Tate motives on Witt vector affine flag varieties

Relying on recent advances in the theory of motives we develope a general formalism for derived categories of motives with Q-coefficients on perfect (ind-)schemes. As an application we give a motivic refinement of Zhu's geometric Satake equivalence for Witt vector affine Grassmannians in this set-up.

1. Introduction 1.1. Motivation. The Satake isomorphism ( [Gro98]) plays a foundational rôle in several branches of the Langlands program. Let p be a prime number, and fix p 1/2 ∈ Q. For a split reductive group G and a non-archimedean local field like Q p or F p ((t)), this isomorphism relates spherical functions pertaining to G to the representation theory of the Langlands dual group G: (1.1) The so-called spherical Hecke algebras at the right and left consist of finitely supported Z[p ±1/2 ]-valued functions on the G(Z p )-(resp. G(F p [[t]])-)double cosets in G(Q p ) (resp. G(F p ((t)))). In the middle, G denotes the Langlands dual group formed over Q using a fixed pinning (T, B, X) of G. By the classification of split reductive groups in terms of their root data, it can be described by switching characters and cocharacters, as well as roots and coroots. For example ( [Cog03]), GL n = GL n , SL n = PGL n and Sp 2g = SO 2g+1 . The ring R( G) is the Grothendieck Z[p ±1/2 ]-algebra of algebraic representations of this dual group. The chain of isomorphisms (1.1) is an instance of the resemblance ( [Wei79]) between Q p and F p ((t)), in which the uniformizer p corresponds to the uniformizer t. Despite being only a superficial similarity -given that the addition and multiplication in the two fields are severely different-the kinship is strong enough so that the characteristic functions of the double cosets G(Z p )p µ G(Z p ) and G(F p [[t]])t µ G(F p [[t]]) correspond to each other under (1.1). Here µ : G m → T is any dominant cocharacter.
The geometric Satake equivalence might be regarded, at the same time, as a geometrization and a categorification of the right hand isomorphism above: the right-most term is geometrized by means of the affine Grassmannian, whose F p -points are the quotient G(F p ((t)))/G(F p [[t]]). The categorification is achieved by relating the entire category of algebraic G-representations (as opposed to its Grothendieck ring) to certain equivariant perverse sheaves on the affine Grassmannian. This circle of ideas has been worked on by various authors including Lusztig [Lus83], Ginzburg [Gin00], Belinson-Drinfeld [BD99] and Mirković-Vilonen [MV07]. For the relation with (1.1) the reader is referred to [RZ15,Zhu17b].
In analogy to the function field case, a geometrization and categorification of the left hand isomorphism in (1.1) was proven by Zhu [Zhu17a]. To begin with, such an endeavour is made possible by [Zhu17a] and Bhatt-Scholze [BS17] who have shown that the cosets G(Q p )/G(Z p ) are the F p -points of an algebrogeometric object known as the Witt vector affine Grassmannian, and denoted Gr G → Spec(F p ) in this paper. For a fixed prime nubmer ℓ = p and a choice of half twist Q ℓ ( 1 2 ), Zhu proves an equivalence of Tannakian categories between semi-simple perverse Q ℓ -sheaves on Gr G of weight 0 which are furthermore equivariant with respect to the action of the positive loop group L + G. This positive loop group has as F p -points the group G(Z p ); it lies inside the full loop group LG, whose F p -points are G(Q p ).
Building upon our earlier work [RS20,RS21], the goal of the present paper is to refine (1.2) into an equivalence between the category of mixed Tate motives on the double quotient L + G\LG/L + G and representations of Deligne's extended dual group G 1 , a certain extension of G m (which records the weights) by G. This equivalence is independent of the choice of ℓ = p present in (1.2) via the use of ℓ-adic cohomology. We refer to Remark 5.8 for the relation with (1.1). Also we remark that Zhu [Zhu18] has explained the construction of a motivic Satake equivalence using numerical motives. This approach is based on an explicit enumeration of algebraic cycles on affine Grassmannians. By comparison, the approach taken in this paper is more strongly relying on the general framework of motives given by Ayoub [Ayo07a,Ayo07b] and Cisinski-Déglise [CD19].
1.2. Results. A prestack over F p is a functor from affine F p -algebras to anima (called spaces in [Lur09], Kan complexes, or ∞-groupoids or, in classical terminology, simplicial sets up to weak equivalences). Examples of prestacks include (ordinary) presheaves such as F p -schemes and ind-schemes (via their functor of points), but also homotopy quotients of such objects by group actions, and more general geometric objects such as higher stacks over F p . In [RS20] we have constructed a stable ∞-category DM(X) of motives with Q-coefficients on any prestack X. For example, we can speak of and conveniently work with DM(L + G\LG/L + G), the category of motives on the indicated double quotient.
In order to state our results, we first need to deal with the general process of perfection, see §6 in the first arXiv version of [RS20], and also [EK20], or the recent preprint of Bouthier-Kazhdan-Varshavsky [BKV20, §2.3.5]. There are two ways to turn a F p -prestack X perfect, i.e., to ensure that pullback along the absolute Frobenius σ is an isomorphism: the limit perfection and the colimit perfection X perf := colim T →X lim σ T . The latter is constructed by glueing the perfections of all affine F p -schemes covering the prestack (see Definition 2.3).
Recall from [Zhu17a,BS17] (or see §2.3 for a brief recapitulation) that perfectly finitely presented (pfp) F p -schemes are precisely those F p -schemes X which arise as X ≃ lim σ X 0 for some finite-type F p -scheme X 0 . Such a scheme X 0 is called a model of X. We have DM(X 0 ) ≃ DM(X) by the previous proposition. A typical example of a pfp scheme is (A 1 Fp ) perf = Spec F p [t p −∞ ], the perfection of A 1 Fp . Theorem 1.2. (Theorems 2.10 and 2.12) The equivalences of Proposition 1.1 can be used to endow motives on pfp schemes over F p with a six functor formalism obeying all the standard properties including base change, localization, h-descent, duality and homotopy invariance with respect (A 1 Fp ) perf .
Similar results also hold for motives on ind-(pfp schemes), i.e., objects presented as N-indexed colimits X = colim X i , where the X i are pfp schemes and the transition maps X i → X i+1 are closed immersions. An example is the Witt vector affine Grassmannian Gr G = (LG/L + G)é t which is shown to be an ind-(pfp scheme) in [BS17].
Such a convenient formalism lends itself to applications in geometric representation theory. Extending previous work of Soergel-Wendt [SW18] for finite type schemes and of [RS20] for ind-schemes of ind-(finite type) to the (ind-)pfp case, we say that a stratification on an ind-(pfp scheme) X is Whitney-Tate if the derived categories of Tate motives on the individual strata "glue" in a meaningful manner (see Definition 3.3 for the precise condition). Theorem 1.3. (special case of Theorem 4.5) i) The stratification on the Witt vector affine Grassmannian Gr G by L + G-orbits is Whitney-Tate. Accordingly, the stable ∞-category DTM(Gr G ) of stratified Tate motives is well-defined: such motives are characterized by the property that their restriction to the L + G-orbits are Tate, i.e., generated (under shifts and extensions) by motives of the form 1(n), n ∈ Z on the L + G-orbits. ii) The category DTM(Gr G ) admits a "motivic" t-structure whose heart MTM(Gr G ) is generated by intersection motives IC µ (n), n ∈ Z as simple objects, where µ : G m → T ranges over the dominant cocharacters of the fixed maximal torus T ⊂ G.
