.4). On the other hand, the base change theorem shows that the
hλ+µ,2ρi Grλ ∩ Sµ , χ∗µ (LK left-hand side identifies with Hc ψ )|Grλ ∩Sµ ; we have therefore proved (3.5) in this case, hence the lemma.  Proposition 3.11. — For k ∈ {K, L}, for any λ, µ ∈ X ∨ + with λ 6= µ we have
HomPervIW (Gr,k) Φ0 (J! (λ, k)), ∇IW µ+ς (k) = 0. Proof. — First, by exactness of Φ we see that the morphism (jλ )! kGrλ [hλ, 2ρi] → J! (λ, k) induces an isomorphism IW HomDIW b (Gr,k) Φ((jλ )! kGrλ [hλ, 2ρi]), ∇µ+ς (k)

∼ −→ HomPervIW (Gr,k) Φ0 (J! (λ, k)), ∇IW µ+ς (k) . Let J be the stabilizer of the point Lς in Iu+ . Then χ∗I + (Lkψ ) is trivial on J, so that we have a forgetful functor b (Gr, k) −→ DJb (Gr, k). DIW By the same considerations as in Section 2.3, this functor admits a left adjoint, denoted (I + ,χ ) IndJ u I + . Moreover, since J is pro-unipotent the forgetful functor DJb (Gr, k) → Dcb (Gr, k) is fully faithful. If we denote by Fς the direct image under the automorphism x 7→ z ς · x of Gr, then from the definition we see that ! (3.6) (I + ,χI + ) Φ(F) = !IndJ u O ◦ Fς ◦ ForG z −ς Jz ς (F)[−hς, 2ρi] b for any F in DG (Gr, k). In the setting of the proposition, we deduce an isomorphism O IW HomDIW b (Gr,k) Φ((jλ )! kGrλ [hλ, 2ρi]), ∇µ+ς (k)

IW ∼ = HomDcb (Gr,k) (jλ )! kGrλ [hλ, 2ρi], F−1 ς (∇µ+ς (k))[hς, 2ρi] . J.É.P. — M., 2019, tome 6 An Iwahori-Whittaker model for the Satake category 723 IW ′ ∗ k Now F−1 ς (∇µ+ς (k)) identifies with the ∗-pushforward of (χµ ) (Lψ )[hµ + ς, 2ρi] under the embedding z −ς Xµ+ς → Gr. Hence, by the base change theorem, the right-hand side identifies with Hhµ+2ς−λ,2ρi (Grλ ∩ z −ς Xµ+ς , a! (χ′µ )∗ (Lkψ )), where a : Grλ ∩ z −ς Xµ+ς ֒→ z −ς Xµ+ς is the embedding. So, we now need to show that Hhµ+2ς−λ,2ρi (Grλ ∩ z −ς Xµ+ς , a! (χ′µ )∗ (Lkψ )) vanishes. If b denotes the unique map z −ς Xµ+ς → pt, then by Verdier duality we have Hhµ+2ς−λ,2ρi (Grλ ∩ z −ς Xµ+ς , a! (χ′µ )∗ (Lkψ ))∗ = Hhµ+2ς−λ,2ρi (b∗ a! (χ′µ )∗ (Lkψ ))∗ ∼ = Hhλ−2ς−µ,2ρi (b! a∗ Dz−ς Xµ+ς ((χ′µ )∗ (Lkψ ))). Now since z −ς Xµ+ς is smooth of dimension hµ + ς, 2ρi we have an isomorphism Dz−ς Xµ+ς ((χ′µ )∗ (Lkψ )) ∼ = (χ′µ )∗ (Lk−ψ )[2hµ + ς, 2ρi], which shows that Hhµ+2ς−λ,2ρi (Grλ ∩ z −ς Xµ+ς , a! (χ′µ )∗ (Lkψ ))∗ ∼ = Hchµ+λ,2ρi (Grλ ∩ z −ς Xµ+ς , a∗ (χ′µ )∗ (Lk−ψ )). The right-hand side vanishes by Lemma 3.10, hence so does the left-hand side, which completes the proof.  We can finally give the proof of Theorem 3.9. Proof of Theorem 3.9. — Let τ : GK → Gr be the projection. Let λ ∈ X ∨ + , and denote by mς,λ the restriction of mGr to τ −1 (Grς ) ×GO Grλ (see Section 2.4 for the notation). Then it is well known that: – mς,λ takes values in Grλ+ς = Xλ+ς ; – its restriction to the preimage of Xλ+ς is an isomorphism; – this preimage is contained in τ −1 (Xς ) ×GO Grλ . These properties imply that the perverse sheaf Φ0 (J! (λ, k)) is supported on Xλ+ς , and that its restriction to Xλ+ς is a perversely shifted local system of rank 1. The same comments apply to Φ0 (J∗ (λ, k)). Hence there exist canonical morphisms 0 fλk : ∆IW λ+ς (k) −→ Φ (J! (λ, k)) and gλk : Φ0 (J∗ (λ, k)) −→ ∇IW λ+ς (k) whose restrictions to Xλ+ς are isomorphisms. We claim that fλk is an isomorphism. First we note that all of our constructions are compatible with extension-of-scalars