The motivic Satake equivalence

We refine the geometric Satake equivalence due to Ginzburg, Beilinson-Drinfeld, and Mirkovi\'c-Vilonen to an equivalence between mixed Tate motives on the double quotient $L^+ G \backslash LG / L^+ G$ and representations of Deligne's modification of the Langlands dual group $\hat G$.

1. Introduction 1.1. Motivation and goals. Split reductive groups are classified by their root data. These come in pairs, consisting of a root datum and its associated dual root datum. Accordingly, to every split reductive group G, there is associated its (Langlands) dual group G.
The work of Kazhdan and Lusztig [KL79,KL80] shows that the representation theory of G is closely related to the singularities arising in certain orbit closures inside (affine) flag varieties associated to G. Building upon [Lus83], the work of Ginzburg [Gin00], Belinson-Drinfeld [BD99] and Mirković-Vilonen [MV07] revealed an equivalence of symmetric tensor categories between the category of finite-dimensional G-representations and the category of certain sheaves on an infinite-dimensional variety Gr G known as the affine Grassmannian of G. This categorical equivalence is called the geometric Satake equivalence. It is an important tool in geometric representation theory which appears in different contexts and has a wide range of applications. For further details on the subject, the reader may refer to the notes of Baumann and Riche [BR18] and of Zhu [Zhu17b], to [Ric14,RZ15] for the relation with the classical Satake isomorphism (for which see [Gro98]), and to [Zhu17a] for a Satake equivalence in the case of mixed characteristic.
The goal of the present manuscript is to provide a motivic refinement of the geometric Satake equivalence. This has both philosophical and concrete consequences: the above papers devoted to the Satake equivalence use different base schemes, and also use different cohomology theories. It is therefore desirable to describe the common content of such different approaches, which is a goal accomplished in this paper. As far as concrete applications are concerned, let us point out that one of our main motivations is the work of V. Lafforgue [Laf18] on the Langlands parametrization for global function fields. V. Lafforgue in particular conjectures [Laf18,Conj. 12.12] that this parametrization is of motivic origin independent of an auxiliary prime number ℓ coming from the use of ℓ-adicétale cohomology. A first evidence for Lafforgue's conjecture is the construction of intersection cohomology motives on moduli stacks of G-shtukas alias IC-Chow groups in [RS20]. The motivic Satake equivalence established in this paper is a second step of an ongoing project whose goal is to provide a motivic approach to V. Lafforgue's Langlands parametrization.
1.2. Results. Let G be a Chevalley group over Z (=split reductive group scheme [Con14]), and fix T ⊂ B ⊂ G, a split maximal torus contained in a Borel subgroup over Z. The loop group of G is the group-valued functor on the category of rings R given by LG(R) = G(R((̟))). Its subgroup functor L + G(R) = G(R[[̟]]) is the positive loop group. Here R[[̟]] ⊂ R((̟)) denotes the ring of power series in a formal variable ̟, contained in its Laurent series. For every finite field F q , the classical Satake isomorphism [Gro98] is an isomorphism of Q( √ q)-algebras where √ q is a fixed square root of q needed in the construction. The left hand side of (1.1) are thus Q( √ q)valued functions supported on finitely many double cosets. The convolution of such functions turns the left hand side into an algebra known as the spherical Hecke algebra. On the right hand side of (1.1) the group G is the Langlands dual group of G formed over Q (with respect to a fixed pinning). Then R G is the Grothendieck Q-algebra of the category of representations of G on finite-dimensional Q-vector spaces. Its ring structure is given by the tensor product of representations. Writing V µ for the simple G-representation of highest weight µ, where µ ∈ X * (T ) + is a dominant cocharacter, their classes [V µ ] form a Q-basis of R G . Under (1.1), these correspond to functions which are related to the singularities of an infinite-dimensional space as follows. The affine Grassmannian is theétale sheaf quotient which is representable by an ind-projective ind-scheme (=infinite union of projective Z-schemes) equipped with a left action of L + G. For each dominant cocharacter µ ∈ X * (T ) + , we denote by Gr ≤µ G the schemetheoretic image of the orbit map L + G → Gr G , g → g · ̟ µ · e where e ∈ Gr G (Z) is the base point. Then Gr ≤µ G → Spec(Z) is a projective scheme, usually singular, which contains the open smooth L + G-orbit Gr µ G ⊂ Gr ≤µ G as a fiberwise dense open subscheme. There is a presentation on the underlying reduced locus Gr G red = colim µ∈X * (T ) + Gr ≤µ G .
For a finite field F q and each auxiliary prime ℓ ∤ q, let IC µ,q,ℓ be the ℓ-adic intersection complex of Gr ≤µ G ⊗ Z F q in the sense of Goresky-MacPherson-Deligne. A surprising observation of [Lus83] is that the class [V µ ] corresponds under (1.1) (up to a power of √ q) to the trace of Frobenius function of IC µ,q,ℓ given by Grothendieck's sheaf function dictionary.
The geometric Satake equivalence is a categorification of (1.1). It is known in several settings using different cohomology theories: in [Gin00,MV07,BR18] the authors work with Gr G ⊗ Z C using Betti cohomology (the latter two with more general coefficients however), whereas [BD99] works with Gr G ⊗ Z C using D-modules, and [Ric14,RZ15] works with Gr G ⊗ Z k for general fields k using ℓ-adicétale cohomology. Here we provide a motivic refinement.
In analogy with the left hand side of (1.1) we consider the double quotient L + G\LG/L + G → Spec(Z) viewed as a groupoid-valued functor on the category of rings. For each such functor we have constructed in [RS20] a category of motives (with rational coefficients) (1.2) DM L + G\LG/L + G = DM L + G\LG/L + G; Q .
The collection of all such categories is equipped with a Grothendieck six functor formalism (with certain restrictions on the * -pullback). The construction in op. cit. builds upon the recent advances in the theory of motivic sheaves due to Ayoub [Ayo07a,Ayo07b,Ayo14] and Cisinski-Déglise [CD19,CD16] as envisioned by Beilinson. In there we consider a much smaller full subcategory of stratified Tate motives To make the connection with (1.1) we base change the groups LG Fq := LG ⊗ Z F q and L + G Fq := L + G ⊗ Z F q to a finite field. Then the analogue of Theorem B holds with MTM(L + G\LG/L + G) replaced by MTM(L + G Fq \LG Fq /L + G Fq ).
Theorem C. For each finite field F q , there is an equivalence of symmetric monoidal categories using the tensor structure from Theorem B. Under this equivalence, the motivic cohomology functor ω corresponds to the forgetful functor Rep Q ( G 1 ) → Vect Q . For each prime ℓ ∤ q, this equivalence gives under the ℓ-adicétale realization the geometric Satake equivalence as explained in [Zhu17b,5.5.14]. (6.8, 6.12) Among other things, Theorem C asserts that the left hand category is semi-simple. This semi-simplicity is inferred, via Lusztig's parity vanishing, from the semi-simplicity of the abelian category MTM(F q ). The latter semi-simplicity holds since higher algebraic K-theory of F q is torsion by Quillen's computation, see Example 6.12. This semi-simplicity is then lifted to the mixed Tate motives on the double quotient over a finite field. Passing to the trace of the Frobenius function as in, say, [Cis19] one recovers the Satake isomorphism similar to (1.1) where one now considers Q-valued functions and a quotient of the representation ring R G1 , cf. §6.4.
In contrast to MTM(F q ), the categories MTM(Z) and, a fortiori, MTM(L + G\LG/L + G) are no longer semi-simple. More generally, if S is a sufficiently nice scheme which satisfies the Beilinson-Soulé vanishing (e.g. the spectrum of finite fields as above; number fields or their rings of integers; function fields over a finite field or their rings of integers; or filtered colimits of these rings) the category of mixed Tate motives (1.5) MTM L + G S \LG S /L + G S is well-defined and satisfies Theorem B where we denote L (+) G S := L (+) G × Spec(Z) S. In the category (1.5) we also have the intersection motives IC µ,S (n) for µ ∈ X * (T ) + , n ∈ Z. We denote by Sat G,S the full semi-simple subcategory of (1.5) generated by the intersection motives by means of direct sums. This subcategory Sat G,S is stable under convolution, and hence inherits a symmetric monoidal structure.
Theorem D. Let p : S → Spec(Z) be a base scheme as above.
i) The pullback of motives induces an equivalence of symmetric monoidal categories Sat G,Z → Sat G,S , IC µ,Z (n) → p * IC µ,Z (n) = IC µ,S (n), and hence Sat G,S ≃ Rep Q ( G 1 ) by Theorem C independently of S. (6.6) ii) Let U S be the pro-unipotent algebraic Q-group arising from extensions in the category MTM(S). Then there is an equivalence of symmetric monoidal categories where on the right-hand side is the category of representations of the pro-algebraic group U S ⋊ G 1 on Q-vector spaces. (6.14, 6.15) Part i) of Theorem D precisely formulates the experimental fact that under the geometric Satake isomorphism the dual side does not depend on the base scheme over which the affine Grassmannian is defined. Part ii) is, in part, an extension of Levine's work [Lev93] which one recovers in the special case where G is the trivial group.
1.3. Related and future work. Zhu [Zhu18] has sketched the construction of a motivic Satake equivalence over F q using the category of numerical motives of Jannsen. Zhu's 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, which we expect to be fruitful also for our upcoming work. One may imagine using the theory of Nori motives to produce an abelian category of motives related to the Satake equivalence. Nori motives, however, depend upon the cohomology theory chosen at the outset. In the case of motives over F q , say, this would in practice mean choosing ℓ-adic cohomology for some ℓ prime to q. Again, the choice of working with motives as developed by Ayoub and Cisinski-Déglise is based on the desire to apply it to a Langlands parametrization over function fields, where we precisely seek to avoid a reference to ℓ-adic cohomology.
Throughout this paper, motives have rational coefficients. Using upcoming work of Spitzweck on tstructures on Tate motives with integral coefficients, it would be very interesting to establish a Satake equivalence in this situation. The reader is referred to [Zhu21] for a result on the level of functions.
There are versions of the geometric Satake equivalence using different affine Grassmannians such as the Witt vector (or p-adic) affine Grassmannian of Zhu [Zhu17a], the B dR -affine Grassmannian of Fargues-Scholze [FS21]. In subsequent work [RS21], we extend the methods of this paper to cover a Satake equivalence for Witt vector Grassmannians.
As was stated above, we conceive the results in [RS20] and the Satake equivalence in this paper to be two steps in a long-term program aiming to prove a motivic version of V. Lafforgue's Langlands parametrization over function fields. The immediate next step, to be addressed in a subsequent paper, is to improve on Theorem C by proving a motivic version of Gaitsgory's factorization (or fusion) version of the geometric Satake equivalence [Gai07]. This will require suitable Whitney-Tate properties of Beilinson-Drinfeld Grassmannians, as opposed to the affine Grassmannian Gr G encountered above. Here the six functor formalism for the categories of motives mentioned in Theorem C will be crucial. Further steps in this program include a motivic Drinfeld lemma, a motivic construction of excursion operators, and their identification with Hecke operators. All these remain to be done as well.