In fact, this result is as a special case of Theorem 4.5 where we prove the same result more generally for stratifications on Witt vector (partial) affine flag varieties Fl f = (LG/P f )é t associated to any parahoric subgroup P f ⊂ LG, in place of L + G. An example is the Iwahori subgroup B ⊂ LG associated to the choice of the Borel subgroup B ⊂ G.
The convolution product of functions in the Hecke algebra is enhanced by the convolution product ( §4.3) By étale descent, motives on LG/L + G are equivalent to motives on the étale sheafification Gr G = (LG/L + G)é t . Therefore, motives on the double quotient L + G\LG/L + G are equivalent to L + G-equivariant motives on Gr G . We define a category of equivariant stratified Tate motives by requiring that the underlying (non-equivariant) motive on Gr G is stratified Tate in the sense above. By Theorem 4.6, the category DTM(L + G\LG/L + G) inherits a t-structure from the t-structure on DTM(Gr G ).
The heart of this t-structure is denoted MTM(L + G\LG/L + G). In relation to the outline in §1.1, this category is an independent-of-ℓ version of perverse L + G-equivariant ℓ-adic sheaves on Gr G subject to the condition that their restriction to the strata are Tate twists of the (possibly shifted) constant sheaf.
Again, this statement is shown in Theorem 4.8 in the greater generality for Witt vector (partial) affine flag varieties as above. The following statements, however, are specific to the choice of L + G. In Proposition 5.3 we prove, as a consequence of the semi-simplicity of MTM(F q ) and Kazhdan-Lusztig's parity vanishing, that the forgetful functor MTM(L + G\LG/L + G) → MTM(Gr G ) (1.4) is an equivalence of semi-simple (abelian) categories whose generators are the simple objects IC µ (n) mentioned in Theorem 1.3 ii). This category is stable under the convolution product ⋆ and hence inherits a monoidal structure by Theorem 4.8 iii).
We write G 1 for the modification of the Langlands dual group introduced by Frenkel and Gross [FG09] following a suggestion of Deligne [Del07]. If G is simply connected, G 1 is the (direct) product of G and G m , where the G m -factor arises from the presence of Tate twists. In general, G 1 is a not necessarily split extension of G m by G, a phenomenon which is related to the necessity of fixing p 1/2 ∈ Q in the Satake isomorphisms (1.1). Our main result is the motivic Satake equivalence for Witt vector affine flag varieties.
Theorem 1.5. (Theorems 5.2 and 5.7) The monoidal structure on (1.4) given by the convolution product is part of the structure of a Q-linear Tannakian category, and as such there is, after extending scalars to Q, an equivalence This statement is independent of ℓ since no category of ℓ-adic (perverse) sheaves is required for its formulation. Furthermore, this equivalence is uniquely determined by its compatibility with (the appropriate version of) the ℓ-adic equivalence (1.2). As mentioned in §1.1 a similar result was also obtained in [Zhu18] using a different route, namely an explicit enumeration of algebraic cycles on the Witt vector affine Grassmannian, and working with the category of numerical motives. As is explained by the material in the sections below, working with the six functor formalism of motives as in this paper offers more flexibility including, say, handling the derived categories DTM(B\LG/B) of doubly Iwahori-equivariant stratified Tate motives on the loop group: the convolution product on this category does not respect the abelian subcategory of equivariant mixed Tate motives, so that working with numerical motives is prohibitive in such a situation.
Acknowledgements. We thank Eugen Hellmann and Thomas Nikolaus for helpful discussions, and the anonymous referee for many suggestions that improved the quality of the manuscript. The authors thank the University of Münster, the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation; under Germany's Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure) and the Technical University of Darmstadt for financial and logistical support which made this research possible.

Motives on perfect (ind-)schemes
2.1. Recollections on prestacks. In order to treat motives on the Witt vector affine Grassmannian and to relate them to motives on ordinary affine Grassmannians, it is very useful to use the definition of a category of motives on arbitrary prestacks developed in [RS20,§2]. We briefly recall the rudiments in this subsection. A very similar theory, for ℓ-adic sheaves instead of motives, has been developed very recently in [BKV20].
For a field k, AffSch ft k is the category of affine k-schemes of finite type. We denote by AffSch k its κpro-completion, i.e., the category of affine k-schemes whose underlying k-algebra is generated by at most κ elements. Here κ is a large enough regular cardinal fixed once and for all. For all purposes in this paper, we may choose κ to be the countable cardinal. The ∞-category of prestacks is defined as the ∞-categorical presheaf category, i.e., PreStk k def = Fun AffSch op k , Ani . Here, following the terminology and discussion in [ČS19, §5.1.4], Ani denotes the ∞-category of anima, i.e., the free completion of the category of finite sets under sifted homotopy colimits, or, equivalently, the ∞-category of spaces [Lur09, §1.2.16], also called ∞-groupoids. Prestacks are an extremely general class of algebro-geometric objects: by means of its functor of points, any k-scheme defines a prestack. In addition, prestacks are closed under all (homotopy) limits and colimits. In particular, any strict ind-scheme X = colim X i is a prestack, as is a quotient (always understood in the ∞-categorical sense, otherwise also known as a homotopy quotient) Y /G, where G is a group-valued presheaf acting on a presheaf Y .
Construction 2.1. In [RS20, Definition 2.2.1], we have defined a category DM(X) in DGCat cont of motives on any prestack X with the following features: i) For a scheme X of finite type over k, the homotopy category of DM(X) (which is a triangulated category) recovers the usual triangulated category of motives with rational coefficients due to Ayoub and Cisinski-Déglise [Ayo07a,Ayo07b], [CD19,§14].
ii) If X = lim n∈N X n is an affine pro-algebraic scheme, i.e., X i /k is affine and of finite type and so are the transition maps X n pn −→ X n−1 , then where the colimit is taken in DGCat cont (we emphasize that this category consists of colimit-preserving functors only). iii) If a prestack X admits a presentation X = colim X i , where X i ∈ AffSch k , then where the maps X i fij −→ X j are the ones in the colimit diagram defining X. More colloquially speaking, a motive on X is a family of motives on all the "covers" X i compatible with the system defining the colimit diagram. iv) For any map of prestacks f : X → Y , there is a functor f ! : DM(Y ) → DM(X Let p be a prime, and let k be a perfect field of characteristic p. For each T = Spec R ∈ AffSch k , let σ T : T → T be the absolute Frobenius morphism given by R → R, x → x p . This defines an endofunctor σ of the identity transformation on AffSch k . Let AffSch perf k be the full subcategory of objects T ∈ AffSch k such that σ T is an automorphism. These are called perfect affine k-schemes. The inclusion AffSch perf k ⊂ AffSch k admits a right adjoint given by T → lim σ T = Spec(colim r →r p R).
Precomposition with σ defines a (co-)limit preserving endofunctor of the identity on PreStk k , denoted by the same letter. The category of perfect prestacks 1 is the ∞-category The natural restriction functor res : PreStk k → PreStk perf k clearly preserves limits and colimits.