functors in the obvious sense (see in particular [MV07, Prop. 8.1] for the case of J! (λ, k); the case of the Whittaker standard object is much easier since no perverse truncation is involved). If k ∈ {K, L}, by Proposition 3.11 we know that Φ0 (J! (λ, k)) has no quotient of the form ICIW µ+ς (k) with µ 6= λ; O k L therefore fλ is surjective. The surjectivity of fλ implies that fλ must be surjective K also. On the other hand, by Remark 3.7(1) the object ∆IW λ+ς (K) is simple; hence fλ O is injective, which implies that ker(fλ ) is a torsion object. Since this object embeds O in the torsion-free object ∆IW λ+ς (O), it must be zero. We finally obtain that fλ is an K L isomorphism, so that fλ and fλ are isomorphisms as well. J.É.P. — M., 2019, tome 6 724 R. Bezrukavnikov, D. Gaitsgory, I. Mirković, S. Riche & L. Rider Once we know that fλk is an isomorphism, by Verdier duality (see the comments preceding Lemma 3.8) we deduce that gλk is an isomorphism as well. (More precisely, we use the claim about fλk in the setting where ψ is replaced by ψ −1 , and the fact that DGr (J! (λ, k)) = J∗ (λ, k), see [MV07, Prop. 8.1(c)].) Now we conclude the proof as follows. Since Φ0 is exact, it induces a functor Db (Φ0 ) : Db PervGO (Gr, k) −→ Db PervIW (Gr, k). We will prove that Db (Φ0 ) is an equivalence, which will imply that Φ0 is an equivalence as well, hence will conclude the proof. It is not difficult to see that the category Db PervGO (Gr, k), resp. Db PervIW (Gr, k), is generated as a triangulated category ∨ IW by the objects {J! (λ, k) : λ ∈ X ∨ + }, resp. by the objects {∆λ+ς (k) : λ ∈ X + }, as well ∨ IW as by the objects {J∗ (λ, k) : λ ∈ X ∨ + }, resp. by the objects {∇λ+ς (k) : λ ∈ X + }. ∨ Hence to conclude it suffices to prove that for any λ, µ ∈ X + and any n ∈ Z the functor Φ0 induces an isomorphism ExtnPervG O (Gr,k) (J! (λ, k), J∗ (µ, k)) ∼ IW −→ ExtnPervIW (Gr,k) (∆IW λ+ς (k), ∇µ+ς (k)). However, this is clear from Remark 3.2(2) and Remark 3.7(2).  Remark 3.12 (1) One can explicitly describe the inverse to Φ0 , as follows. In view of (3.6), the functor (I + ,χI + ) −1 u O Ψ := ∗IndG z −ς Jz ς ◦ Fς ◦ ForJ b b [hς, 2ρi] : DIW (Gr, k) −→ DG (Gr, k) O is right adjoint to Φ. Since Φ is exact, Ψ is left exact, and the functor Ψ0 := pH0 ◦ Ψ| PervIW (Gr,k) is right adjoint to Φ0 . Since Φ0 is an equivalence, Ψ0 must be its inverse. (2) From the point of view suggested by the Finkelberg-Mirković conjecture (see Section 1.2), the isomorphisms fλk and gλk are geometric analogues of the isomorphism stated in [Jan03, Prop. II.3.19]. 4. Applications We continue with the assumptions of Sections 2–3; but from now on (except in Remark 4.17) for simplicity we assume that k is a field. 4.1. Some perverse sheaves associated with regular W -orbits in X ∨ . — Consider the flag variety B = G/B − , and let U − be the unipotent radical of B − . Recall that the category PervU − (B, k) of U − -equivariant perverse sheaves on B has a natural structure of highest weight category, see [BGS96]. Moreover, the projective cover Pe of the skyscraper sheaf at the point B − /B − is also an injective and a tilting object; see e.g. [BR18] for details and references. − For any λ ∈ X ∨ ++ , in the notation of Section 2.2 we have Pλ = B , so that the map pλ defined there has codomain B. We set Pλ := (pλ )∗ (Pe )[dim(Grλ ) − dim(B)]. J.