Motives on affine flag varieties
In this section, we recollect and extend some material from [RS20] as is needed throughout this manuscript. In §2.1, we state some facts on loop groups and their affine flag varieties. The next §2.2 treats motives on prestacks which is applied in § §2.3-2.4 to affine flag varieties. §2.5 gathers some facts pertaining to Kazhdan-Lusztig parity vanishing.
Notation 2.1. Throughout this manuscript, S is an irreducible, regular scheme which is separated of finite type over a Noetherian, excellent, separated and at most 2-dimensional scheme. Further, we assume that S satisfies the Beilinson-Soulé vanishing conjecture (cf. [RS20, (3.2.2)]), and admits an ℓ-adic realization functor in the sense of [RS20, §2.1.2, Rmk. 3.2.9].
Examples include finite fields, number fields and function fields of curves over finite fields, their rings of algebraic integers and filtered colimits of these rings.
2.1. Loop Groups and their affine flag varieties. We refer the reader to [RS20,§4] for further details and references on the following material.
We denote by AffSch S the category of affine schemes Spec(R) → S equipped with a map to S. Let G be a split reductive group scheme over S, for example G = GL n,S . The loop group LG is the presheaf LG : AffSch op S → Groups, Spec(R) → G(R((̟))), where R((̟)) is the ring of Laurent series in the formal variable ̟. It is represented by an ind-affine ind-scheme over S, and in particular LG is an fpqc sheaf on AffSch op S . We fix T ⊂ B ⊂ G over S, a split maximal torus contained in a Borel subgroup. Let A = A (G, B, T ) be the standard apartment with origin 0 defined by G and standard alcove a defined by B. We only consider facets f ⊂ A which are contained in the closure of a. Attached to f is the parahoric subgroup P f ⊂ LG which is an S-affine, S-flat closed subgroup scheme. For this paper, the most important case is f = 0, in which case P 0 =: L + G is the positive loop group given by the presheaf ). If f = a, then P a =: B is the standard Iwahori subgroup defined as the preimage of B under the map L + G → G, ̟ → 0.

5
Theétale sheafification of the quotient Fl f := (LG/P f ) et is called the partial affine flag variety associated with f . It is represented by an ind-projective ind-scheme over S. For f = 0, it is denoted by Gr = Gr G , and called the affine Grassmannian.
Given two facets f ′ , f ⊂ā ⊂ A , the orbits of the P f ′ -left-action on Fl f are enumerated by the double quotient W f ′ \W/W f of the Iwahori-Weyl (or extended affine Weyl) group W = W (G, T ) by the subgroups W f ′ , W f ⊂ W generated by the reflections preserving f ′ resp. f . The choice of a defines a length function , and a Bruhat partial order ≤ on the double coset. For each w ∈ W f ′ \W/W f , the locally closed immersion of the P f ′ -orbit of w is denoted by Then Fl ≤w f → S is a proper scheme called the (affine) Schubert scheme. It contains Fl w f as an open Ssmooth subscheme which is fibrewise dense and which is called the (affine) Schubert cell. For each map Spec(k) → S from a field, the base change Fl w f × S Spec(k) ⊂ Fl ≤w f × S Spec(k) identifies on the underlying reduced locus with the Schubert cell, resp. Schubert variety over k attached to the class w and the k-group is the partially ordered set of dominant cocharacters with length function l = l(0, 0) : X * (T ) + → Z ≥0 , µ → 2ρ, µ where ρ denotes the half sum of the B-positive roots and -, -: X * (T ) × X * (T ) → Z is the natural pairing. In the case of the affine Grassmannian, we denote the orbits (resp. orbit closures) by Gr µ ⊂ Gr ≤µ for µ ∈ X * (T ) + .