Lemma 2.2. There is an adjunction incl : PreStk perf k ⇆ PreStk k : res which under the Yoneda embedding extends the adjunction AffSch perf k ⇆ AffSch k . Both functors incl and res preserve colimits, and satisfy res • incl = id. In particular, incl is a full embedding, and σ is an equivalence on objects in the essential image.
Definition 2.3. The (colimit) perfection is the endofunctor on PreStk k defined by . For X ∈ PreStk k , X = colim T →X T with T ∈ AffSch k , it is computed as X perf = colim T →X lim σ T .
1 Recall that we fixed a regular cardinal κ in §2.1.
In the following, we identify PreStk perf k ≃ incl(PreStk perf k ) ⊂ PreStk k . This is the full subcategory of all objects X such that the counit of the adjunction X perf → X is an equivalence. Note that the perfection of ind-objects produces ind-(perfect objects), so that the functor (-) perf is well-suited for working with indschemes.
The following lemma appeared in an early arxiv version of [RS20]. A more refined statement for SH[ 1 p ], the stable A 1 -homotopy category localized at the characteristic (as opposed to DM with rational coefficients) is due to Elmanto and Khan [EK20].
i) The pullback σ ! : DM(X) → DM(X) is equivalent to the identity.
Proof. For i), first let X ∈ AffSch ft k . Invariance of DM under pullback along finite surjective radicial maps [CD19, Proposition 2.1.9] shows that the natural map id → σ * is invertible. Thus, we get id ≃ σ ! by adjunction (using σ * = σ ! ). This equivalence extends to PreStk k by continuity.
For ii), we reduce to the case X ∈ AffSch k by continuity. Then X perf = lim σ X = lim X→T lim σ T for T ∈ AffSch ft k , and we reduce further to the case X = T ∈ AffSch ft k . In this case, π ! : DM(X) → DM(X perf ) = colim σ ! DM(X) is the natural map which is an equivalence by i).
Definition 2.5. The (limit ) perfection is the endofunctor on PreStk k defined by where the limit is indexed by the positive integers.
Restricted to schemes, this functor is used in [Zhu17a,BS17]. It gives another way of constructing prestacks on which σ acts as an equivalence. In general, the natural map is not an equivalence. For a counter-example, take the ind-scheme . For a k-algebra R, the natural map is bijective if R is reduced (so that Frobenius is injective), but not in general (e.g., take R = k[̟ 1 p ∞ ]/(̟)). However, as DM is invariant under Nil-thickenings, the categories of motives on these two prestacks are equivalent: Corollary 2.6. For any prestack X ∈ PreStk k , there is a commutative diagram of prestacks where π is the natural projection. In particular by Lemma 2.4 ii), the !-pullback induces equivalences Proof. We have to show that π perf is an equivalence, and we claim that res • π is already an equivalence. Namely, Hom(T, lim σ X) ≃ Hom(T, X) for any T ∈ AffSch perf k . Indeed, for any map f : T → X and any r ≥ 0, the diagram Remark 2.7. If X is a scheme, then the canonical map X perf → lim σ X induces an isomorphism after Zariski sheafification. Indeed, if U → X is an open affine subscheme, then U perf = lim σ U → lim σ X is an open affine subscheme by [BS17, Lemma 3.4 (ix)]. This easily implies (X perf ) Zar ∼ = lim σ X.
2.3. Motives on perfect schemes. In order to obtain a six functor formalism, we need to put suitable finiteness assumptions on the objects.
The full subcategory of Sch k spanned by the qcqs pfp perfect k-schemes, or just pfp schemes, is denoted by Sch pfp k .
The functor Sch ft k → Sch pfp k , X → lim σ X is full and essentially surjective. Thus, each object X ∈ Sch pfp k and similarly each morphism f admits a model, i.e., a finite type k-scheme X 0 such that X = lim σ X 0 .
Tags 04DF, 01WJ, 01S4] or, equivalently by [BS17, Lemma 3.8], iff lim σ f is an isomorphism. It remains to show that any functor ϕ : Sch ft k → C to a category C that sends maps in S to isomorphisms factors uniquely over the perfection functor. By Zhu's result in i) ψ : Sch pfp k → C is unique if it exists. Also, to show the existence we may replace Sch pfp k by the full subcategory of Sch pfp k spanned by the image of the perfection functor. We define ψ on objects by fixing a model X = lim σ X 0 for each X, and set ψ(X) := ϕ(X 0 ). By definition [BS17, Proposition 3.11], given any pfp map f : X → Y , there is some n ≫ 0 such that we have the diagram below, where X (n) 0 is the n-th stage of the limit defining X (it is abstractly isomorphic to X 0 , but the structural map to k differs by σ n ): We define ψ(f ) as the composite ϕ(f (n) 0 )•ϕ(σ n ) −1 . This is well-defined independently of n. One immediately checks that it is also compatible with composition. This shows the existence of a factorization, hence ii).
For iv), let X 0 be a reduced model of X. Then |X| = |X 0 | on the underlying topological spaces, and we define Z 0 (resp. U 0 ) as the reduced closed (resp. open) subscheme X 0 defined by the closed subset |Z| (resp. open subset |U |) in |X| = |X 0 |. The limit perfection preserves open and closed immersions by [BS17, Theorem 2.10. Motives on pfp schemes enjoy a six-functor formalism with the following properties. Throughout, let X, Y denote objects, resp. f a morphism in Sch pfp k . Let X 0 , Y 0 , resp. f 0 be models thereof.
i) The presheaves DM * and DM ! on Sch ft k (given on objects by X → DM(X), and on morphisms by f → f * , resp. f ! ) factor uniquely over functors On objects X ∈ Sch pfp k , these two presheaves take the same values, denoted simply DM(X). The functor DM ! agrees with the restriction of the presheaf DM in Construction 2.1.iv ) from PreStk k to Sch pfp k . ii) The natural map p : X → X 0 induces an equivalence of ∞-categories which preserves the monoidal unit 1 and is compatible with all functors below. In particular, if X has a regular model X 0 /k, then where the terms at the right are the n-th Adams eigenspace in the algebraic K-theory of X and Bloch's higher Chow group, both tensored with Q. iii) There are adjunctions If f has a smooth model, there is an adjunction These functors are functorial in f . Under the equivalence (2.1), they correspond to More generally, if f has a smooth model f 0 of relative dimension n, there is a functorial equivalence (called relative purity) vi) The category DM(X) is compactly generated by the objects t ♯ 1(n), t : T → X pfp smooth and n ∈ Z.