É.P. — M., 2019, tome 6 An Iwahori-Whittaker model for the Satake category 725 Then Pλ is a perverse sheaf on Grλ , and it is Iu− -equivariant. We will consider the objects Π!λ := (jλ )! Pλ , Π∗λ := (jλ )∗ Pλ . Lemma 4.1. — The objects Π!λ and Π∗λ are Iu− -equivariant perverse sheaves on Gr. Proof. — As recalled above, Pe admits both a standard filtration and a costandard filtration. It follows that Π!λ , resp. Π∗λ , admits a filtration (in the sense of triangulated Gr categories) with subquotients of the form ∆Gr v(λ) , resp. ∇v(λ) , for v ∈ Wf . Since these objects are perverse sheaves, it follows that Π!λ and Π∗λ are perverse. The fact that these perverse sheaves are Iu− -equivariant readily follows from the fact that Pλ is Iu− -equivariant.  Lemma 4.2. — There exists a canonical isomorphism Π!ς ∼ = Π∗ς . Proof. — This claim is proved in the D-modules setting in [FG06, Prop. 15.2]. The arguments apply verbatim in the present context.  In view of this lemma, the object Π!ς = Π∗ς will be denoted Πς . Recall now that we have the “negative” Iwahori subgroup I − (associated with the negative Borel B − ), but also the “positive” Iwahori subgroup I + (associated with the positive Borel B + ) which was used to define the Iwahori-Whittaker category. Let I◦ be the kernel of the morphism GO → G. Then I◦ = Iu− ∩ Iu+ , and the morphism χI + is trivial on I◦ . It follows that there exists a natural forgetful functor b b ForIW I◦ : DIW (Gr, k) −→ DI◦ (Gr, k). We also have a forgetful functor I− ForIu◦ : DIb− (Gr, k) −→ DIb◦ (Gr, k) u I− I− which admits both a left and a right adjoint, denoted !IndIu◦ and ∗IndIu◦ respectively, see Section 2.3. We set I− b b AvIu− ,∗ := ∗IndIu◦ ◦ ForIW I◦ : DIW (Gr, k) −→ DI − (Gr, k); u AvIu− ,! := ! I− IndIu◦ ◦ ForIW I◦ : b DIW (Gr, k) Lemma 4.3. — For any λ ∈ X ∨ ++ we have
! ∼ IW Πλ = AvIu− ,! ∆λ (k) [− dim U − ], −→ DIb− (Gr, k). u
− Π∗λ ∼ = AvIu− ,∗ ∇IW λ (k) [dim U ]. Proof. — Consider the constructible equivariant derived categories b DU − (B, k) b and D(U + ,χ∗ (Lk )) (B, k) ψ − + of sheaves on B which are U -equivariant and (U , χ∗ (Lkψ ))-equivariant respectively. These categories are related by functors b b AvU − ,∗ , AvU − ,! : D(U + ,χ∗ (Lk )) (B, k) −→ DU − (B, k). ψ J.É.P. — M., 2019, tome 6 726 R. Bezrukavnikov, D. Gaitsgory, I. Mirković, S. Riche & L. Rider (U + ,χ) Moreover, if ∆e denotes the !-extension of the shift by dim U + of the unique + ∗ k simple (U , χ (Lψ ))-equivariant local system on the orbit U + B − /B − ⊂ B (which also coincides with the ∗-extension of this local system), then it is well known that we have isomorphisms

+ + ,χ) (4.1) AvU − ,! ∆e(U ,χ) [− dim U − ] ∼ [dim U − ], = AvU − ,∗ ∆(U = Pe ∼ e see [BY13, §4.6] or [AR16, Lem. 5.18]. Now, the functors AvIu− ,∗ and AvIu− ,! have versions for the variety Grλ , which we will denote similarly. Clearly we have isomorphisms of functors (4.2) AvIu− ,∗ ◦ (jλ )∗ ∼ = (jλ )∗ ◦ AvIu− ,∗ , AvIu− ,! ◦ (jλ )! ∼ = (jλ )! ◦ AvIu− ,! . Moreover, the map pλ induces a morphism Iu− ×I◦ Grλ → U − × B compatible with the action maps in the obvious way. Using the base change theorem (and the fact that pλ is smooth), we deduce isomorphisms of functors (4.3) AvIu− ,∗ ◦ (pλ )∗ ∼ = (pλ )∗ ◦ AvU − ,∗ , AvIu− ,! ◦ (pλ )∗ ∼ = (pλ )∗ ◦ AvU − ,! . Since ∗ (U ∆IW λ (k) = (jλ )! (pλ ) ∆e + ,χ) [dim Grλ − dim B] and ∗ (U ∇IW λ (k) = (jλ )∗ (pλ ) ∆e + ,χ) [dim Grλ − dim B] the isomorphisms of the lemma finally follow from (4.2), (4.3) and (4.1).  