Motives on prestacks.
We refer the reader to [RS20,§2] for further details and references on the following material.
We consider the triangulated category of motives with rational coefficients where Sch ft S is the category of finite type schemes over S. This category is denoted by DA(X, Q) in [Ayo14] and by D A 1 ,et (X, Q) in [CD19]. Categories of motives with rational coefficients admit a full six functor formalism: there are pairs of adjoint functors (f * , f * ), (f ! , f ! ) for a map f ∈ Sch ft S and (-⊗ -, Hom(-, -)) satisfying the usual compatibilities such as smooth/proper base change, Poincaré duality, Künneth/projection formula etc. Following Hoyois [Hoy17] and Khan [Kha16], this can be upgraded to a presheaf of ∞-categories where DGCat cont is the category of presentable, stable, Q-linear, dg-∞-categories with colimit-preserving functors. The ∞-category DGCat cont is complete and cocomplete, i.e., admits all (homotopy) limits and (homotopy) colimits, so that the following Kan extensions are available.
Definition 2.4. i) Let AffSch ft S ⊂ AffSch S be the full subcategory of objects of finite type over S. Throughout, we will replace this category by a small skeleton containing the objects of interest to us. ii) Fix some regular cardinal κ, and let AffSch κ S := Pro κ-small (AffSch ft S ) be the category of κ-small pro-objects in AffSch ft S . iii) The ∞-category of prestacks is defined as PreStk κ S := Fun((AffSch κ S ) op , ∞-Gpd) where ∞-Gpd is the ∞-category of ∞-groupoids (also called spaces). Hereafter, we will usually drop the κ from the notation so that PreStk := PreStk κ S , AffSch S := AffSch κ S . iv) Define the functor to be the right Kan extension of the preceding functor along the Yoneda embedding AffSch S ⊂ PreStk S .
We emphasize that DM ! in (2.5) encodes a category of motives DM(X) (with rational coefficients) for each prestack X, and for each map f : X → Y in PreStk S a colimit-preserving functor f ! : DM(Y ) → DM(X). This definition follows the approach of Gaitsgory-Rozenblyum and Raskin. We refer to [RS20, §2.2] for references and also for further discussion of the definition.
Theorem 2.6. i) The presheaf DM ! : PreStk op S → DGCat cont is a sheaf in theétale topology. For each prestack X ∈ PreStk S the ∞-sheafification X → X et induces an equivalence on categories of motives DM(X et ) Lemma 2.10. Let π : X → S be a smooth surjective map of schemes of relative dimension d with connected fibers. We assume that X admits a stratification in the sense of [RS20, Def. 3.1.1] by schemes of the form V(E) (where E is a vector bundle over S), e.g., by affine spaces over S. Then there is an equivalence of categories where the Tateness of motives on X is with respect to the stratification by a single stratum.
Proof. By the conventions in Notation 2.1, S is connected and hence so is X [Sta17, Tag 0378]. The functor is fully faithful by [RS20, Lem. 3.2.12]. For essential surjectivity, we first claim that Hom S (M, N [1]) = Hom X (π * M, π * N [1]) for M, N ∈ MTM(S). We prove this by induction on the number of strata in X. If X = V(E) is a single stratum, then this holds even for all N ∈ DTM(S). For the inductive step we use as in loc. cit. the localization sequence for a minimal stratum Let π Z := π • i, π U := π • j. Since X is connected and is assumed to have at least two strata, the codimension c := codim X Z is positive. By induction, the composite ) is an isomorphism. By the localization sequence, the kernel of the right hand map is mapped onto by Hom Z (π * Z M, π * Z N (−c)[1 − 2c]) which vanishes by the Beilinson-Soulé condition for Z (equivalently, for S). Hence the left hand map above is an isomorphism as well, showing our claim.
The generators 1(n)[d], n ∈ Z of MTM(X) trivially lie in the image of our functor, so we are done by using that π * is an isomorphism on the level of extensions of mixed Tate motives by the above claim.
2.4. Changing the base scheme. For facets f ′ , f ⊂ā ⊂ A , we show that the category DTM(Fl f ) for the stratification in left-P f ′ -orbits is, to a certain extent, insensitive to the choice of the base scheme S, cf. Theorem 2.14 below. In §6.1, we will sharpen this idea by introducing the (abelian) Satake category Sat G ⊂ DTM(Gr G ) and showing that this category is completely independent of the base scheme S.
Let f : T → S be a map of schemes, where T is Noetherian and of finite Krull dimension, so that f * : DM(S) → DM(T ) is well-defined. (An important example to have in mind is T = Spec F p → S = Spec Z.) We indicate base changes to T by a subscript, e.g., G T := G × S T . We still write f for all maps obtained using such base changes, e.g., f : Fl f ,T → Fl f ,S . The condition in [RS20, Thm. 2.4.2] is satisfied, so that we obtain a functor f * : DM(Fl f ,S ) → DM(Fl f ,T ). As before, write ι w : Fl w f → Fl f for the inclusion of the P f ′ -orbits, both over S and over T . We clearly have an equivalence (ι w ) * f * ≃ → f * (ι w ) * by functoriality, so that f * restricts to a functor Here is the key lemma concerning the change of the base scheme.
Lemma 2.12. Let w ∈ W f ′ \W/W f . The following natural transformations of functors, when restricted to the indicated categories of Tate motives, are equivalences.
Proof. The claim for (ι w ) ! results from base change. The claim for (ι w ) ! will follow from the others using an induction argument based on the localization fiber sequence for any complementary closed (resp. open) embedding i (resp. j).
In order to show that f * commutes with (ι w ) * , we may assume that f ′ = a (the base alcove), since the stratification by P f ′ -orbits is coarser than the one by Iwahori orbits so that the claim for the Iwahori stratification together with a localization argument implies the one for the stratification by P f ′ -orbits.
We first show the claim for f = a. By [RS20, Prop. 5.2.2], DTM(Fl a ) is the smallest cocomplete full subcategory of DM(Fl a ) which contains the twists of the unit motives supported at the base points {τ } for each τ ∈ Stab a ⊂ W and which is stable under the operation π ! s π s,! along the smooth proper projection maps π s : Fl a → Fl s := Fl fs for all simple reflections s ∈ S. We proceed by induction on the length of w, the case l(w) = 0 being trivial since ι w is a closed embedding of a base point τ in this case. For l(w) > 0, let w = v · s be a reduced expression with s ∈ S. We obtain a fibre sequence (2.13) (ι v ) * 1 → π ! s π s,! (ι v ) * 1 → (ι w ) * 1, which is the dual of the fibre sequence [RS20, (5.1.2)]. Using induction l(v) < l(w), the functor f * commutes with (ι v ) * . Since π s is smooth, f * also commutes with π ! s , and hence with (ι w ) * by (2.13). This finishes the case f = a. Now, for a general facet f ⊂ā, we reduce the claim to the one previously considered using the map π : Fl a → Fl f . This map is smooth, proper, surjective and a stratified map with respect to the Iwahori stratification on both ind-schemes [RS20, Lem. 4.3.13]. Therefore, in the cartesian diagram is itself stratified by some Iwahori strata on Fl a as indicated above. Using that π is smooth, (ι w ) * π * = π * (ι w ) * . Moreover, π * commutes with f * . Finally, π * is conservative. Thus, to show that f * commutes with (ι w ) * , we may replace the inclusion ι w (both on T and on S) byι w . Using again a localization argument, we then reduce this statement to the one for the inclusions ι v : Fl v a → Fl a of the Iwahori strata in the flag variety refining the preimage stratification under π.
Recall from [RS20, Thm. 2.4.2] that both categories carry weight structures. The aim of this section is to prove: Theorem 2.14. Let f : T → S be a map of schemes both satisfying the conditions in Notation 2.1. Then the functor (2.11) has the following properties: i) it is conservative. ii) it creates weights, i.e., M ∈ DTM(Fl f ,S ) is of weights ≥ n (resp. ≤ n) iff f * M has the corresponding property. iii) it creates the t-structure, i.e., M ∈ DTM(Fl f ,S ) is of in the "≥ n" (resp. "≤ n") part of the t-structure iff f * M has the corresponding property.
We need some preparation for the proof.
Proposition 2.15. In the situation of Theorem 2.14, the functor (2.11) is t-exact with respect to the perverse motivic t-structures (cf. Theorem 2.8), and commutes with the intermediate extension functors (j w ) ! * defined in (2.2). In particular, Proof. For both base schemes S and T , the subcategory DTM(Fl f ) ≤0 consists by definition precisely of those objects M such that ι * M ∈ DTM(Fl + f ) ≤0 where ι : Fl + f = ⊔ w Fl w f → Fl f denotes the disjoint union of the inclusions of all strata. Likewise with "≥ 0" and ι ! instead. To show the exactness of f * we may by Lemma 2.12 replace f by the induced map f + : Fl + f ,T → Fl + f ,S . It then remains to observe that the following diagram is cartesian and has smooth vertical maps to the base scheme (which is the same for S, resp. T ). The remaining claim now follows from Lemma 2.12 which ensure that f * commutes with all functors involved in the formation of j w,! * := im( p H 0 (j w,! ) → p H 0 (j w, * )).
Proof of Theorem 2.14. For i), let M ∈ DTM(Fl f ,S ). For the conservativity, we have to show f * M = 0 implies M = 0. Using the non-degeneracy of the perverse motivic t-structure [RS20, Cor. 3.2.6] and the t-exactness of f * , it is enough to show the conservativity of f * | MTM(Fl f ,S ) . Any M ∈ MTM(Fl f ,S ) is the filtered colimit of its compact subobjects, so we may assume M is also compact. Then, M has a Jordan-Hölder series with simple constituents given by twisted intersection motives [RS20, 3.3.8]. We may thus assume that M is an intersection motive, so we are done by (2.16). For ii), we need to show that f * is weight-exact and detects weights. As in the proof of Proposition 2.15, to show that f * is weight-exact, we may replace Fl f by Fl + f over both base schemes S and T , which is again clear by definition of the weight structures. The detection of weights then follows from Lemma 2.17 below using part i). For iii), we use likewise the conservativity and t-exactness of f * .
Lemma 2.17. A conservative, weight-exact functor F : C → D between triangulated categories with weight structures detects weights: if F (M ) has weights < n (resp. ≥ n) for some M ∈ C, then the same is true for M .
Proof. We use that M has weights ≥ n (resp. < n) iff for any weight truncation triangle E : M <n s<n → M s ≥n → M ≥n , the maps s ≥n (resp. s <n ) are isomorphisms. Indeed, the "⇐" direction holds by definition, the converse also follows from elementary applications of the axioms, see [Fon17,Cor. 2 .
This functor f * preserves the subcategories of equivariant Tate motives.
Proof. By construction, f * is the unique functor which is given by the usual f * on the level of finite type S-schemes and compatible with the insertion functors DM(Fl ≤w f ) → DM(Fl f ) (both over S and T ). By [RS20, Cor. 2.3.4], it is therefore enough to construct a functor There is a split pro-unipotent subgroup U ⊂ P f ′ such that the quotient K := P f ′ /U is smooth and of finite type and the P f ′ -action on Fl ≤w Finally, in order to check the existence of f * on this level, it is enough to observe that the maps in the bar construction Bar(K, Fl ≤w ) are all smooth, and hence !-pullback along them commutes with f * . Hence the f * -functors in all levels of the diagram DM ! (Bar(K, Fl ≤w f )) glue to a functor on the limit of this diagram, which is DM(K\ Fl ≤w f ). Given that the underlying non-equivariant functor of f * is just f * , the preservation of equivariant Tate motives follows from (2.11).
2.5. Kazhdan-Lusztig parity vanishing. We now apply Proposition 2.15 to prove the Kazhdan-Lusztig parity vanishing [KL80, Thm. 5.5] (see also [Lus83,Thm. 11.c)]) for the intersection motives. Our main tool is the ℓ-adic realization functor which exists by assumption on S (Notation 2.1). We continue with the notation and assumptions from §2.4. In particular, we fix two facets f , f ′ contained in the closure of the standard alcove a, and denote by DTM(Fl f ) the category of (f ′ , f )-stratified Tate motives whose heart is the abelian category MTM(Fl f ) (Theorem 2.8).
Theorem 2.19. ([RS20, Thm. 5.2.3]) The restriction of the ℓ-adic realization functor The following corollary is useful in lifting results from the ℓ-adic to the motivic setting.
Corollary 2.20. For each geometric point f :s → S, the composition of functors is well-defined, exact, conservative and faithful.
Proof. Each object in MTM(Fl f ) c admits a Jordan-Hölder series (Theorem 2.8 i)) whose simple constituents are the intersection motives IC w (n) for w ∈ W f ′ \W/W f and n ∈ Z. Using the same method as in Proposition 2.15 we deduce that these are mapped under ρ := f * • ρ ℓ to the corresponding ℓ-adic intersection complex on Fl ≤w f ,s . Since the subcategory Perv( Being the restriction of an exact functor between triangulated categories, ρ is exact as well. For the conservativity of ρ it is therefore enough to show that the simple objects, namely the IC w (n) are not mapped to 0, which holds true by the above.
Being an exact conservative functor between abelian categories, ρ is also faithful: If a morphism p : A → B maps to 0 under ρ, then ker ρ(p) = ρ(ker p) = A by exactness. Hence, the natural map ker p → A is mapped to an isomorphism, and therefore is an isomorphism by conservativity. This shows p = 0.
We can now prove the Kazhdan-Lusztig parity vanishing for intersection motives. Recall that for each where H i denotes the truncation with respect to the classical motivic t-structure (cf. [RS20, Rem. 3.2.7]) which agrees on S also with the perverse motivic t-structure (Theorem 2.8).
Proof. We may assume n = 0. By Corollary 2.20, it is enough to show that, for S being the spectrum of a separably closed field, the ℓ-adic intersection complex IC w,ℓ on Fl ≤w f satisfies the parity vanishing (2.22) where H i denotes the classical cohomology functor. This case is certainly well-known; we recall the part of the argument where we did not find a reference for the reader's convenience.
Reduction to the case f ′ = f = a. By refining the orbit stratification on Fl f we may assume that f ′ = a is the standard alcove. Now consider the projection π : Fl a → Fl f which is a smooth surjective map of relative dimension d := dim(P f /P a ) by [RS20,Prop. 4.3.13]. The preimage π −1 (Fl ≤w f ) is a Schubert scheme in Fl a , and it follows from e.g. [RS20, Lem. 4.3.7 iii)] that where w max is the unique representative of right maximal length with respect to l a := l(a, a) in w · W f . Its length is l a (w max ) = dim(Fl ≤wmax a ) = l(w) + d by loc. cit.. As taking intermediate extensions commutes with smooth pullback, we have π * [d]IC ℓ,w = IC ℓ,wmax . Taking the cohomological shift into account and using the conservativity of pullback of surjective maps, we see that it is enough to prove (2.22) in the case f ′ = f = a.

The convolution product
In this section, we will discuss the tensor structure on the category DM(P f \LG/P f ) given by convolution. We start with the definition and basic properties in §3.1. In §3.2, we show that the convolution product preserves stratified Tate motives.