In particular, the monoidal unit 1 X ∈ DM(X) is compact. The functors f ♯ , f * , f * , f ! , and f ! preserve compact objects and preserve arbitrary (homotopy) colimits. vii) For any pfp map f , there is a projection formula is a contravariant involution on the subcategory DM(X) c of compact objects, i.e., D X • D X = id. Furthermore, on compact objects, there are equivalences Here laxlim denotes the lax limit [GHN17] whose objects can be colloquially described as triples where α is any map (and the non-lax limit would be the subcategory where this map is an isomorphism). Equivalently, the (co)units of the adjunctions above assemble into so-called localization homotopy fiber sequences (2.5) x ) For a cartesian diagram in Sch pfp S (necessarily consisting of pfp maps) xi) The category DM is perfectly homotopy-invariant in the sense that for the projection map p : (A n ) perf × k X → X for any n ∈ Z ≥0 , the counit and unit maps p ♯ p * → id and id → p * p * are functorial equivalences in DM(X) [CD19, 2.1.3]. xii) The presheaves DM * and DM ! are sheaves for the h-topology on pfp schemes, i.e., if f : X → Y admits a model which is an h-covering in Sch ft k , then the natural map is an equivalence, and likewise with !-pullbacks instead. We refer to this property as h-descent. xiii) Suppose X is separated. The category DM(X) is equipped with a weight structure If X admits a regular model, 1 X is in the heart DM(X) w=0 = DM(X) w≤0 ∩ DM(X) w≥0 of this weight structure. Moreover, f * and f ! are weight-left exact (preserve "w ≤ 0") while f * and f ! are weight-right exact (preserve "w ≥ 0"). xiv ) Let ℓ be a prime number different from p = char k. There is a ℓ-adic realization functor Proof. i) follows from Lemma 2.9 and the above-mentioned fact that f * and f ! are equivalences for any finite radicial surjective map f [CD19, Proposition 2.1.9].
ii): The prestack approach to DM(X) yields DM(X) ≃ DM(X 0 ) by Corollary 2.6. With i) and ii) in hand, the remaining statements mostly follow directly from the classical assertions for finite-type k-schemes, which for the large majority are due to Ayoub and Cisinski-Déglise and, for functoriality for non-separated maps and h-descent proven in [RS20, Theorem 2.1.13, Proposition 2.1.14]. We will use these classical properties below without further comment, and refer to [RS20, Synopsis 2.1.1] for a detailed list of references.
The equivalence (2.1) is immediate from Lemma 2.9 using invariance of DM under finite radicial surjective maps (hence DM is well-defined on the localization) and Corollary 2.6. The statements in iii)-vi), x), xi) are then direct consequences of the corresponding assertions for finite type k-schemes.
For vii) note that the functoriality f ! and f * and also the tensor product are induced, by definition, from (f 0 ) ! , (f 0 ) * and the tensor product on DM(X 0 ) etc. Similarly for the internal Hom in viii).
The purity equivalence in the case of a perfectly smooth map f follows from the smooth case and the étale descent property in xii).
In the definition of D X as stated in (2.3), note that the pullback functor p ! exists by virtue of the general formalism of motives on prestacks. Under the equivalence DM(X) = DM(X 0 ), it can also be computed as Hom(−, p ! 0 1), where p 0 : X 0 → k is the structural map. This again reduces all the assertions to the classical duality statements for motives on finite type k-schemes.
In the situation of ix), apply Lemma 2.9.iv) to get the localization fiber sequences (2.4), (2.5) from the corresponding statements for DM on Sch ft k due to Ayoub. In the presence of the base change equivalence in x), these fiber sequences are equivalent to the formulation involving the lax limit, cf. the argument in [AG18, Example 4.1.6].
For xiii) note that X is separated iff it admits a separated model X 0 , so that the weight structure on DM(X 0 ) can be used.
In the same vein, xiv) follows from the equivalence (2.1) and its ℓ- Similarly to [Ric19, Lemma 1.10] one shows that this category has final object Spec(k), is closed under fibre product and countable disjoint unions.
As an example we note that if X = colim X i is an ind-scheme of ind-(finite type) over k, then the Zariski sheafification of the perfection (X perf ) Zar = colim lim σ X i is an ind-(pfp scheme) by Remark 2.7. We have the following basic lemma.
Lemma 2.11. Every quasi-compact map between ind-(pfp schemes) is schematic and pfp.
Proof. Let X → Y be a quasi-compact map of ind-(pfp schemes). Choosing a presentation Y = colim Y i by pfp schemes, it is enough to show that each X × Y Y i is a pfp scheme. We reduce to the case where Y = Y i is a pfp scheme, and in particular quasi-compact. By assumption X = colim X i is quasi-compact as well. Being an object in IndSch pfp k the underlying topological space |X| = colim |X i | is Jacobson. Hence, the sequence of reduced schemes X i = X i,red stabilizes by [Ric19, Corollary 1.24] which shows that X = X i , i > > 0 is a pfp scheme.
Theorem 2.12. Motives on ind-(pfp schemes) X ∈ IndSch pfp k satisfy the properties listed in Theorem 2.10, with the following adjustments: • The statement in ii) has no analogue for ind-(pfp schemes), since they need not have models among ind-schemes of ind-(finite type). • The description of compact generators (cf. vi) above) is as follows: the category DM(X) is compactly generated by the images of the compact objects in DM(X i ), where X = colim X i is a presentation. • The weight structure on DM(X) exists if there is a presentation X = colim X i , where the X i ∈ Sch pfp k are separated.
• The functor f * exists (and satisfies the properties as stated in the remaining items) if f is quasicompact (hence schematic and pfp). • If X is componentwise quasi-compact, then p * 1 is a monoidal unit, for the structural map p : X → Spec k. In general, DM(X) only carries a non-unital symmetric monoidal structure.
• The condition that f has a smooth model (resp. is a pfp closed immersion, pfp open immersion) has to be replaced with "f has a schematic smooth model f 0 " (has a model which is a schematic pfp closed, resp. open immersion). The h-covering in xii) has to be schematic. The term "proper" has to be replaced by "ind-(perfectly proper )" in iv ).
Proof. Recall from [RS20, Theorem 2.4.2] that the corresponding assertions hold for motives on ind-schemes. From there, the theorem as stated follows from Lemma 2.9, in the same way as the assertions for motives on pfp schemes follow from those on finite-type k-schemes.

Stratified Tate motives for perfect ind-schemes
With this convenient extension of the six functor formalism for the categories DM of motives on pfp schemes and their ind-variants, we can quickly adapt the contents of [RS20, § §3-5].
Definition 3.1. A stratified ind-(pfp scheme) is a map of ind-(pfp schemes) where ι is bijective on the level of underlying sets, each X w is a pfp scheme, each restriction ι| Xw is an immersion and the topological closure of each ι(X w ) is a union of strata. Here W is a countable index set which is partially ordered by the closure relations on the strata, i.e., ι(X v ) lies in the closure of ι(X w ) if and only if v ≤ w.
Since each X w is pfp (hence quasi-compact), each restriction ι| Xw : X w → X is a quasi-compact map of ind-(pfp schemes) and hence schematic and pfp by Lemma 2.11. A map of stratified ind-(pfp schemes) is required to be quasi-compact (hence, again, schematic and pfp), and to map strata to strata.
Definition 3.2. A perfect cell is a k-scheme isomorphic to the perfection of a cell A n k × G r m,k . A pfp scheme X is perfectly cellular if it admits a smooth model over k, and admits a stratification into perfect cells. (We do not require that the smooth model admits a stratification into cells.) A perfectly cellular stratified, or just pcs ind-scheme is a stratified ind-(pfp scheme) X where each X w , w ∈ W is a perfectly cellular k-scheme.