The following proposition is the main result of this subsection. Proposition 4.4. — For any λ ∈ X ∨ + , we have isomorphisms Πς ⋆GO J! (λ, k) ∼ = Π!λ+ς , Πς ⋆GO J∗ (λ, k) ∼ = Π∗λ+ς . Proof. — The first isomorphism is obtained by applying the functor AvIu− ,! [− dim U − ] to the isomorphism ∼ ∆IW (k) ∆IW ⋆GO J! (λ, k) = ς λ+ς (see the proof of Theorem 3.9) and then using Lemma 4.3 and the fact that AvIu− ,! b commutes with the functor (−) ⋆GO F for any F in DG (Gr, k). The proof of the O second isomorphism is similar, using AvIu− ,∗ instead of AvIu− ,! .  Remark 4.5. — Consider the restrictions Av0Iu− ,! , Av0Iu− ,∗ : PervIW (Gr, k) −→ DIb− (Gr, k) u of AvIu− ,! and AvIu− ,∗ to the heart of the perverse t-structure. Then there exists an isomorphism of functors ∼ Av0Iu− ,! [− dim U − ] −→ Av0Iu− ,∗ [dim U − ], and moreover these functors take values in PervIu− (Gr, k) and send tilting perverse sheaves to tilting perverse sheaves. (Here the highest weight structure on J.É.P. — M., 2019, tome 6 An Iwahori-Whittaker model for the Satake category 727 PervIu− (Gr, k) is the standard one, as considered e.g. in [BGS96, §3.3].) In fact, as in the proof of Proposition 4.4, for any F in PervGO (Gr, k) we have

Av0Iu− ,! [− dim U − ] ◦ Φ0 (F) ∼ = Πς ⋆GO F ∼ = Av0Iu− ,∗ [dim U − ] ◦ Φ0 (F), and then the isomorphism follows from the fact that Φ0 is an equivalence of categories, see Theorem 3.9. Once this fact is established, it follows from Lemma 4.3 that this functor sends standard perverse sheaves, resp. costandard perverse sheaves, to perverse sheaves admitting a standard filtration, resp. a costandard filtration (see the proof of Lemma 4.1); the other claims follow. 4.2. Interpretation in terms of the Weyl character formula. — The isomorphisms in Proposition 4.4 can be considered a geometric version of the Weyl character formula as stated by Lusztig in [Lus83, (6.3)], in the following way. Let Z : PervGO (Gr, k) −→ PervI − (Fl, k) be the “central” functor constructed (in terms of nearby cycles) in [Gai01]. Lemma 4.6. — There exists a canonical isomorphism of functors ∗ O ∼ ∗ IndG I− ◦ Z = π . O Proof. — By definition, the functor ∗IndG I − is given by convolution with kGO /I − on the left. Since Z (F) is central for any F in PervGO (Gr, k) (see [Gai01, Th. 1(b)]), ∗ O IndG I − ◦ Z is therefore the composition of Z with convolution on the right with kGO /I − , which itself identifies with the functor π ∗ π∗ . The claim follows, since π∗ ◦ Z ∼  = id by [Gai01, Th. 1(d)]. Using this lemma we obtain the following reformulation of Proposition 4.4. Proposition 4.7. — For any λ ∈ X ∨ + there exist canonical isomorphisms

− π ∗ Πς [dim B] ⋆I Z (J! (λ, k)) ∼ = π ∗ Π!λ+ς [dim B] ,
−
π ∗ Πς [dim B] ⋆I Z (J∗ (λ, k)) ∼ = π ∗ Π∗λ+ς [dim B] . Proof. — By Lemma 2.6 we have
−
G π ∗ Πς [dim B] ⋆I Z (J! (λ, k)) ∼ = Πς [dim B] ⋆GO ∗IndI −O Z (J! (λ, k)) . Using Lemma 4.6, we deduce an isomorphism
− π ∗ Πς [dim B] ⋆I Z (J! (λ, k)) ∼ = Πς [dim B] ⋆GO π ∗ (J! (λ, k)).