Definition and associativity.
Definition 3.1. Let f ′ , f , f ′′ be three facets in the closure of the standard alcove, see §2.1. The convolution product is the functor Here the maps are the natural maps of prestacks induced by the identity on LG × LG (for p) and the multiplication LG × LG → LG (for m). LG ii) The exterior product For clarity, we will momentarily denote the functor in (3.3) by ⊠ R and the convolution product stemming from this choice by ⋆ R . Alternatively, we may consider The resulting convolution product functor is denoted by ⋆ L . Proposition 3.4. On the level of the homotopy categories, the two functors ⋆ R and ⋆ L are naturally isomorphic, i.e., one has ⋆ R ∼ = ⋆ L as functors LG(S) is any representative of w, and likewise for LG ≤v . Note that this agrees with the preimage of Fl ≤w f under the quotient map LG → Fl f , resp. the preimage of Fl op,≤w The labels ≈ at the horizontal arrows indicate maps of prestacks which become equivalences afterétale sheafification and therefore descent equivalences upon applying DM (Theorem 2.6).
By Corollary A.15, we have an exterior product, denoted by ⊠, for motives on placid prestacks such as the top middle term. Under the descent equivalence, it is compatible with the exterior product ⊠ L for motives on prestacks of the form as in the top left term, and similarly with the top right term. Of course, the same applies for the middle row as well. Moreover, these identifications are compatible with the pushforwards along the maps (induced by the closed embeddings i v,w ) between the top and middle row, i.e., there is a natural equivalence where ⊠ w stands for an exterior product on terms as in the middle row of the diagram, and likewise for ⊠ v . For yet another u ≤ v in W f ′ \W/W f , this equivalence and the one for i u,v and i u,w are compatible.
We obtain that the equivalence of ∞-categories is compatible with exterior products (⊠ L and ⊠ R , respectively), provided that we pass to the homotopy category and restrict to objects which are supported on some Fl ≤w,op , resp. Fl ≤w . In particular, this is true for compact objects. We may drop this compactness condition, since the homotopy category of a compactly generated category, such as the categories DM on the above prestacks, is again compactly generated by [Lur17, Rem. 1.4.4.3], and since the exterior product preserves filtered (homotopy) colimits separately in both variables.
Remark 3.6. The point of passing to the homotopy categories Ho(DM) is that these are ordinary categories, as opposed to ∞-categories DM. For this reason, it is enough to check the compatibility of α v,w for two composable maps, as opposed to verifying higher coherences. We do not expect this loss of information to be necessary though: a more full-fledged approach would be to establish that DM ! is a symmetric lax monoidal functor on the ∞-category of ind-placid prestacks, such as P ′ f \LG/P f . Hereafter, we will write ⋆ for ⋆ R above. Since our main interest in this paper lies in the convolution product on the abelian (in particular ordinary) category MTM(L + G\LG/L + G), Proposition 3.4 shows that there is no ambiguity in the definition of the convolution product on this category.
As is well-known, the associativity of the convolution product is a consequence of the base-change formula: The square in the above diagram is (homotopy) cartesian. Moreover, the map m is ind-proper, so that proper base change [RS20, Prop. 2.3.3] yields an equivalence Thus, the convolution (A⋆B)⋆C can be computed by pullback and pushforward along the pictured composite correspondence. Considering instead the composition of the correspondences computing A⋆(B⋆C), we obtain the same composition, which yields a zig-zag of equivalences.
3.1.1. Reformulation in terms of schemes. We now spell out the above definition in terms of ordinary schemes as opposed to prestacks. This relates to the classical definition of the convolution product as in [Ric14,§2], and is used to show that the convolution product preserves Tate motives (Theorem 3.17 below).
Definition 3.9. Let f ′ , f , f ′′ be facets as in Definition 3.1. We define which is an ind-proper S-ind-scheme. Consider the following commutative diagram of prestacks: The left-hand horizontal maps such as u are the natural quotient maps. By [RS20, Lem. 2.2.7], the !pullback along such a map can be regarded as forgetting the P f ′ -action on some motive. According to [RS20, Thm. 2.2.16], the horizontal maps labelled "≈" induce descent type equivalences after applying DM ! . We will use similar equivalences without further comment; for example we identify motives on the double quotient P f ′ \LG/P f with those on P f ′ \ Fl f . The terms in the right hand column will be explained further below. For Letm : Fl f × Fl f ′′ → Fl f ′′ be the map of ind-schemes induced by multiplication, i.e., the map m above is the non-sheafified version obtained by passing to the left-P f ′ -quotients. By virtue of the following lemma, we will denotem simply by m (it will be clear from the context which version we mean).
Proof. We only need to show that the functor m ! commutes with the forgetful map to its non-equivariant versionm ! . This is precisely the characterization of m ! in [RS20, Lem. 2.2.9], see also Remark 3.2 i) for its existence.
Both functors, -⋆ -and -⊠ -preserve colimits separately in each variable. They therefore factor over the Lurie tensor product DM(P f ′ \LG/P f ) ⊗ DM(P f \LG/P f ′′ ). Since categories of motives are compactly generated [RS20, Lem. 2.3.6], the functors are therefore determined by their values on compact objects. Suppose then that A and B are compact objects, so they are supported on closed, finite type subschemes X ⊂ Fl f , Y ⊂ Fl f ′′ which are finite unions of Schubert schemes (these are the objects in the right vertical column in (3.10)). The right-most vertical maps in the diagram, such as ι are induced by the closed embeddings of these subschemes. In this case, A ⊠B admits the following description: The vertical map e labelled ≈ in the above diagram yields an equivalence upon applying DM ! (this stems from A 1 -invariance, using that U f ,i is split pro-unipotent, see [RS20, Prop. 2.2.11]). In particular, we can regard B ∈ DM(P f \Y ) as an object in DM(P f ,i \Y ). By the support setup, we can write Note that the schemes X, Y , P f ,i and Z intervening in the correspondence X × P f ,i \Y e•p ← X ×Ym → Z, are of finite type over S (unlike the remaining terms in the diagram). We also see that the above definition of ⊠ agrees with the definition of ⊠ used for example in [Ric14, Lem. 2.20, Rmk. 2.21].
Finally, writing Z :=m(X ×Y ) (scheme-theoretic image, again a finite type S-scheme), A ⋆ B has as its underlying non-equviarant objectm ! (A ⊠B), which is, by proper base change, the !-pushforward along For the second statement note that the map r := e •p in (3.10) is a P f ,i -torsor, in particular a smooth map (of finite type S-schemes). Therefore r ! commutes with the exterior product and with the !-pushforward along the embeddings 3.1.2. Compatibility with the ℓ-adic realization. We denote by D b ct,P f (Fl f ′′ , Q ℓ ) the category of P f -equivariant ℓ-adic sheaves on Fl f ′′ . In [MV07] (see also [Ric14,§2] and [PZ13, §10.2]), the convolution product for ℓ-adic sheaves Proposition 3.14. Under the ℓ-adic realization functor ρ ℓ (cf. [RS20, Thm. 2.3.7]) the convolution product corresponds to the convolution product ⋆ ℓ considered in the context of the ℓ-adic Satake equivalence, i.e., there is a natural isomorphism Proof. The twisted box product ⊠ for motives (Definition 3.9) is formed using descent along !-pullbacks. We need to compare its ℓ-adic realization with ⊠ ℓ , formed using * -pullbacks.
Since the objects A, B are compact, hence supported on S-finite type closed subschemes X ⊂ Fl f , Y ⊂ Fl f ′′ , we can replace the maps p, q in (3.13) by the diagram of P f ,i -torsors for some i ≥ 0 as in §3.1.1 above. Let G := P f ,i which is a smooth affine S-group scheme acting on Z := X i × Y either via the torsor p or q. By definition, the category of G-equivariant ℓ-adic sheaves on Z is defined as where we emphasize that the functors in this limit are the * -pullbacks along the maps in the bar complex. The motivic analogue of that category is DM * (G\Z) c := lim DM * (Bar(Z, G)) c , where again we use * -pullbacks to form the limit. (See also [RS20, Rem. 2.2.2, iv)] for further discussion of the presheaf DM * .) The vertices (G) ×n × Z of the bar construction are separated S-schemes of finite type, and the action and projection maps in this diagram are smooth and affine, noting that G → S is so. We can therefore use the equivalence of DM * with DM ! applied to the smooth morphisms p, q, see Corollary A.8. Under this equivalence the functor p ! (resp. q ! ) corresponds to p * (resp. q * ). Since p, q have the same relative dimension dim(G/S), we can equivalently form A ⊠B using descent along * -pullbacks. Moreover, the map m is ind-proper, so that m * = m ! . We conclude using that ρ ℓ is compatible with the six functors.

Convolution product and change of base scheme.
Lemma 3.15. Let f : T → S be a map of schemes satisfying the assumptions in Notation 2.1. Then, for M 1 ∈ DM(P f ′ \LG/P f ) and M 2 ∈ DM(P f ′ \LG/P f ), there is a natural isomorphism Proof. All functors involved in the definition of ⋆ are compatible with f * . For p ! , this holds true since (thé etale sheafification of) p is a P f -torsor, in particular pro-smooth.
3.2. Preservation of Tate motives. In this section, we show that the convolution product on partial affine flag varieties respects stratified Tate motives. A key point in this proof is the well-knwon distinguished triangle (3.21) below (cf. [KL79, App.]), which is a geometric incarnation of the following formula for the multiplication in the Iwahori-Hecke algebra over a finite field F q given in [Bou68, IV, §2, Ex. 24]: where φ s is the characteristic function of the Iwahori double coset B(F q )sB(F q ) for a simple reflection s, and φ e is the characteristic function of the base point. Here B := P a denotes the standard Iwahori subgroup associated with the choice of the alcove a.
Theorem 3.17. Let the base scheme S be as in Notation 2.1. For any three facets f ′ , f , f ′′ contained in the closure of a, the convolution product restricts to a functor In particular, taking f ′ = f = f ′′ = 0, there is a convolution product on DTM(L + G\LG/L + G).
We show (3.18) in several steps starting with the following key case.
Proposition 3.19. If P f ′ = P f = P f ′′ =: B is the standard Iwahori subgroup, then Theorem 3.17 holds.
Proof. In this case, where ι s ×s and ι s are the inclusion of the open strata and q is the projection onto the second factor. Writing a :=m • ι s ×s and using Lemma 3.12 for ι s ×s = ι s ×ι s , we have to provẽ or equivalently that where the Tateness of the motive is with respect to the stratification of P 1 ≃ Fl ≤s by {∞} ⊔ A 1 ≃ Fl e ⊔ Fl s .
To check this, we use the localization sequence, noting that The outer terms are in DTM(P 1 ), hence so is the middle. This finishes the first case.
Second case. Let w, w ′ ∈ W , and assume l(ww ′ ) = l(w) + l(w ′ ) for the Bruhat length. Then the composition of the open inclusion ι w ×w ′ : Fl w × Fl w ′ → Fl ≤w × Fl ≤w ′ with the multiplication map m is an isomorphism This implies that ι w,! 1 ⋆ ι w ′ ,! 1 ≃ ι ww ′ ,! 1 ∈ DTM(Fl) is Tate and finishes the second case.
General case. Let w, w ′ ∈ W be arbitrary. Fix a reduced expression w ′ = s 1 · . . . · s n where s i are simple reflections and n = l(w ′ ). By the second case, we have ι w ′ ,! 1 ≃ ι s1,! 1⋆. . .⋆ι sn,! 1 where we omit the parenthesis in view of Lemma 3.7. By repeated use of the third case, we conclude that ι w,! 1 ⋆ ι w ′ ,! 1 ∈ DTM(Fl) is Tate. This finishes the general case, and the theorem follows.
Remark 3.24. If k = F q , (3.21) gets mapped by the ℓ-adic realization to Taking the alternating trace of the geometric Frobenius, we obtain the identity (3.16) using the relation trace(Frob |Q ℓ (−1)) = q.
Remark 3.25. The method used in the proof of Theorem 3.17 works more generally for not necessarily split reductive groups G defined over k((̟)) which are residually split, i.e., Fl s ≃ A 1 k whenever s ∈ W is a simple reflection. However, we will not need these non-split cases in this manuscript.
Proposition 3.26. Theorem 3.17 holds in the case that P f ′ = P f ′′ = B is the standard Iwahori subgroup of LG, and any facet f in the closure of the standard alcove (so that B is a subgroup of P f ).
Proof. Let P := P f . For any w ∈ W/W f , w ′ ∈ W f \W , consider the diagram To construct the map s with b•s = id, it suffices to construct a section to the composition of quotient maps which is equivariant for right B-action on the second factor. It follows from [RS20, Prop. 4.3.9] that there exists a closed subscheme U ⊂ B, a finite direct product of some affine root groups (depending on w), such that the multiplication Uẇ × P → L w is an isomorphism. Hereẇ ∈ W is any representative of w ∈ W/W f .
The map Fl w a × Fl w ′ a → Fl w f × Fl w ′ a is stratified with respect to the stratification by B × P-orbits. Hence a ! (ι w,! 1 ⊠ ι w ′ ,! 1) is stratified Tate. Therefore the motive m ! b ! b ! M is stratified Tate by Proposition 3.19, and so is ι w,! 1 ⋆ ι w ′ ,! 1 as a direct summand. This proves (3.18).
We now prove Theorem 3.17 in the general case. By definition, the category DTM(P f ′ \LG/P f ′′ ) consists of those P f ′ -equivariant motives on Fl f ′′ = (LG/P f ′′ ) et whose underlying non-equivariant motive is a Tate motive with respect to the stratification by P f ′ -orbits. Such an orbit is the form X = (P f ′ ,i /H) et , where P f ′ → P f ′ ,i is a smooth S-affine quotient (i.e., of finite type) and H ⊂ P f ′ ,i is a smooth closed subgroup scheme with connected fibers over S. Let e : S → X be the unit section. The composition • M is Tate with respect to the stratification by P f ′ -orbits; • M is Tate at the base point of each P f ′ -orbit; • M is Tate with respect to the (finer) stratification by B-orbits.
We may therefore assume that P f ′ = B is the standard Iwahori. Using that DTM(B\LG/P f ′′ ) consists of P f ′′equivariant motives whose underlying motive on Fl op = (B\LG) et is stratified Tate [RS20,5.3.4], we similarly reduce to the case that P f ′′ = B and therefore deduce Theorem 3.17 in general from Proposition 3.26.