For a pfp scheme X, or a disjoint union of those, the category of Tate motives DTM(X) ⊂ DM(X) is the full subcategory generated under arbitrary shifts and colimits by 1 X (n), n ∈ Z. By Theorem 2.10 ii), it is equivalent to DTM(X 0 ) as defined in [RS20, Definition 3.1.8], where X 0 is any model of X. Indeed, the equivalence σ ! : DM(X 0 ) → DM(X 0 ) restricts to an equivalence of the DTM-subcategories by Theorem 2.10 ii).
For the next definition, we note that by quasi-compactness of ι the pullback ι * is well-defined, cf. Theorem 2.12.
Definition 3.3. A stratified ind-(pfp scheme) X is Whitney-Tate if ι * ι * : DM(X + ) → DM(X + ) preserves the subcategory DTM(X + ), i.e., (ι w ) * (ι v ) * induces DTM(X v ) → DTM(X w ) for all v, w ∈ W . In this case, the category of stratified Tate motives is well-defined. Here the transition functors used to form the lax limit is (ι w ) * (ι v ) * . Since DM(X) is equivalent to the lax limit of the categories DM(X w ), the category DTM(X, X + ) is a full subcategory of DM(X). A map π between stratified ind-(pfp schemes) is a Whitney-Tate map if π * preserves stratified Tate motives.
The characterization of Whitney-Tate stratifications in [RS20, Definition and Lemma 3.1.11] holds literally. Also the perfection of a cellular stratified k-scheme of finite type in the sense of [RS20, Definition 3.1.5] is pcs. Conversely, every pcs scheme ⊔X w → X admits a model ⊔X w,0 → X 0 where each X w,0 is smooth. We now show that the t-structures on the individual strata X w , which are induced from a model X w,0 , glue together for any pcs (ind-)scheme.
Recall from [Lev93] that the Beilinson-Soulé vanishing conjecture for k is equivalent to the existence of a tstructure on DTM(Spec k) such that 1 Spec k (r) is in the heart for all r ∈ Z. By Quillen's work, this conjecture holds for k = F q . By Harder's work, it holds for a function field F q (t) and therefore, by the compatibility of algebraic K-theory with filtered colimits of rings, it also holds for its perfection k = F q (t) perf . Definition and Lemma 3.4. Suppose k satisfies the Beilinson-Soulé vanishing conjecture. Let X be a pcs Whitney-Tate ind-scheme over k. Then each DTM(X w ) carries a unique t-structure such that 1 Xw (n)[d w ], d w := dim(X w ) is in the heart.
The subcategories define a compactly generated t-structure on DTM(X, X + ). The heart of this t-structure is denoted by MTM(X, X + ) or just by MTM(X) if the stratification is clear from the context.

Proof.
By assumption there exists a finite stratification ⊔ i X (i) w → X w into perfect cells. We may choose a model ⊔ i X (i) w,0 → X w,0 where both source and target are smooth. (Here the schemes X (i) w,0 are cells, but this map is only a stratification after perfection.) Then the first claim follows from [RS20, Lemma 3.2.4] using that the equivalence DTM(X w ) = DTM(X w,0 ) in Theorem 2.10 ii) commutes with the six functors (by construction). The second statement is also shown similarly as in op. cit.: for any w ∈ W , the t-structure on DTM(X ≤w , X + ) c arises by glueing the t-structures on the strata X v , v ≤ w. Given that the compact objects in DM(X, X + ) are supported on some finite union of X ≤w , this gives a t-structure on DTM(X, X + ) c and therefore a compactly generated t-structure on DTM(X, X + ).
Restricting the ℓ-adic realization functor to Tate motives continues to have the pleasant properties -such as conservativity and faithfulness-we know from the case of cellular Whitney-Tate stratified ind-schemes. The proof of the following statement is a verbatim copy of [RS20, Lemma 3.2.8] and [RS21, Corollary 2.20].
Proposition 3.5. Let X be a pcs Whitney-Tate ind-scheme over k and suppose that k satisfies the Beilinson-Soulé vanishing conjecture. The restriction of the ℓ-adic realization functor ρ ℓ from DM(X) to DTM(X) := DTM(X, X + ) has the following properties: i) it is conservative, ii) it creates the motivic t-structure in the sense that M ∈ DTM(X) c is in positive (resp. negative) degrees of the motivic t-structure iff ρ ℓ (M ) has the corresponding property for the perverse t-structure. In particular, it restricts to a functor ρ ℓ : MTM(X) c → Perv(X, Q ℓ ). (3.1) Furthermore, this latter functor is faithful.

Motives on Witt vector affine flag varieties
Let K/Q p be a finite extension with uniformizer ̟ ∈ O K and residue field k/F p . Let G be a split reductive group over O K . In this section, we introduce Tate motives on the Witt vector affine flag varieties for G and prove basic properties thereof.   LG is representable by a group ind-(perfect affine scheme). ii) For every smooth, affine O K -group scheme G, the functor L + G = lim i≥0 G i is representable by the perfection of an inverse system of smooth, affine group schemes with smooth, surjective transition maps. For each i ≥ 0, the kernel ker(G i+1 → G i ) is the perfection of a vector group. In particular, every L + G-torsor in the fpqc-topology admits sections étale locally. If G = G f for some facet f ⊂ A , then each G i = G f ,i is moreover the perfection of a cellular k-scheme, i.e., the perfection of a smooth (finite type) k-scheme which can be stratified by pieces isomorphic to A r k × G s m,k . Proof. For the representability statements see [BS17, Proposition 9.2], and the references cited therein. For the étale local triviality, we note that L + G is already representable as a functor on the category of all kalgebras by the same formula by Greenberg [Gre61] (cf. also [Zhu17a, §1.1.1], [BS17, Remark 9.3]). It is the perfection of the pro-algebraic group lim i≥0 H i where H i : R → G(W OK (R)/V i ̟ W OK (R)) and V ̟ denotes the Verschiebung on the ramified Witt vectors. Consequently,    where w ranges through the double quotient W f ′ \W/W f of the Iwahori-Weyl group by the reflection subgroups associated with the facets, defined analogously to [RS20,(4 There is a stratification where the closure relations of the strata are given by the Bruhat   ii) The stabilizers of the P f ′ -left action on Fl f are geometrically connected.
iii) If P f ′ ⊂ P f , the projection π : Fl f ′ → Fl f is schematic, perfectly proper, and étale locally on the target isomorphic to the projection P f /P f ′ ×Fl f → Fl f . The induced map on the Iwahori strata π + : Fl + f ′ → Fl + f is a Whitney-Tate map and admits a section s + : Fl + f → Fl + f ′ which is an open and closed immersion. iv ) In iii), the étale quotient P f /P f ′ is the perfection of an homogenous space.