Now the right-hand side is clearly isomorphic to π ∗ Πς [dim B] ⋆GO J! (λ, k) , and then the first isomorphism of the proposition follows from Proposition 4.4. The proof of the second isomorphism is similar.  The Grothendieck group of the category PervIu− (Fl, k), resp. PervGO (Gr, k), identifies naturally with the (integral) group ring Z[W ] of W , resp. with its center Z[X ∨ ]Wf , and under this isomorphism the right convolution with objects of the form Z (−) corresponds to the natural multiplication map, see [Gai01, §0.1]. (See also [AB09] for J.É.P. — M., 2019, tome 6 728 R. Bezrukavnikov, D. Gaitsgory, I. Mirković, S. Riche & L. Rider this point of view.) Under these identifications, the isomorphisms of Proposition 4.7 are categorical incarnations of the identity [Lus83, (6.3)]. 4.3. Tilting objects in the Satake category. — Recall the notion of parity comb plexes in DIW (Gr, k) considered in Remark 3.5. In certain proofs of this subsection we will also consider the (Iu+ , χ∗I + (Lkψ ))-equivariant constructible derived category b of Fl, which we will denote DIW (Fl, k). Of course, we can also consider parity complexes in this category, as well as in the I − -equivariant derived category DIb− (Fl, k), b or in the GO -constructible derived category D(G (Gr, k). (Note that, by definition, O) b an object of DI − (Fl, k) is parity iff its image in the I − -constructible derived category ∨ b D(I − ) (Fl, k) is parity.) In particular, for any λ ∈ X + , we denote by Eλ the unique b indecomposable parity complex in the category D(G (Gr, k) supported on Grλ and O) whose restriction to Grλ is kGrλ [dim Grλ ] (see [JMW14, Th. 2.12 & §4.1]). Since H•GO (pt; k) might not be concentrated in even degrees, in general the theory b of [JMW14] does not apply in DG (Gr, k). This difficulty will be remedied by the O following lemma. b Lemma 4.8. — Any parity object E in D(G (Gr, k) is a direct summand of a parity O) b ′ (Gr, k) → object E which belongs to the essential image of the functor ForGO : DG O b D(GO ) (Gr, k). Proof. — Of course we can assume that E = Eλ for some λ ∈ X ∨ + . Recall that the forgetful functor ForI − sends indecomposable parity objects to indecomposable parity objects (see [MR18, Lem. 2.4]). In view of the classification of such objects in the I − -equivariant and I − -constructible derived categories, this means that any I − -constructible parity complex on Gr belongs to the essential image of ForI − . In particular, there exists a parity complex F in DIb− (Gr, k) such that Eλ ∼ = ForI − (F). ′ O (F)). Then E is parity as a convolution of parity Now we set E′ := ForGO (∗IndG I− complexes, see [JMW14, Th. 4.8]. And since this object is supported on Grλ and has nonzero restriction to Grλ , it must admit a cohomological shift of Eλ as a direct summand.  Remark 4.9. — If char(k) is not a torsion prime for G, then H•GO (pt; k) is concentrated b in even degrees; see [JMW14, §2.6]. In this case the parity objects in DG (Gr, k) are O b well behaved, and one can easily show that in fact any parity object in D(G (Gr, k) O) belongs to the essential image of the functor ForGO . b b Recall that the forgetful functor ForGO : DG (Gr, k) → D(G (Gr, k) restricts O O) to an equivalence between GO -equivariant and GO -constructible perverse sheaves, b see [MV07, Prop. 2.1] (or [BR18, Prop. 1.10.8]). Therefore for any F in D(G (Gr, k) O) p n and any n ∈ Z, the perverse sheaf H (F) is GO -equivariant. The main result of this section is the following. J.