Purity of Tate motives
Throughout §4, the base scheme S is as in Notation 2.1. 4.1. Intersection motives are pure. In this section, we show that the intersection motives IC w for the stratification of Fl f given by the P f ′ -orbits (for arbitrary facets f , f ′ contained in the closure of the base alcove a 0 ) are pure. This will be proven by lifting the corresponding fact for ℓ-adic intersection complexes to motives over F p , which is then extended to more general base schemes using the results of §2.4. ct (Fl f , Q ℓ ) takes values in the subcategory D et,mix (Fl f , Q ℓ ) of mixed complexes. With respect to the standard weight structure on that category (and the motivic weight structure on DM), the functor is weightexact. (This follows from the definition of the motivic weight structure and the fact that in the realization these functors preserve weights, see [Bon14, Prop. 3.6.1.2] for details.) Its restriction to DTM(Fl f ) is texact and conservative by [RS20, Lem. 3.2.8]. It therefore creates the t-structure and the weight structure (Lemma 2.17). Now recall the notation (2.2). Since ρ ℓ also commutes with (ι w ) ! and (ι w ) * , the motive IC w is mapped under ρ ℓ to the ℓ-adic intersection complex on the Schubert variety Fl ≤w f , i.e., to ρ ℓ (IC w ) = (i w ) * (j w ) ! * (Q ℓ [dim(Fl w f )]). By [KW01, Ex. III.10.3], it is pure of weight + dim Fl w f , hence so is IC w itself. For general S, consider the zig-zag S → Spec Z ← Spec F p . By Theorem 2.14 and (2.16), the purity of IC w,S ∈ MTM(Fl f ,S ) is equivalent to the one of IC w,Z ∈ MTM(Fl f ,Spec Z ), which in turn is equivalent to the one of IC w,Fp ∈ MTM(Fl f ,Spec Fp ).

Convolution preserves weights.
In this section, we show that the convolution product preseves the subcategories of motives of weight ≤ n and ≥ n. To prove this, we need the following lemma: Lemma 4.3. For separated schemes X 1 , X 2 of finite type over a perfect field k, the exterior product is weight-exact.
Proof. By the description of the ≤ 0-and ≥ 0-part of the weight structure in [Héb11, Rem. 1.17], we have to show that exterior products of motives of the form (f i ) ! 1(n)[2n], where f i : T i → X i is a proper map and T i is regular for i = 1, 2, is again pure of weight 0.
Since k is perfect, a finite type k-scheme is regular iff it is smooth over k [Sta17, Tag 0B8X]. Hence T 1 × k T 2 is again regular. We conclude using the formula f 1,! 1 ⊠ f 2,! 1 = (f 1 × f 2 ) ! 1 whose proof is straightforward using the base change formula for f * vs. g ! and the projection formula (see [RS20, Synopsis 2.1.1, vii) and x)]). . This equivalence is compatible with ⊠ and !-pushforwards, so we obtain our claim in this case.
In order to state that the convolution product preserves weights, we need to talk about weights on equivariant motives. The idea is simple: a G-equivariant motive is declared to be of weights ≥ 0 or ≤ 0 if its corresponding underlying non-equivariant motive has the corresponding property. The following definition makes this precise. Note that we only define a pair of full subcategories of DM(G\X). We do not claim they constitute a weight structure, i.e., we do not assert the existence of weight truncation triangles.
Remark 4.6. The smooth transition maps preserve the subcategory of objects of weight ≥ 0 and also ≤ 0, so the limits make sense.
The definition is independent of the choice of the presentation of G\X: if G\X = G ′ \X ′ , then M ∈ DM(G \ X) is of weights ≤ 0, say, if it is so on X (under !-pullback) and therefore on G ′ × X = G × X ′ . Here and in the following we use the standard weight preservation properties under smooth pullback and (in (4.7) below) also under proper pushforward [Héb11, Thm. 3.8]. The projection map G × X ′ → X ′ is smooth and surjective, hence M | X ′ is of weights ≤ 0 by Lemma 2.17.
The following weight preservation property will be central to the stability of the Satake category Sat G under convolution (see Lemma 6.5). We only consider compact objects since it is enough for our purposes, but the statement could be extended to arbitrary ones, at the expense of a more lengthy discussion of weights in that case.
where Fl f = colim Fl f ,i is a presentation as an ind-scheme (with transition maps denoted by t ij ) and P f ,i is an appropriate finite-type quotient of P f acting on Fl f ,i , and colim denotes the colimit in the ∞-category of ∞-categories, which (see loc. cit.) can in this case just be thought of as the union of the above ∞-categories, as i grows. Using that (t ij ) ! is weight-exact, we define and likewise for ≥ 0.
Proposition 4.8. The convolution product for Tate motives is weight-exact, i.e., for any objects A, B ∈ DTM(P f \ Fl f ) c,w≤0 , their convolution A ⋆ B is also of weights ≤ 0 and likewise with ≥ 0.
Proof. We first assume S is (the spectrum of) a perfect field k or just S = Spec F p , in which case we show the stronger statement that the convolution product on DM(P f \ Fl f ) c (as opposed to DTM) preserves weights. We use the notation in the diagram (3.10) and the discussion around it. The functor − ⊠ − : DM(X) × DM(P f ,i \ Y ) → DM(X × P f ,i \ Y ) preserves weights by Definition 4.5 and Lemma 4.3. Similarly, (e •p) ! preserves weights: e •p is a smooth map of prestacks, i.e., it admits a smooth covering on which the map is a smooth map of finite type k-schemes. Hence (e •p) ! preserves weights by the same argument as in Definition 4.5. Finally,m is proper so thatm ! preserves weights again. Hence the non-equivariant motive underlying A ⋆ B, namelym ! (A ⊠B) has the same weights as A ⊠ B, so that the same holds for A ⋆ B itself.
For general S, we consider parallely the structural map f : T := S → Spec Z and the closed immersion f : T := Spec F p → Spec Z. The functor f * : DTM(P f ,Z \LG Z /P f ,Z ) → DTM(P f ,T \LG T /P f ,T ) creates the weight structure by Theorem 2.14. We conclude by using Lemma 3.15.

Mixed Tate motives on the affine Grassmannian
In this section, we endow the category MTM(L + G\LG/L + G) with a Tannakian structure, cf. Theorem 5.14. A recurrent idea is that the conservativity of the ℓ-adic realization functor restricted to stratified Tate motives allows us to lift many statements from the ℓ-adic to the motivic setting.
Synopsis 5.1. Throughout §5, the base scheme S is as in Notation 2.1. We fix a split reductive group G → S and a Borel pair T ⊂ B ⊂ G over S where T is a split maximal torus contained in the Borel subgroup B. We start by listing some basic properties as needed in the following, see §2 for more details and the references cited there.
where X * (T ) + is endowed with the topology given by the dominance partial order "≤", i.e., for each µ ∈ X * (T ) + the subset {λ | λ ≤ µ} is closed. iii) Theétale descent equivalence DM(L + G\LG/L + G) = DM(L + G\ Gr) (cf. Theorem 2.6) will be used freely throughout. We have the full subcategory Here H µ → S is a fibrewise connected, strictly pro-algebraic closed subgroup of L + G by [RS20, Lem. 4.3.7 ii)], that is, it can be written as a sequential limit of fiberwise connected, S-smooth, S-affine group schemes with surjective transition maps. iv) The category of stratified Tate motives admits a non-degenerate t-structure whose heart is the full subcategory of mixed Tate  is exact, conservative and faithful. Under the realization, the motivic convolution product corresponds to the classical convolution product in the ℓ-adic setting (Proposition 3.14).