Proof. Part i) follows from the discussion above, and the following observation: is the length in W/W f . This is proven as in [RS20, (4.3.10)], but working with perfections everywhere: for each affine root α, there is a root subgroup U α ⊂ LG, U α ≃ (A 1 k ) perf , and an isomorphism where the product (taken in any order) ranges over all affine roots α such that (wα)| a0 takes positive values and α| f takes negative values. Part ii) follows from the presentation Fl f ,w = P f /P f ′ ∪wf and the geometrical connectedness of P f ′ ∪wf explained above. Part iii) is proven analogously to [RS20, Proposition 4.3.13]; note that s + is open since it is an inclusion of a disjoint union of certain Iwahori strata. For iv), let P f ,0 = H perf 0 , P f ′ ,0 = (H ′ 0 ) perf be the Greenberg realization as in the proof of Lemma 4.1. LetP f ′ ,0 = im(P f ′ ,0 → P f ,0 ), and letH ′ 0 = im(H ′ 0 → H 0 ) which is a parabolic subgroup of H 0 . Then on étale quotients For the second isomorphism, we use that these étale quotients are already fpqc sheaves, and that taking perfections preserves faithfully flat maps [BS17, Lemma 3.4 (xii)], henceP f ′ ,0 ∼ = (H ′ 0 ) perf . See also [BGA18, Theorem 6.1].
Example 4.4. Let s ∈ S be a simple affine reflection. Let f s be the unique facet of maximal dimension in the closure of a 0 such that s(f s ) = f s , i.e., W fs is the subgroup of W generated by s. We specialize Lemma 4.3 iii) to the case f ′ = a 0 , and f = f s , so that π : Fl := Fl a0 → Fl fs is the projection from the full affine flag variety. In this case, π has general fiber P fs /B = lim σ P 1 k by Lemma 4.3 iv) (cf. also [Zhu17a, above (1.4.1)]). If w ∈ W , and w = v · s is a reduced decomposition, then Here π + | Fl v is an isomorphism, and π + | Fl vs is isomorphic to the projection The Q-linear abelian category MTM(Fl f ) is generated (by means of extensions and direct sums) by the simple objects called intersection motives Third, for both f , f ′ arbitrary, we use [RS20, Proposition 3.1.23]: a P f -equivariant motive is Tate with respect to the stratification by P f -orbits as soon as it is Tate with respect to the finer stratification by Iwahori-orbits (i.e. P a0 -orbits). This latter property holds by the second step. The category DTM(Fl f ,w ) of (unstratified) Tate motives on the strata Fl f ,w carries a t-structure whose heart MTM(Fl f ,w ) is an abelian category whose simple objects are of the form 1(n)[dim Fl f ,w ] (Definition and Lemma 3.4). These t-structures are glued together, as is classical in the context of perverse sheaves [BBD82,§1.4]; see also [SW18,Theorem 10.3] for an application of this idea in the context of stratified Tate motives.
The description of the simple generators then follows from the description of the simple generators in the categories that are glued together [RS20, Theorem 5.2.3].
ii) is then a direct consequence of Proposition 3.5.
Theorem 4.6. i) The full subcategories of DM(P f ′ \LG/P f ), (For example, the first category consists of those motives on P f ′ \LG/P f whose underlying, i.e., non-equivariant motive on Fl f is a stratified Tate motive with respect to the stratification by P f ′ -orbits.) This category is denoted DTM(P f ′ \LG/P f ).
ii) For f = f ′ , there is a t-structure on DTM(P f ′ \LG/P f ) such that the forgetful functors to DTM(Fl f ) (and also to DTM(Fl op f )) are t-exact. The heart of this t-structure is the category of mixed Tate motives on the double quotient, denoted MTM(P f ′ \LG/P f ). The functor forgetting the P f -equivariance of a mixed Tate motive, MTM(P f \LG/P f ) → MTM(Fl f ), is fully faithful, and induces a bijection of isomorphism classes of simple objects as described in Theorem 4.5.i).
Proof. i) is shown by exhibiting an explicit set of generators, in the same way as in [RS20, Theorem 5.3.4]. ii): The t-structure on DTM(P f ′ \LG/P f ) is an instance of a t-structure on a limit of a diagram of stable 4.3. The convolution product. In this section, we discuss the convolution product on the category MTM(P f \LG/P f ) by indicating how the corresponding arguments in the equi-characteristic situation carry over to the case considered here.
We define the convolution product by the same formula as in [RS21, Definition 3.1]: where p and m are the projection and multiplication maps:

As in [RS20, Proposition 2.4.4], the exterior product
is defined using that the kernels of P f → P f ,i are perfectly split pro-unipotent (i.e., of the form lim G i , where the underlying schemes of G 0 and all of the ker(G i+1 → G i ) are perfections of vector groups, cf. the proof of Lemma 4.1) and the perfect homotopy invariance of DM (Theorem 2.10.xi)). The pushforward m ! exists by Theorem 2.12 (and the descent of functoriality expressed in [RS20, Lemma 2.2.9]) since the multiplication map induces a schematic map of ind-schemes Fl f × Fl f := (LG × P f LG/P f )é t → Fl f . Note that the convolution of two compact objects is again compact.
Remark 4.7. The exterior product for motives on Fl op f /P f can be used to define another convolution product functor. At least on the level of the homotopy categories Ho(DM(P f \LG/P f )), these two functors are isomorphic as can be shown like in [RS21, Proposition 3.4], using that LG ≤w = P f wP f is perfectly placid, i.e., a countable filtered limit of affine pfp schemes whose transition maps are perfections of smooth maps.
Theorem 4.8. The convolution product ⋆ has the following properties: i) It is compatible with the convolution product defined in the ℓ-adic situation [Zhu17a, p. 432] under the ℓ-adic realization functor ρ ℓ : cons,P f (Fl f ) (taking values in the bounded derived P f -equivariant category of constructible sheaves on Fl f ).
ii) It admits an associativity isomorphism.
iii) It preserves the subcategory DTM(P f \LG/P f ) ⊂ DM(P f \LG/P f ). For P f = L + G, it also preserves the abelian subcategory MTM(L + G\LG/L + G).
Proof. Part i) holds since both convolution product functors are defined as the concatenation of the same functors (motivic, resp. ℓ-adic) and Theorem 2.10.xiv). Part ii) results from the base-change isomorphism for proper maps of ind-(pfp schemes) (cf. (2.6)) as in [RS21,Lemma 3.7]. For iii), we review the key geometric arguments in case P f is the Iwahori group B, and refer to [RS21, Proposition 3.26] for the reduction to this case and for further details concerning the case P f = B. Writing ι w : Fl w → Fl := Fl a0 for the inclusion of the stratum, the claim (ι w ) ! 1 ⋆ (ι w ′ ) ! 1 ∈ DTM(Fl) is first shown in case w = w ′ is a simple reflection s. In this case the map Fl ≤s × Fl ≤s (pr 1 ,m) → Fl ≤s × Fl ≤s is an isomorphism because it is a closed immersion of 2-dimensional irreducible perfectly proper schemes. There is a stratification of Fl ≤s = Fl s ⊔ Fl e ∼ = (A 1 ) perf ⊔ Spec k. Furthermore, the multiplication map m : Fl ≤s × Fl ≤s → Fl ≤s is a Whitney-Tate map with respect to the this stratification. This can be seen as in loc. cit., except that all the affine lines (intervening as strata) are replaced by their perfections. The second step uses the isomorphism (see loc. cit.) Fl w × Fl w ′ → Fl ww ′ for w, w ′ ∈ W such that l(ww ′ ) = l(w) + l(w ′ ). This isomorphism is a consequence of (4.3) and Bruhat-Tits theory, applied to the Witt vector affine flag variety. The remaining steps (still for P f = B) are then formal consequences.