É.P. — M., 2019, tome 6 An Iwahori-Whittaker model for the Satake category 729 Theorem 4.10. — For any n ∈ Z and λ ∈ X ∨ + , the GO -equivariant perverse sheaf p n H (Eλ ) is tilting in the highest weight category PervGO (Gr, k). In particular, the indecomposable tilting object associated with λ is a direct summand of pH0 (Eλ ). Remark 4.11 (1) Theorem 4.10 was stated as a conjecture (in the case n = 0) in [JMW16]. (2) If char(k) is good for G, it is known that the objects Eλ are actually perverse, see [MR18]. (This property was proved earlier in [JMW16] under slightly stronger assumptions; it is known to be false in bad characteristic, see [JMW16].) Hence in Theorem 4.10 we in fact know that the indecomposable tilting object associated with λ is pH0 (Eλ ) = Eλ . In general, it seems natural to expect that pH0 (Eλ ) is indecomposable; but we do not have a proof of (or strong evidence for) this fact. (3) Since our proof of Theorem 4.10 relies on Theorem 3.9, we have stated it with the same assumptions on G. However, once it is known in this generality standard arguments allow to extend its validity to general connected reductive groups; see e.g. [JMW16, §3.4] for details. Similarly, the analogous claim in the setting of the classical topology on the complex counterpart of Gr follows from its étale version using the general considerations of [BBDG82, §6.1]. The proof of Theorem 4.10 requires a few preliminaries. We start with the following observation, which will be crucial for us. b (Gr, k) are exactly the direct sums of Proposition 4.12. — The parity objects in DIW cohomological shifts of tilting perverse sheaves. Proof. — As already noticed in the proof of Corollary 3.6, the strata Xλ ⊂ Gr supporting (Iu+ , χ∗I + (Lkψ ))-equivariant local systems (i.e., those with λ ∈ X ∨ ++ ) have dimensions of constant parity on each connected component of Gr. Therefore, the tilting objects in the highest weight category PervIW (Gr, k) are also parity. By unicity, they must then coincide with the “parity sheaves” (or, in another terminology, normalized indecomposable parity complexes) of [JMW14, Def. 2.14]. The claim follows, since any parity complex is a direct sum of cohomological shifts of such objects.  Next we observe that the parity property is preserved under convolution, in the following sense. b (Fl, k) and G ∈ DIb− (Fl, k) are parity complexes, then Lemma 4.13. — If F ∈ DIW − b (Fl, k) is a parity complex. F ⋆I G ∈ DIW Proof. — In view of the description of parity complexes in [JMW14, §4.1], the claim follows from standard arguments going back at least to [Spr82]. In fact it suffices to treat the case G = kFlw when ℓ(w) ∈ {0, 1}, which can be done “by hand” as in [Spr82].  b b Lemma 4.14. — If F ∈ DIW (Gr, k) is parity and G ∈ DG (Gr, k) is such that O GO b ForGO (G) is parity, then F ⋆ G ∈ DIW (Gr, k) is parity. J.É.P. — M., 2019, tome 6 730 R. Bezrukavnikov, D. Gaitsgory, I. Mirković, S. Riche & L. Rider Proof. — The natural projection π : Fl → Gr (a smooth and projective morphism) is Iu+ -equivariant, hence induces functors b b π ∗ : DIW (Gr, k) −→ DIW (Fl, k), b b π∗ : DIW (Fl, k) −→ DIW (Gr, k). The projection formula shows that F is a direct summand in π∗ (π ∗ F), and by Lemma 2.5 we have − G π∗ (π ∗ F) ⋆GO G ∼ = π ∗ (F) ⋆I ForI −O (G). O Hence to conclude it suffices to prove that π ∗ (F) ⋆I ForG I − (G) is parity. However we have
− I− ∗ O O ∼ ∗ π (ForG π ∗ π ∗ (F) ⋆I ForG I − (G) = π (F) ⋆ I − (G)). − O Since π ∗ F and π ∗ (ForG are parity (because π is smooth), Lemma 4.13 implies I − (G))
− GO ∗ I− ∗ O that π π (F) ⋆ ForI − (G) is parity. We deduce that π ∗ (F) ⋆I ForG I − (G) is parity, as expected.  b Corollary 4.15. — Let E be in DG (Gr, k), and assume that ForGO (E) is parity. O b Then Φ(E) is parity in DIW (Gr, k). In particular, Φ0 (pHn (E)) is a tilting perverse sheaf for any n ∈ Z. Proof. — Since ∆IW ς (k) is parity (see (3.2)), the first claim follows from Lemma 4.14. The second claim follows from the facts that Φ is t-exact and that the perverse b (Gr, k) are tilting perverse sheaves, see cohomology objects of parity objects in DIW Proposition 4.12.  We can finally give the proof of Theorem 4.10. Proof of Theorem 4.10. — Since Φ0 is an equivalence of highest weight categories, to prove the first claim it suffices to prove that Φ0 (pHn (Eλ )) is tilting in the highest weight category PervIW (Gr, k). This follows from Lemma 4.8 and Corollary 4.15, since a direct summand of a tilting perverse sheaf is tilting. The second claim follows,  since pH0 (Eλ ) is supported on Grλ , and has nonzero restriction to Grλ . 