Indecomposable objects. The affine Grassmannian admits a decomposition into open and closed sub-
ind-schemes Gr = ⊔ τ ∈π1(G) Gr τ where π 1 (G) is the algebraic fundamental group, i.e., the finitely generated abelian group given by the quotient of the cocharacter lattice by the coroot lattice. Within each component Gr τ , every Schubert cell Gr µ → S has either even-or odd-dimensional fibers: indeed if λ, µ ∈ X * (T ) + with the same class in π 1 (G), then 2ρ, λ ≡ 2ρ, µ in Z/2 because ρ takes integer values on the coroot lattice. This defines the locally constant function where the source is equipped with the topology given by the dominance order as in Synopsis 5.1 ii). An object A ∈ MTM(Gr) is said to have constant parity p(A) ∈ Z/2 if the restriction of (5.2) to the closure of its support is a constant function, i.e., the object is supported either on a union of even components, or is supported on a union of odd components. The following result is a direct consequence of our discussion and the Kazhdan-Lusztig parity vanishing. where cl H i denotes the truncation with respect to the classical motivic t-structure on Gr.
iii) If ι : X ⊂ Gr is a finite union of Schubert schemes, then where m H i denotes the truncation with respect to the perverse motivic t-structure on X.
Proof. Part i) is immediate from the definitions using that each connected component of Gr is either of constant even or constant odd parity. Part ii) and iii) for ℓ-adic sheaves are certainly well-known, and hence are immediate from the conservativity of ρ ℓ,s (Synopsis 5.1 vi)). Let us give an argument by reduction to Theorem 2.21. The category MTM(Gr) is compactly generated, so A is the filtered colimit of its compactly generated subobjects. Moreover, cl H and m H commute with filtered colimits, so we may assume A is compact. For ii), we use that the length function on X * (T ) + = W 0 \W/W 0 is computed as l(µ) = 2ρ, µ which is also the relative dimension of each Gr µ → S. Hence, for each µ ∈ X * (T ) + , the vanishing of cl H i (ι * µ A) ∈ DTM L + G (Gr µ ) = DTM Hµ (S) (Synopsis 5.1 iii)) in degrees i as above follows from Theorem 2.21. Here we use that the forgetful functor DM Hµ (S) → DM(S) is conservative. By the compactness of A, its support is a finite-type subscheme of Gr. We can therefore invoke the localization property of DM to see that the condition for all µ ∈ X * (T ) + implies the vanishing cl H i (A) = 0. For iii), the argument is similar now taking the dimension shifts in the construction of the perverse motivic t-structure into account.
The next lemma adresses the interplay of Tate motives on the base scheme S and intersection motives. For L ∈ MTM(S) and µ ∈ X * (T ) + , we write for the intersection motive twisted by the motive L (more precisely by its * -pullback along the projection Gr µ → S). We have IC 1S(n),µ = IC µ (n) for any n ∈ Z.
Proof. This is immediate from Corollary 5.3 iii), and we refer to [Gai01, Prop. 1 ff.] and [Ric14,Prop. 3.1] for more details. Here is a sketch for the reader's convenience: First assume µ = λ, and denote IC := IC L,µ , We have a long exact localization sequence We have the following isomorphisms: where the last one is a consequence of the equivalence MTM(S) ≃ MTM(Gr µ ) (Synopsis 5.1 iv)). It is therefore enough to show that the outer groups in the above exact sequence vanish. The Kazhdan-Lusztig parity vanishing implies that i * IC, resp. i * IC ′ lives in perverse degree ≤ −2. Indeed, by general properties it lives in degree ≤ 0, degree 0 vanishes because it is an IC-sheaf, degree −1 vanishes by Corollary 5.3 iii). By duality i ! IC ′ lives in perverse degrees ≥ 2, and taking the shifts [1], resp. [2] into account, we see that the outer groups vanish by the axioms of a t-structure. This implies the corollary in the case µ = λ. Now let µ < λ, and denote i : Gr ≤µ → Gr ≤λ the closed embedding. Again the group vanishes for t-structure reasons as above using the parity vanishing. The case λ < µ is similar. Now if both µ ≤ λ and λ ≤ µ, there are no extensions between IC-sheaves. This is shown in loc. cit. without appealing to parity vanishing.
The category of compact objects MTM(Gr) c is both Noetherian and Artinian (Theorem 2.8), i.e., each object has finite length. Thus, it is a Krull-Remak-Schmidt category by [Kra15], so that each object A ∈ MTM(Gr) c admits a direct sum decomposition into indecomposable objects A = A 1 ⊕ . . . ⊕ A n which is unique up to permutation of the factors.
Proof. By the discussion above, we may assume that A is indecomposable in which case we have to show A ≃ IC L,µ for some necessarily indecomposable L ∈ MTM c (S) and µ ∈ X * (T ) + . We proceed by induction on the length l(A). The condition l(A) = 1 is equivalent to A being simple, and thus A ≃ IC L,µ with L = 1(n) for some n ∈ Z (Synopsis 5.1 iv)). Let l(A) ≥ 2, and let 0 = A ′ ⊂ A be a subobject of length l(A ′ ) = 1. In particular, A ′ is indecomposable. The quotient A/A ′ is also indecomposable. By induction A ′ ≃ IC L ′ ,µ , A/A ′ ≃ IC L,λ , and thus [A] ∈ Ext 1 MTM(Gr) (IC L,λ , IC L ′ ,µ ). As A is indecomposable, the class [A] = 0 which by Corollary 5.5 implies that λ = µ and that A is of the desired form.
The following result is similar to [MV07, Prop. 2.1].
Corollary 5.7. The forgetful functor is an equivalence of Q-linear abelian categories.
Proof. The functor is fully faithful by [RS20, 5.3.4 iii)]. As every object in MTM(Gr) is isomorphic to a direct sum of objects IC L,µ , it is essentially surjective as well.
23 5.2. Tensor structure. In this section, we show that the convolution product on DM(L + G\LG/L + G) preserves the subcategory MTM(L + G\LG/L + G).
Lemma 5.8. For A, B ∈ MTM(L + G\LG/L + G) one has A ⋆ B ∈ MTM(L + G\LG/L + G), i.e., the category MTM(L + G\LG/L + G) is stable under the convolution product.
Proof. By Theorem 3.17, convolution preserves Tateness, i.e., A⋆B ∈ DTM(L + G\LG/L + G). The remaining property A ⋆ B ∈ MTM(L + G\LG/L + G), equivalently m H i (A ⋆ B) = 0 for all i = 0 is shown using the isomorphism For the classical ℓ-adic convolution functor ⋆ ℓ and the rightmost isomorphism see around Proposition 3.14.
We now use that at least over a separably closed base field the convolution product of perverse equivariant sheaves is again perverse [Ric14, Thm. 2.1]. We then conclude using the conservativity of the composite ρ ℓ,s , cf. Synopsis 5.1 vi).
The following proposition is a subtle part of the geometric Satake equivalence in the different settings [Gin00, BD99, MV07, Ric14, Zhu17a, BR18]. Here we benefit from the existence of the symmetric monoidal structure in these settings and the faithfulness of the ℓ-adic realization in Synopsis 5.1 vi) to check the required compatibilities between the commutativity and associativity constraints. i) The isomorphisms are colimit-preserving in each argument. ii) For any geometric points → S, the constraints map under the composition of functors (Corollary 5.7, Synopsis 5.1 vi)) to the usual constraints used in geometric Satake as, e.g., in [Zhu17a, Prop. 2.21]. In particular, the category MTM(L + G\LG/L + G) ≃ MTM(Gr) is a symmetric monoidal tensor category with respect to these constraints.
Proof. Uniqueness. By property i) it is enough to characterize the constraints on the subcategory of compact objects. For anys → S as above, the functor ρ ℓ,s : MTM(L + G\LG/L + G) c → Perv(Grs, Q ℓ ) is faithful by Synopsis 5.1 vi). This implies uniqueness. Existence. Note that once c A,B , a A,B,C with properties i) and ii) exist, these constraints have to satisfy the axioms required for a symmetric monoidal category (hexagon axiom etc.). This follows from the corresponding identities for the ℓ-adic Satake equivalence, and the faithfulness of the functor in ii). It remains to construct the constraints. The associativity constraint was constructed in Lemma 3.7. For the commutativity constraint, we use the categorical analogue of Gelfand's trick whose construction is explained in [BD99, §5.3.8] and [Zhu17a, §2.4.3]. The use of prestacks simplifies the construction a little bit: Fix a pinning (G, B, T, X). Define the anti-involution θ : G → G, g → (g * ) −1 = (g −1 ) * where (-) * denotes the Cartan involution. The latter is characterized by the fact that it maps a dominant cocharacter λ ∈ X * (T ) + to −w 0 λ, where w 0 is the longest element in the finite Weyl group. By functoriality, we obtain an anti-involution on L := LG preserving L + := L + G, and thus an equivalence of prestacks, still denoted by For all A, B ∈ MTM(L + G\LG/L + G) we construct a canonical isomorphism θ ! (A ⋆ B) ≃ (θ ! B) ⋆ (θ ! A) as follows: there is a (homotopy) Cartesian diagram of prestacks where sw is induced from the switch L × L → L × L, (g 1 , g 1 ) → (g 2 , g 1 ). Hence, we obtain The isomorphism labelled ( * ) follows from (θ × θ) ! (A ⊠ B) = (θ ! A) ⊠ (θ ! B), which holds since θ is a placid map and DM ! is symmetric lax monoidal as a functor on placid prestacks with placid maps. Next we define an isomorphism of (plain) endofunctors on MTM(L + G\LG/L + G) denoted by For this we fix a square root i ∈ C of −1, and work temporarily with coefficients in Q(i). We define e on each indecomposable object IC L,µ (Corollary 5.6) to be the map corresponding to i (2ρ,µ) · id under obtained by restriction to the open orbit L + G\LG µ /L + G ⊂ L + G\LG ≤µ /L + G (this orbit is invariant under θ). Since i (2ρ,µ) · id is a central endomorphism, one checks that e is functorial. We leave the details to the reader. Next define  [LY13], that the ℓ-adic realization ρ ℓ (c A⋆B ) is the (modified) commutativity constraint coming from the fusion interpretation of the convolution product. This finishes the construction of the constraints.
Remark 5.10. If S is the spectrum of a field, then the above commutativity constraint can also be constructed as e.g. in [BD99] and [MV07] (see also [Gai01]) by using the fusion interpretation of the convolution product and the motivic nearby cycles functor constructed in [Ayo07c], [Ayo14, §10].

Tannakian structure.
In this section, we show that the category MTM(L + G\LG/L + G) ≃ MTM(Gr), which admits a symmetric monoidal structure with respect to the convolution product ⋆ by Proposition 5.9, has in fact a Tannakian structure with fibre functor being the global motivic cohomology functor. 5.3.1. The fiber functor. The fiber functor is a motivic analogue of the augmentation map for the spherical Hecke algebra.
Definition 5.11. The fiber functor is the composition of the forgetful functor σ ! (which is an equivalence of categories by Corollary 5.7), the pushforward along the structural map ǫ : Gr → S (which preserves Tate motives by [RS20, Lem. 3.1.19], using that the stratification of Gr by L + G-orbits is cellular), followed by the grading functors for the classical motivic (which agrees in this case with the perverse motivic) t-structure and the weight structure (Corollary 4.2): and finally the equivalence [Lev93] of pure Tate motives of weight 0 with the category of Q-vector spaces (here we use that S is connected).
As a consequence of the Kazhdan-Lusztig parity vanishing one obtains: Corollary 5.12. Let A ∈ MTM(L + G\LG/L + G) be of constant parity p(A) ∈ Z/2 (Corollary 5.3). Then Proof. This is immediate from the conservativity of the ℓ-adic realiztaion as in Synopsis 5.1 vi) using the well-known statement for ℓ-adic sheaves. Here is an argument by reduction to Corollary 5.3 according to which A has only, say, even classical cohomology, i.e., cl H i (A) = 0 for i odd.
As in the proof of Corollary 5.3, we may assume that A is compact. The functors (ι µ ) ! and (ι µ ) * between DTM(Gr G ) and DTM(Gr µ G ) are exact with respect to the classical motivic t-structure since the corresponding statement is true for ℓ-adic sheaves. Moreover, by localization, σ ! A is an iterated extension of the A µ := (ι µ ) ! (ι µ ) * σ ! A where µ runs over the finitely many dominant cocharacters in the support of A. Using the long exact cohomology sequence for the cl H-cohomologies of ǫ ! σ ! A, we may thus replace A by A µ . We now use that the Iwahori stratification of Gr µ consists of affine spaces. By the same localization argument, we may replace Gr µ by such a stratum A n S . We are left to showing that for the structural map It is a classical fact that the augmentation map from parahoric Hecke algebras to the coefficient field respects the multiplicative structure. The corresponding fact is also well-known in a categorified situation. The proof below is thus similar to, say, [Zhu17a, Prop. 2.20]. These two maps agree: this can be checked after precomposing with the epimorphismX → (P × ∆)\X = ∆\X, where it boils down to using that the structural mapX → S agrees withX m → LG/P ǫ → S, since S is is the final object, and in particular is acted upon trivially by all copies of P.
This shows that the following diagram of prestacks is cartesian, as soon as we omit the dotted map. Note that the map ǫ ′ × ǫ ′ arises as (P op × P)\(X → S). Once we do include the dotted map, the small bottom left square is still cartesian (but the top left square does not commute): As was already noted in the proof of Lemma 3.11, ǫ ! exchanges with the forgetful functor σ ! by ind-proper base change. Similarly, σ ! S exchanges with (ǫ ′ • m) ! , so the above is equivalent to (ǫ × ǫ) ! (σ × σ) ! (A ⊠ B). The !-pullback along the map X → (P op × P)\X commutes with ⊠ by construction of ⊠, see Remark 3.2. Furthermore, ⊠ also commutes with the !-pushforward along the structural map X → S. After reducing this claim to the case of finite-type S-schemes (instead of the ind-finite type ind-scheme X), this is a consequence of the projection formula. Hence the above object is equivalent to ω(A) ⊗ ω(B).
i) The functor ω has a monoidal structure. By Proposition 5.13, it remains to observe that gr perv and gr W have monoidal structures: both functors and the respective tensor products preserve colimits, so we may consider the subcategories of compact objects instead. Then we are in the situation of ,µ ′′ for some µ ′ , µ ′′ ∈ X * (T ) + and some indecomposable motives L ′ , L ′′ ∈ MTM(S). By Corollary 5.5, the extension splits unless µ ′ = µ ′′ =: µ in which case M ≃ IC L,µ , where L is an extension in MTM(S) of L ′ and L ′′ . We conclude that ω is exact since ω(IC L,µ ) ≃ ω(L ⋆ IC 1,µ ) ≃ ω(L) ⊗ ω(IC 1,µ ) using i) above. The faithfulness of ω follows from the conservativity of ω, which in its turn follows from the conservativity of the ℓ-adic realization at some geometric point s → S (Synopsis 5.1 vi)) and the conservativity of the fiber functor in the ℓ-adic situation.