The preservation of MTM as stated holds because of the corresponding assertion in the ℓ-adic case [Zhu17a, Proposition 2.2] together with i). Here we use Theorem 4.5.ii) to reduce to ℓ-adic (perverse) sheaves.

The motivic Satake equivalence
We keep the notation from §4: K/Q p is a finite extension with ring of integers O K and finite residue field k/F p . Let G be a split reductive group over O K . Fix a maximal split torus contained in a Borel subgroup Let G denote the Langlands dual group over Q formed with respect to the pair (T, B) and a fixed pinning X ∈ Lie(B). Recall that ( G, T , B, X) is a split reductive Q-group scheme equipped with a pinning whose root datum is dual to the root datum of (G, T, B). Denote by T ad the image of T in the adjoint group G ad . Via X * (T ) = X * ( T ), we can view the half-sum of the B-positive roots in G as a cocharacter ρ : G m,Q → T ad . Then G m,Q acts from the right on G via conjugation as Ad ρ : (g, λ) → ρ(λ) −1 gρ(λ). This action preserves the pair ( T , B).
Definition 5.1. The extended Langlands dual group is the Q-group scheme G 1 := G ⋊ Adρ G m,Q equipped with the pinning ( T 1 , B 1 , X 1 ).
As an example take G = PGL 2 in which case G 1 = SL 2 × µ 2 G m,Q = GL 2 , and the character d identifies with the determinant. We write Rep Q ( G 1 ) for the category of algebraic G 1 -representations on Q-vector spaces. The subcategory Rep fd Q ( G 1 ) of finite-dimensional representations is a semi-simple abelian category with (absolutely) simple objects where B op 1 ⊂ G 1 denotes the Borel opposite to B 1 , and µ n : B op 1 → T 1 → G m,Q is the composition of the projection with the character (µ, n) ∈ X * (T ) + × Z = X * ( T 1 ) + . Note that the restriction of V µ (n) along Extending scalars from Q to Q we obtain the categories Rep fd Q ( G 1 ) ⊂ Rep Q ( G 1 ) of algebraic G 1,Q -representations on Q-vector spaces which have the same simple objects (5.1).
The aim of this section is to prove the following theorem due to Zhu [Zhu17a] in the context of ℓ-adic sheaves and in the context of numerical motives [Zhu18,§2]. It will be stated and proven in a slightly sharper form in Theorem 5.7 and Remark 5.9 below.
Theorem 5.2. There is an equivalence of Q-linear abelian tensor categories under which IC µ (n) → V µ (n) for all µ ∈ X * (T ) + , n ∈ Z. For each prime number ℓ = p, this equivalence is compatible under the ℓ-adic realization with the geometric Satake equivalence constructed in [Zhu17a].
At the left hand side, MTM L + G\LG/L + G denotes the category of stratified mixed Tate motives on the loop group double quotient constructed in Theorems 4.5 and 4.6. The subscript Q indicates that we consider this category with Q-coefficients (as opposed to Q-coefficients like in the preceding sections). The simple objects IC µ (n), µ ∈ X * (T ) + , n ∈ Z are Tate twists of intersection cohomology motives of the Schubert varieties. We already know that the convolution product ⋆ for motives on L + G\LG/L + G preserves mixed Tate motives (Theorem 4.8). In order to prove the above theorem, we will perform the following steps: we show in Theorem 5.6 that is a neutral Tannakian category. The fiber functor ω is given by taking global motivic cohomology. We show that the Tannaka dual group is reductive by showing that the category is semi-simple (Proposition 5.3). This step relies on the Kazhdan-Lusztig parity vanishing and on the semisimplicity of MTM(k). In particular, every object in MTM (L + G\LG/L + G) is a (possibly infinite) direct sum of IC µ (n)'s. We further identify, at least over Q, the Tannaka dual group G 1 := Aut ⋆ (ω) of MTM (L + G\LG/L + G) c with G 1 , by relying on the ℓ-adic Satake equivalence for the Witt vector affine Grassmannian due to Zhu. Once this is done, we obtain Theorem 5.2 by passing to ind-completions, cf. Remark 5.9. 5.1. Semisimplicity. The Witt vector affine Grassmannian Gr G = (LG/L + G)é t admits a stratification by left L + G-orbits which yields a presentation as ind-(perfectly proper pfp k-schemes) by Witt vector Schubert varieties: Here µ ranges of the partially ordered semigroup X * (T ) + of B-dominant cocharacters. Recall that Gr G,≤λ ⊂ Gr G,≤µ if and only if λ ≤ µ in the dominance order, i.e., µ − λ is a sum of positive coroots with non-negative integral coefficients. The dense open stratum Gr G,µ ⊂ Gr G,≤µ is an irreducible perfectly smooth pfp kscheme of dimension 2ρ, µ . Here ρ denotes the half-sum of the positive roots in B and -, -is the natural pairing between the characters and cocharacters. In particular, the L + G-orbit stratification satisfies the parity property Gr G,λ ⊂ Gr G,≤µ =⇒ dim(Gr G,µ ) ≡ dim(Gr G,λ ) mod 2, (5.3) because 2ρ, µ−λ = 2 ρ, µ−λ is an even integer. Recall from Theorem 4.5 the abelian category MTM(Gr G ) of mixed stratified Tate motives with respect to the L + G-orbit stratification. Its simple objects are the Tate twists of the intersection motives IC µ (n), µ ∈ X * (T ) + , n ∈ Z of the Schubert varieties Gr G,≤µ . By Theorem 4.6 the forgetful functor is fully faithful and yields a bijection on simple objects. The following proposition is analogous to [MV07, Proposition 2.1].
Proof. The argument follows closely [RS21, §5.1]. We outline the key steps for the reader's convenience. If MTM(Gr G ) is semisimple, then MTM(L + G\LG/L + G) is semisimple as well by the full faithfulness of (5.4). In this case (5.4) is an equivalence because the simple objects agree.
To prove the semisimplicity of MTM(Gr G ) it is enough to show that all extensions between simple objects vanish, i.e., that Ext 1 MTM(GrG) IC λ (m), IC µ (n) = 0 (5.5) for all λ, µ ∈ X * (T ) + , m, n ∈ Z. By the conservativity and t-exactness of the ℓ-adic realization (Proposition 3.5) and (5.3), the Kazhdan-Lustig parity vanishing holds, i.e., where A ∈ MTM(Gr G ) and ι : X ⊂ Gr G is a finite union of Schubert varieties and where m H i denotes the truncation with respect to the motivic t-structure on X. This formally implies where we have used that MTM(Gr µ ) = MTM(k), cf. the argument in [RS21, Lemma 6.3]. Finally, the higher algebraic K-groups of the finite field k are torsion (i.e., their rationalizations vanish), so that MTM(k) is semisimple. This shows (5.5).
5.2. The tensor structure. By Theorem 4.8 the category MTM(L + G\LG/L + G) is stable under the convolution product ⋆ compatible with the ℓ-adic realization, i.e., for each pair of such motives A, B, there is a functorial isomorphism of ℓ-adic perverse sheaves where ρ ℓ : MTM(L + G\LG/L + G) c → Perv L + G (Gr G , Q ℓ ) (5.7) denotes the ℓ-adic realization taking values in the category of L + G-equivariant perverse ℓ-adic sheaves on Gr G (cf. Theorem 4.8), and where ⋆ ℓ denotes the convolution of ℓ-adic complexes.