4.4. Convolution and restriction of tilting objects. — In this subsection we will consider the affine Grassmannian for several reductive groups, so we write GrG instead of Gr. For P ⊂ G a parabolic subgroup containing B + , with Levi subgroup containing T denoted L, we denote by b b RG L : DGO (GrG , k) −→ DLO (GrL , k) the “renormalized” hyperbolic localization functor defined as follows. The connected ∨ components of GrL are in a canonical bijection with X ∨ /Z∆∨ L , where ∆L is the coroot system of (L, T ); the connected component associated with c will be denoted GrcL . We denote by UP+ the unipotent radical of P . Then for c ∈ X ∨ /Z∆∨ L we consider the subvariety Sc := (UP+ )K · GrcL of GrG . We denote the natural maps by sc σ GrG ←−− − Sc −−−c→ GrcL . J.É.P. — M., 2019, tome 6 An Iwahori-Whittaker model for the Satake category 731 Then if ∆L ⊂ ∆ is the root system of (L, T ), the functor RG L is defined as Ei h D P L α, c . (σc )! (sc )∗ − + α∈∆ r∆L c∈X ∨ /Z∆∨ L By work of Beı̆linson-Drinfeld [BD] this functor is known to be exact for the perverse t-structures; see [BR18, Lem. 1.15.1] for a more detailed proof. As a consequence of Theorem 4.10 (and its proof) we obtain the following result, which is a geometric version of a celebrated result on tilting modules due in full generality to Mathieu [Mat90]. (See [JMW16, §1.1] for more historical remarks and references on this result). In fact (as noted in the introduction), reasoning in the opposite direction, combining this result with the geometric Satake equivalence, our work can also be considered as providing a new proof of this representation-theoretic result. Theorem 4.16 (1) If F, G are tilting objects in PervGO (Gr, k), then so is F ⋆GO G. (2) If F is a tilting object in PervGO (GrG , k), then RG L (F) is a tilting object in PervLO (GrL , k). Proof (1) In view of Theorem 4.10, it suffices to show that if F, G are parity objects in b D(G (Gr, k), then pH0 (F)⋆GO pH0 (G) is a tilting perverse sheaf. In view of Lemma 4.8, O) b (Gr, k). Then it suffices to consider the case when G = ForGO (G′ ) for some G′ in DG O by exactness of convolution with GO -equivariant perverse sheaves (see Lemma 2.3) we have p H0 (F) ⋆GO pH0 (G) = pH0 (pH0 (F) ⋆GO G′ ). Hence, using the t-exact functor Φ of Section 3.3, we see that to conclude it suffices to prove that Φ(pH0 (pH0 (F) ⋆GO G′ )) ∼ = pH0 (Φ(pH0 (F) ⋆GO G′ )) ∼ = pH0 (Φ(pH0 (F)) ⋆GO G′ ) ∼ (where the second identification uses the canonical isomorphism Φ(M ⋆GO N) = GO b Φ(M) ⋆ N for M, N in DGO (Gr, k)) is a tilting object in PervIW (Gr, k). However Φ(pH0 (F)) is a tilting perverse sheaf by Theorem 4.10, hence it is also parity by Proposition 4.12. By Lemma 4.14, it follows that Φ(pH0 (F)) ⋆GO G′ is parity, hence that its perverse cohomology objects are tilting (see again Proposition 4.12), which finishes the proof. b (2) As in (1), it suffices to prove that if F is a parity object in D(G (Gr, k), then O) G p 0 G RL ( H (F)) is a tilting perverse sheaf. However, since RL is t-exact we have p 0 ∼p 0 G RG L ( H (F)) = H (RL (F)). By [JMW16, Th. 1.6], RG L (F) is a parity complex. Then the claim follows from Theorem 4.10.  J.É.P. — M., 2019, tome 6 732 R. Bezrukavnikov, D. Gaitsgory, I. Mirković, S. Riche & L. Rider Remark 4.17. — For simplicity, we have stated Theorem 4.10 only in the case k is a field. But the Satake equivalence also holds when k is the ring of integers in a finite extension of Qℓ , and the notion of tilting objects also makes sense for split reductive group schemes over such rings, see [Jan03, §§B.9–B.10]. Therefore we can consider the tilting objects in PervGO (Gr, k). On the other hand, the notion of parity objects b also makes sense in D(G (Gr, k), and their classification is similar in this setting; O) see [JMW14]. We claim that Theorem 4.10 also holds for this choice of coefficients. In fact, if k0 is the residue field of k, then it follows from [Jan03, Lem. B.9 & Lem. B.10] and the compatibility of the Satake equivalence with extension of scalars that an object F in PervGO (Gr, k) is tilting if and only if k0 ⊗L k F belongs to b (Gr, k), PervGO (Gr, k0 ) and is tilting therein. Now if E is a parity object in D(G O) then we have (4.4) L