The dual group
In this final section we determine the Tannaka dual of the categories MTM(L + G S \LG S /L + G S ) and the so-called Satake category Sat G,S ⊂ MTM(L + G S \LG S /L + G S ), (6.1) which can be thought of as the semi-simplification of the latter category. We show in Theorem 6.8 that the Tannaka dual of Sat G,S is Deligne's modified Langlands dual group G 1 /Q as constructed in [FG09,§2]. In particular, this group is independent of the (connected) base scheme S.
For S = Spec F q , the inclusion (6.1) is an equivalence. For more general bases S, we show in Theorem 6.14 that the Tannakian group of MTM(L + G S \LG S /L + G S ) is the semi-direct product of G 1 with a pro-unipotent affine group scheme coming from extensions between Tate motives on S.
Throughout §6, the base scheme S is as in Notation 2.1. Also recall Synopsis 5.1.
Definition 6.2. The Satake category Sat G = Sat G,S is the full subcategory of MTM(L + G\LG/L + G) generated by means of arbitrary direct sums (as opposed to allowing extensions) by the intersection motives IC µ (n), µ ∈ X * (T ) + , n ∈ Z.
Lemma 6.3. For L, L ′ ∈ MTM(S) and λ, µ ∈ X * (T ) + , we have natural identifications Proof. For λ = µ, this is a standard property of intermediate extensions, see [KW01,Cor. III.5.11]. To show the vanishing in case λ = µ, we may assume L is a simple object of MTM(S). In this case, IC L,µ is also simple, so any non-zero morphism would need to be an isomorphism, which is impossible if λ = µ.
Corollary 6.4. The category Sat G is abelian. Its subcategory of compact objects Sat c G is semi-simple.
Lemma 6.5. The full subcategory Sat G ⊂ MTM(L + G\LG/L + G) is stable under the convolution product.
Proof. We have to show that M := IC µ ⋆ IC λ is a direct sum of some intersection motives of the form IC κ (n).
(A priori we only know it is a successive extension of twists of some IC κ .) By Corollary 5.6, M = ⊕ (L,µ) IC L,µ with L indecomposable. The intersection motives IC µ and IC λ are pure by Theorem 4.1, hence so is M by Proposition 4.8. Therefore, each direct summand IC L,µ is also pure. Let j : Gr µ → Gr ≤µ be the open stratum. Since j * = j ! , the motive j * IC L,µ = L[d µ ] is also pure, which implies L is pure. Since L is also indecomposable, it is of the form L = 1 S (n) for some n ∈ Z, hence IC L,µ = IC µ (n) ∈ Sat G .
Corollary 6.6. The subcategory Sat c G ⊂ Sat G spanned by the compact objects in the Satake category has the following properties: i) Sat c G is a neutral Tannakian subcategory. ii) For any map f : T → S of connected schemes as in Notation 2.1, there is an equivalence of neutral Tannakian categories f * : Sat c G,S ≃ −→ Sat c G,T , having the property f * IC µ,S (n) = IC µ,T (n) for all µ ∈ X * (T ) + , n ∈ Z.
Proof. Part i) is immediate from Lemma 6.5 and Theorem 5.14. For ii), we use Proposition 2.15 which gives f * IC µ,S (n) = IC µ,T (n), so that f * is an equivalence of Q-linear abelian categories. The compatibility of f * with the convolution product was checked in Lemma 3.15. Also ω(IC µ,S (n)) = ω(IC µ,T (n)) is immediate from Definition 5.11. The rest is clear from the characterization of the constraints in (5.9). Now fix a pinning (G, B, T, X), and denote by ( G, B, T , X) the dual group in the sense of Langlands formed over Q. By definition G is a split reductive Q-group with split maximal torus T , and Borel subgroup B. Denote by T ad the image of T under the map G → G ad to the adjoint group. Then we may view the half sum ρ of the roots in B (=coroots in B) as a cocharacter ρ : G m,Q → T ad ⊂ G ad . We let G m,Q act through ρ by inner automorphisms on ( G, B, T ) from the right. Colloquially speaking, this action is given by the formula g · λ = ρ(λ) −1 gρ(λ). We consider the semi-direct product G 1 := G ⋊ G m,Q which is again a split reductive Q-group with Borel pair T × G m,Q =: For each µ ∈ X * (T ) + , and n ∈ Z we get an irreducible algebraic G 1 -representation [Jan87, Ch. II.5] 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 ) + . Then V µ (n) is the representation of G 1 of highest weight (µ, n). We denote by Rep fd Q ( G 1 ) the category of algebraic G 1 -representations on finite-dimensional Q-vector spaces. This category is semi-simple with simple objects the highest weight representations as above.
Remark 6.7. The split reductive group G 1 is Deligne's modified Langlands dual group constructed in [FG09], see also [Del07]. More precisely, one checks that the map (g, λ) → (g · (2ρ)(λ), λ 2 ) induces a short exact sequence of Q-group schemes where µ 2 ≃ Z/2 is the constant subgroup scheme generated by the element (ǫ, −1), ǫ := (2ρ)(−1). It follows that the semi-direct product G 1 = G⋊G m,Q is (canonically) a direct product if ǫ = 1. The latter condition is also equivalent to ρ being a cocharacter of T (as opposed to T ad ). For example, this is the case if G is simply connected, so that G ad = G is adjoint. We note that the difference of G versus G 1 relates to the notions of L-algebraic versus C-algebraic as introduced by Buzzard and Gee in [BG14]. For further discussion and examples we refer to [BG14, Prop. 5.39 ff.].
Theorem 6.8. There is an equivalence of Tannakian categories Proof. We denote by Aut ⋆ SatG (ω) the affine Q-group scheme of tensor automorphisms of ω provided by the neutral Tannakian category (Sat c G , ⋆, ω), cf. [DMOS82, Ch. II, Thm. 2.11]. The Satake category Sat G = Sat G,S is independent from the (connected) base scheme S by Corollary 6.6 ii), i.e., for any map T → S of schemes as in Notation 2.1 we have Aut ⋆ Sat c G,S (ω) = Aut ⋆ A.1. The exterior product. We equip the categories Sch ft S , Sch S and Cat, Cat ∞ , the (∞-)category of all small (∞-)categories, with their cartesian symmetric monoidal structure [Lur17, §2.4.1].
In particular, we consider the cocartesian fibration (Sch ft S ) × → Fin taking values in the category of finite pointed sets. Recall that the objects of (Sch ft S ) × are sequences (X 1 , . . . , X n ) with X i ∈ Sch ft S and, among others, the category has morphisms of the form (X 1 , . . . , X n ) → X 1 × S · · · × S X n , corresponding to id X1×···×Xn . Let (Sch ft S ) ×,∨ → Fin op be the associated cartesian fibration as constructed in [BGN14]. The opposite of this, which is again a cocartesian fibration, encodes the usual symmetric monoidal structure on (Sch ft S ) op . We will abbreviate the source of this map as (Sch ft S ) op,× or even just (Sch ft S ) op . The subcategory AffSch ft S ⊂ Sch ft S (consisting of affine finite type S-schemes, Spec R → S) is closed under the product since S is by assumption separated. We further endow DGCat cont and Pr L (presentable ∞-categories with colimit-preserving functors) with the Lurie tensor product, see e.g. [GR17, Ch. 1, §6].
Proof. The functor Sch ft,op S → Cat, X → Sm/X is symmetric lax monoidal (with respect to the cartesian monoidal structures on both categories) by means of the exterior product. The inclusion Cat → Cat ∞ is symmetric monoidal. The presheaf functor (in the ∞-categorical sense) P : Cat ∞ → Pr L is symmetric monoidal [Lur17, Rem. 4.8.1.8]. Thus, the composite X → P(Sm/X) is symmetric lax monoidal. In addition, the (non-full) subcategory W X ⊂ P(Sm/X) consisting of the usual A 1 -projections andétale hypercoverings are monoidal subcategories, so that the functor X → (P(Sm/X), W X ) is a symmetric monoidal functor taking values in the ∞-category WCat ∞ of relative ∞-categories. The localization functor WCat ∞ → Cat ∞ , The stabilization process, i.e., turning P 1 into an invertible object, is also a symmetric lax monoidal functor. This is readily apparent from the description of this process in [Rob14, §4.1]: abbreviating the notation of loc. cit. as P := P(free ⊗ (∆[0])) ⊗ and P inv := P(L ⊗ free ⊗ (∆[0]), * ) (free ⊗ (∆[0]))) ⊗ , let CAlg(Pr L,⊗ ) pt be the undercategory CAlg(Pr L,⊗ ) P/ . Its objects are pairs (C, X) consisting of a presentable symmetric monoidal ∞-category C and an object X ∈ C. Similarly, consider the undercategory CAlg(Pr L,⊗ ) pt,inv := CAlg(Pr L,⊗ ) Pinv/ whose objects consist of similar pairs (C, X), but where X is a ⊗-invertible object. The objects P and P inv have natural comonoid structures stemming from the comonoid structure present on any object in a cartesian symmetric monoidal category such as Cat × ∞ . Thus, the undercategories under these two objects have a natural symmetric monoidal structure in such a way that the functor (C, X) → X is symmetric monoidal. The natural functor, arising from the map P → P inv , CAlg(Pr L,⊗ ) Pinv/ → CAlg(Pr L,⊗ ) P/ is symmetric monoidal. Hence its left adjoint, which by [Rob14, Def. 4.1.8] is the functor mapping a pointed category (C, X) to (C[X −1 ], X), is symmetric lax monoidal. This abstract observation is applied to the functor Sch ft,op S → CAlg(Pr L,⊗ ) P/ , X → (P(Sm/X)[ A 1 , et −1 ], P 1 X ) which is symmetric lax monoidal by the above (and P 1 X × P 1 Y = P 1 X×Y ). The composite, denoted by SH * : Sch ft,op S → Pr L,⊗ takes values in Pr L,⊗ stb , the ∞-category of stable presentable symmetric monoidal ∞-categories and colimit-preserving functors and is by the above a symmetric lax monoidal functor. Finally, DM arises from composing with the symmetric monoidal functor Pr L stb = Mod Sp (Pr L ) −⊗Q −→ Mod Q (Pr L stb ) =: DGCat cont . We now retrace the construction of DM * ! as a functor out of the category of correspondences, by keeping track of the symmetric lax monoidal structure and thus of projection formulas, including all their higher coherences. We use, in the same vein as Hoyois [Hoy17] and Khan [Kha16], the universal property of the category of correspondences. An alternative approach for coherently encoding projection formulas avoiding the category of correspondences appears in [AGV20, §4.5].