Proposition 5.4. The convolution product ⋆ can be upgraded to a unique symmetric monoidal structure on MTM(L + G\LG/L + G) such that the ℓ-adic realization functor Here we only sketch the main ideas. The anti-involution θ : G → G is defined by g → (g * ) −1 = (g −1 ) * , where (-) * denotes the Cartan involution. By functoriality, we obtain an anti-involution on LG preserving L + G, and thus an equivalence of prestacks, still denoted Then for A, B ∈ MTM(L + G\LG/L + G) we have a functorial isomorphism θ ! (A ⋆ B) ∼ = (θ ! B) ⋆ (θ ! A). Also we have a (carefully chosen) equivalence θ ! ≃ id of endofunctors on MTM(L + G\LG/L + G). The point being that θ preserves Schubert varieties and hence also intersection motives. However, to obtain the correct equivalence θ ! ≃ id compatible with the ℓ-adic realization one needs to introduce a carefully chosen sign, cf. the aforementioned references. Combining the above equivalences leads to the commutativity constraint c ′ A,B : A ⋆ B ∼ = B ⋆ A. We finally define c ICµ(n)⋆IC λ (m) := (−1) 2ρ,µ+λ c ′ ICµ(n)⋆IC λ (m) , and extend it linearly to all objects in MTM(L + G\LG/L + G) using the semisimplicity, cf. Proposition 5.3. 5.3. The Tannakian structure. Next we equip the Q-linear, abelian, symmetric monoidal, semisimple category (MTM(L + G\LG/L + G), ⋆) with a tensor functor valued in vector spaces (Vect Q , ⊗) leading to the aforementioned Tannakian structure. Following [RS21, §5.3] we define the fibre functor as the composition of the forgetful functor (which is an equivalence by Proposition 5.3; note that mixed Tate motives are with respect to the stratification by L + G-orbits), the pushforward along the structural map ǫ : Gr G → Spec(k) (which preserves Tate motives by [RS20, Lemma 3.1.19], using that the stratification of Gr G by L + G-orbits is perfectly cellular), followed by the grading functors for the classical (which agrees in this case with the motivic) t-structure gr cl and the weight structure gr W , and finally the equivalence [Lev93] of pure Tate motives of weight 0 with the category of Q-vector spaces.
Lemma 5.5. The fibre functor ω : MTM(L + G\LG/L + G) → Vect Q has a natural monoidal structure with respect to the convolution product on the source and the ordinary tensor product on the target which is compatible with the ℓ-adic realization.
Proof. At least on the level of homotopy categories the composition of functors has a natural monoidal structure with respect to convolution on the source and the tensor product on the target, cf. [RS21, Proposition 5.13] whose arguments translate literally. Next the conservativity of the ℓ-adic realization implies the Kazdhan-Lusztig parity vanishing cl H i (ǫ ! IC µ (n)) = 0 whenever i ≡ 2ρ, µ mod 2. admits a natural monoidal structure as well. The vanishing is needed in order to obtain a monoidal structure as opposed to a Z/2-graded monoidal structure coming from a sign in the formation of tensor products of complexes. Using the natural monoidal structure on the weight graduation functor gr W we obtain a monoidal structure on ω. The compatibility with the ℓ-adic realization follows from Theorem 2.10.xiv) and the fact that the ℓ-adic fiber functor is defined by the analogous functors.
Theorem 5.6. The category MTM(L + G\LG/L + G) c of compact mixed Tate motives on the double quotient, endowed with the convolution product, the constraints from Proposition 5.4 and ω as fibre functor is a neutral Tannakian category over Q ([DMOS82, Chapter II, Definition 2.19]).
Proof. Note that the restriction of ω to compact objects takes values in finite-dimensional vector spaces. As in [RS21, Theorem 5.14] we check the conditions in [DMOS82, Chapter II, Proposition 1.20]: i) The functor ω has the structure of a tensor functor. This is Lemma 5.5.
ii) The functor ω is Q-linear, exact and faithful. The functor ω is clearly Q-linear and additive. Hence, it is exact because MTM(L + G\LG/L + G) c is semi-simple. The faithfulness follows from the compatibility with the ℓ-adic realization, the faithfulness of the latter, and the faithfulness of the ℓ-adic fiber functor ω ℓ [Zhu17a, Corollary 2.10]. iii) The constraints constructed in Proposition 5.4 give the usual constraints in Vect Q after applying ω. This is immediate from Lemma 5.5 and the ℓ-adic case, cf. [Zhu17a, Proposition 2.21]. iv) Neutral object. This is the skyscraper IC 0 supported at the base point. Theorem 5.7. There is an equivalence of Tannakian categories MTM(L + G\LG/L + G) c Q , ⋆, ω ≃ Rep fd Q ( G 1 ), ⊗, v , where the subscript Q indicates the category with the same objects, but Hom-spaces are tensored with Q. At the right, v : Rep fd Q ( G 1 ) → Vec Q denotes the forgetful functor. The intersection motives IC µ (n) correspond to the simple G 1 -representations V µ (n) for (µ, n) ∈ X * (T ) + × Z = X * ( T 1 ) + .
Proof. The argument is very similar to the proof of [RS21, Theorem 6.8]; it relies on the ℓ-adic geometric Satake equivalence proven in [Zhu17a]. Let us recall the key points. Denote by G 1 := Aut ⋆ (ω) the affine Q-group scheme of tensor automorphisms provided by Theorem 5.6. We need to prove an isomorphism G 1 ≃ G 1 of Q-group schemes.
For a prime number ℓ ∤ p, the ℓ-adic realization induces an equivalence of Tannakian categories where Sat G,ℓ denotes the full semi-simple subcategory of L + G-equivariant Q ℓ -adic perverse sheaves sheaves on Gr G generated by the Tate twisted intersection complexes IC µ,ℓ (n), µ ∈ X * (T ) + , n ∈ Z. The affine Q-group scheme of tensor automorphisms Aut ⋆ (ω ℓ ) is isomorphic to G 1 ⊗ Q ℓ by the ℓ-adic geometric Satake equivalence as in [Zhu17a]. This reference is for the ordinary affine Grassmannian, but the arguments carry over to the case of Witt vectors, see also [Zhu18,§2]. Hence, (5.8) implies that there is an isomorphism of Q ℓ -group schemes G 1 ⊗ Q ℓ ≃ G 1 ⊗ Q ℓ for all prime numbers ℓ ∤ p. In particular, G 1 is a reductive Q-group scheme by fpqc descent along the extension Q ℓ /Q for one fixed prime number ℓ ∤ p. We conclude G 1 ≃ G 1 by the isomorphism theorem for split reductive groups.
Remark 5.9. Theorem 5.7 implies Theorem 5.2 by passing to ind-completions. Indeed, the category of algebraic representations Rep Q ( G 1 ) is compactly generated by its subcategory Rep fd Q ( G 1 ) of finite-dimensional representations. Similarly, MTM(L + G\LG/L + G) is compactly generated by MTM(L + G\LG/L + G) c .