L k0 ⊗k pH0 (E) ∼ = pH0 (k0 ⊗k E). Indeed, assume that E is even, and supported on a connected component of Gr containing GO -orbits of even dimension. (The other cases are similar.) By [JMW16, Th. 1.6 and its proof], the complex L L ∼ G k0 ⊗ k R G T (E) = RT (k0 ⊗k E) is an even complex on the affine Grassmannian GrT ; therefore so is the complex RG T (E) by [JMW14, Prop. 2.37]. In view of [BR18, Lem. 1.10.7], this shows that pHn (E) = 0 and pHn (k0 ⊗L k E) = 0 unless n is even. Then (4.4) is an easy consequence of this observation. From (4.4) and the comments above we obtain the desired extension of Theorem 4.10. 4.5. Interpretation in terms of Donkin’s tensor product theorem. — In this subsection we assume that ℓ = char(k) is good for G. Recall the triangulated category b (Fl, k) introduced in Section 4.3. The Iu+ -orbits in Fl are parametrized in a natDIW ural way by W , and those which support an (Iu+ , χ∗I + (Lkψ ))-equivariant local system are the ones corresponding to the elements w ∈ W which are of minimal length in Wf w. In this case, we denote by EIW w the corresponding indecomposable parity object. As observed in Section 4.3 (see in particular Remark 4.11), under our present assumption, for any λ ∈ X ∨ + the object Φ(Eλ ) is indecomposable and parity. Therefore its pullback to Fl is also parity (by Lemma 4.13) and indecomposable (by [ACR18, Lem. A.5]). We deduce that (4.5) π ∗ Φ(Eλ )[dim B] ∼ = EIW tλ+ς . Using the functor Z considered in Section 4.2, this formula can also be interpreted as follows. Proposition 4.18. — For any λ ∈ X ∨ + , we have I− EIW Z (Eλ ) ∼ = EIW tς ⋆ tλ+ς . J.É.P. — M., 2019, tome 6 An Iwahori-Whittaker model for the Satake category 733 Proof. — As in the proof of Proposition 4.7, the claim follows from (4.5) using Lemma 2.6 and Lemma 4.6.  Let k be an algebraic closure of k, and assume that ℓ is strictly bigger than the Coxeter number of G. Then the formula of Proposition 4.18 is related to Donkin’s tensor product theorem for tilting modules of the Langlands dual k-group Gk∨ as follows. In [RW18, AR18a, AMRW19] the authors construct a “degrading functor” ), η : ParityIW (Fl◦ , k) −→ TiltPrin (G∨ k where Fl◦ is the connected component of the base point in Fl, ParityIW (Fl◦ , k) is the category of (Iu+ , χ∗I + (Lkψ ))-equivariant parity complexes on Fl◦ , and TiltPrin (Gk∨ ) denotes the category of tilting objects in the (non-extended) principal block of the category of finite-dimensional Gk∨ -modules. We expect that Donkin’s tensor product theorem (see [Jan03, §E.9]) can be explained geometrically by an isomorphism of complexes involving the functor Z (see also [AR18b, §9.3] for more details). In fact, from this point of view Proposition 4.18 is the geometric statement that underlies the isomorphism (4.6) T(ℓς) ⊗ T(λ)(1) ∼ = T(ℓς + ℓλ), where T(ν) is the indecomposable tilting Gk∨ -module of highest weight ν. 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Springer – “Quelques applications de la cohomologie d’intersection”, in Séminaire N. Bourbaki, Vol. 1981/82, Astérisque, vol. 92, Société Mathématique de France, Paris, 1982, Exp. no. 589, p. 249–273. J. Wang – “A new Fourier transform”, Math. Res. Lett. 22 (2015), no. 5, p. 1541–1562. Manuscript received 9th July 2018 accepted 11th July 2019 Roman Bezrukavnikov, Department of Mathematics, Massachusetts Institute of Technology Cambridge, MA 02139, USA E-mail : bezrukav@math.mit.edu Url : http://math.mit.edu/~bezrukav/ Dennis Gaitsgory, Harvard University 1 Oxford St, Cambridge, MA 02138, USA E-mail : gaitsgde@math.harvard.edu Url : http://www.math.harvard.edu/~gaitsgde/ Ivan Mirković, University of Massachusetts Amherst, MA, USA. E-mail : mirkovic@math.umass.edu Url : http://people.math.umass.edu/~mirkovic/ Simon Riche, Université Clermont Auvergne, CNRS, LMBP F-63000 Clermont-Ferrand, France E-mail : simon.riche@uca.fr Url : http://math.univ-bpclermont.fr/~riche/ Laura Rider, Department of Mathematics, University of Georgia Athens Georgia 30602, USA E-mail : laurajoy@uga.edu Url : https://faculty.franklin.uga.edu/laurajoy/ J.É.P. — M., 2019, tome 6