Recall the (∞, 2)-category of correspondences Corr((Sch ft S ) ×,∨ ) adm horiz,vert defined in [GR17,§7]. Here horiz, vert and adm are certain subcategories of (Sch ft S ) ×,∨ , to be specified below more concretely. The objects of 33 this category are the objects X ∈ (Sch ft S ) ×,∨ (which are, in their turn, finite collections of objects in Sch ft S ); 1-morphisms from X to Y are spans of the form Y g ← Z f → X with g ∈ vert and f ∈ horiz, and 2-morphisms between such a morphism and another similar correspondence is a map Z → Z ′ in adm that is compatible with the maps to X and Y . This describes the low-dimensional data of this category, we refer to loc. cit. for the full definition including the (∞, 2)-categorical structure. is cartesian in (Sch ft S ) ×,∨ and the left vertical map is again in "open". In order to check the Beck-Chevalley condition, it suffices to separately consider the case where (g i ) is inert, respectively active, since these form a factorization system. For inert morphisms, this is clear. For active morphisms, we may assume that m = 1 and n = 2 above, in which case we consider g : Z 1 → Y 1 × Y 2 . Then the Beck-Chevalley condition is the assertion that the following diagram commutes, which follows from the construction of ⊠: The functor DM * ♯ obtained in this way clearly preserves edges that are cocartesian over Fin, thus giving a symmetric lax monoidal functor.
Lemma A.7. Let sep, resp. proper be the subcategory of (Sch ft S ) × spanned by morphisms that are dormant (i.e., map to an identity in Fin), and are componentwise separated (resp. proper). The functor DM * ♯ in (A.5) extends uniquely to a symmetric lax monoidal functor: Thus the Beck-Chevalley condition reduces to an assertion similar to the commutativity of (A.6), except that (−) ♯ is replaced by (−) ! , in other words, the classical projection formula as recalled in (A.1).
Corollary A.8. Write Sm ft S,sm∩sep for the category consisting of smooth separated finite-type S-schemes and smooth separated morphisms. There is a natural isomorphism of functors Tw : DM * | Sm ft S,sm∩sep ⇒ DM ! | Sm ft S,sm∩sep : (Sm ft S,sm∩sep ) op → DGCat cont , whose evaluation at a map f : X → Y is the natural transformation stemming from the projection formula. Here, ω X := p ! X 1 with p X : X → S the structural map.
Proof. Any map X f → Y of schemes naturally gives rise to a pair (Y, X), where Y is a comonoid and X is a Y -comodule (both with respect to the cartesian monoidal structure) on Sch. The coaction is given by Applying the symmetric lax monoidal functor DM * ! to this object, we obtain an object in LM(DGCat cont ), namely the commutative algebra object DM(Y ), and the DM(Y )-module DM(X), where the action arises as This can also be computed as the natural action of DM(Y ), via f * , on the symmetric monoidal category DM(X) (equipped with its usual ⊗).
The map f also gives rise to a map of comodule objects (Y, X) → (Y, Y ). Applying the functor DM * ! to the map of induced left module objects, namely the pair (id, f ! ) : (DM(Y ), DM(X)) → (DM(Y ), DM(Y )).
(Such an interpretation of the projection formula was observed by Khan [Kha16]. Note, however, that the approach to projection formulas laid out in op. cit. does not seem to work as is, since the morphisms in Ch. 2, §4.1.6 there cannot be composed.) This map admits a left adjoint separately for each object in LM, namely id and f ! , respectively. By [Lur17, Cor. 7.3.2.7], the functor therefore admits a right adjoint relative to LM ⊗ , still denoted f ! . This in particular expresses the existence of a natural map f * A ⊗ f ! B → f ! (A ⊗ B) that is functorial in A and B. It follows from the naturality of the construction that it is also functorial in f . The sought-for transformation is defined as the restriction of this map to B = ω Y . By relative purity, this map is an isomorphism whenever f is smooth. Moreover, for X and Y smooth, ω X and ω Y are ⊗-invertible. • This functor is symmetric lax monoidal if S = Spec k is a field.
• The restriction of this functor to (Sch ft S,sm∩sep ) ×,∨ is symmetric lax monoidal for general S. Here the subscript sm ∩ sep refers to the (non-full, symmetric monoidal) subcategory comprising all finite type S-schemes, but only smooth separated maps.
The fiber over 1 of (Sch ft S,sm∩sep ) ×,∨ identifies with (Sch ft S,sm∩sep ) op , and we also denote this symmetric lax monoidal functor by DM ! : (Sch ft S,sm∩sep ) op → DGCat cont .
Proof. The functor DM ! exists as stated, since the right adjoints happen to preserve colimits as well. (This is well-known and uses the assumptions that S is Noetherian and of finite Krull dimension.) To check it is a symmetric lax monoidal functor it remains to check that for two maps f 1 , f 2 in Sch ft S the natural map (f 1 ) ! M 1 ⊠ (f 2 ) ! M 2 → (f 1 × f 2 ) ! (M 1 ⊠ M 2 ) is an isomorphism. If S is a field, this holds by [JY18,Prop. 2.3.5] (note this is nontrivial and uses alterations). For general S, but smooth maps f i , this holds by relative purity and (A.1) for * -pullbacks.
A.2. Motives on placid prestacks. For a regular cardinal κ, recall from Definition 2.4 the category AffSch κ S . It consists of those affine schemes that can be presented as κ-small cofiltered limits X = lim X i , (A.10) where the X i are affine finite type S-schemes. ). An object in AffSch κ S is called placid if it admits a placid presentation, i.e., one such that the transition maps X i → X j in (A.10) are smooth (and necessarily affine). A map between two such placid affine S-schemes is called placid if for any pair of placid presentations X = lim X i , Y = lim Y j , and any j, there is some i such that X → Y → Y j factors as X → X i → Y j , where the second map is smooth. (It follows from [Ras,Lemma 4.5.1] that this condition only needs to be checked for any fixed presentations of X and Y .) The non-full subcategory of AffSch κ S consisting of placid affine schemes and placid maps is denoted by AffSch κ,pl S . (Equivalently, [Ras,Rem. 4.10.3], AffSch κ,pl S can be defined as the pro-category Pro κ-small (AffSch ft S,sm ), the pro-completion of affine schemes of finite type over S with smooth maps.) We call the category PreStk pl S := Fun((AffSch κ,pl S ) op , ∞-Gpd) the category of placid prestacks. It is the free completion of AffSch κ,pl S under arbitrary colimits. The restriction of prestacks to AffSch κ,pl S induces an adjunction PreStk pl S ⇆ PreStk S . A prestack is called placid if it lies in the essential image of the functor PreStk pl S → PreStk S .
Example A.12. The groups P f (in particular the positive loop group L + G) are placid affine S-schemes, provided that S itself is affine. So the preimage of any open affine subscheme under the P f -torsor LG ≤w → Fl ≤w (see §3.2) is placid affine. Thus, for any pair of facets f , f ′ , the double quotient of the P f ′ × P f -action (from the right and the left) on LG ≤w exists as a placid prestack P f ′ \ LG ≤w /P f . More generally, for any quasi-compact closed subscheme X ⊂ LG the prestack P f ′ \ X/P f is placid.
Our goal is to have ⊠-products for motives on placid prestacks. By means of the following two results, this is a formal consequence of the symmetric lax monoidality of DM ! on AffSch ft S,sm ⊂ Sch ft S,sm∩sep .
Lemma A.13. Let C be a small symmetric monoidal ∞-category and D be a cocomplete symmetric monoidal ∞-category whose tensor product preserves colimits separately in each variable. Fix a regular cardinal κ.
Then the restriction functor Fun(Ind κ-small (C), D) ⊗ → Fun(C, D) ⊗ admits a symmetric monoidal left adjoint, where the monoidal structure on the functor categories is given by Day convolution. Thus, any symmetric lax monoidal functor F : C → D can be left Kan extended to a symmetric lax monoidal functor on the ind-completion Ind κ-small (C).
Proof. This can be proven as in [Nik16,Cor. 3.8]. The assumption in loc. cit. that D is accessible is, for the particular situation considered here, not needed since the invokation of the adjoint functor theorem in the proof of loc. cit. can be replaced by the universal property of the ind-completion [Lur09, Prop. 5.3.5.10]. The last statement follows from the equivalence of symmetric lax monoidal functors and commutative monoid objects in the functor category under the Day convolution [Gla16, Prop. 2.12].
In the next statement, PreStk (pl) S is equipped with the cartesian symmetric monoidal structure.
Corollary A.14. The functor DM * on (AffSch ft S ) op admits a natural symmetric lax monoidal extension to AffSch κ S and to PreStk S . The same is true for DM ! provided that S = Spec k is a field. Finally, the functor DM ! admits a symmetric lax monoidal extension to AffSch κ,pl S and to PreStk pl S .

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Proof. First, apply Lemma A.13 to C = (AffSch ft S ) op . Second, in order to extend DM * from AffSch S to a symmetric lax monoidal functor on PreStk × S we use the argument in [GR17, Chapter 9, Prop. 3.2.4], according to which it is enough to observe that for any prestacks F 1 , . . . , F n , the map lim X∈AffSchS ,X→ Fi

DM(X) → lim
Xi∈AffSchS ,Xi→Fi is an equivalence for cofinality reasons. This shows the claim for DM * . The one for DM ! (and placid prestacks, or arbitrary ones for S being a field) is done the same way, using Lemma A.9 instead.
Corollary A.15. The restriction of DM ! to (PreStk pl S ) op is symmetric lax monoidal. In particular, for any two placid prestacks X 1 , X 2 , there is a natural functor DM(X 1 ) ⊗ DM(X 2 ) → DM(X 1 × S X 2 ).
Proof. The restriction of DM ! to PreStk pl S is the unique colimit-preserving functor extending the restriction of DM ! to (AffSch pl S ) op . This latter functor is the unique extension, preserving (κ-small) cofiltered limits, of the restriction of DM ! to (AffSch ft S,sm ) op . We can conclude using Corollary A.14.