Constant term functors with $\mathbb{F}_p$-coefficients

We study the constant term functor for $\mathbb{F}_p$-sheaves on the affine Grassmannian in characteristic $p$ with respect to a Levi subgroup. Our main result is that the constant term functor induces a tensor functor between categories of equivariant perverse $\mathbb{F}_p$-sheaves. We apply this fact to get information about the Tannakian monoids of the corresponding categories of perverse sheaves. As a byproduct we also obtain geometric proofs of several results due to Herzig on the mod $p$ Satake transform and the structure of the space of mod $p$ Satake parameters.

1. Introduction 1.1.Constant term functors with Q -coefficients.In the Langlands program over a global field F , the constant term and Eisenstein series operators relate automorphic functions with respect to a reductive group G/F and its Levi subgroups.When F is the function field of a smooth curve C over a finite field F q of characteristic p, it is possible to upgrade these operators to functors on sheaves, cf.[BG02] and [DG16].
For simplicity suppose G arises from a split connected reductive group over F q .For each x ∈ C(F q ) a local Hecke algebra acts on automorphic functions.After choosing an isomorphism C Q and a uniformizing element at x, this local Hecke algebra can be identified with the unramified Hecke algebra H G, of G(F q ((t))) with Q -coefficients.
In order to geometrize H G, , one considers the following functors on F q -algebras: LG : R → G(R((t))), The affine Grassmannian is the fpqc-quotient Gr G := LG/L + G, which is representable by an ind-scheme.Then in the context of the geometric Langlands program, the algebra H G, is replaced by the tensor category (P L + G (Gr G , Q ), * ) of L + G-equivariant perverse Q -sheaves on Gr G for = p.
If P is a parabolic subgroup of G/F q with Levi factor L there is a diagram Gr L Gr G . (1.1) The local analogue of the constant term functor is for a certain locally constant function deg P : Gr P → Z, cf.(6.2).The function-sheaf dictionary sends CT G L, to the Satake transform H G, → H L, up to a normalization factor.Remarkably, the functor CT G L, takes values in P L + G (Gr L , Q ), and is compatible with the tensor structures.
1.2.Constant term functors with F p -coefficients.Let k be an algebraically closed field of characteristic p > 0 and let G be a connected reductive group over k.Let Gr G be the affine Grassmannian of G over k, and let (P L + G (Gr G , F p ), * ) be the abelian symmetric monoidal category of L + G-equivariant perverse F p -sheaves on Gr G as defined in [Cas19].
Fix a maximal torus and a Borel subgroup T ⊂ B ⊂ G. Let B ⊂ P ⊂ G be a standard parabolic subgroup and L be its Levi factor containing T .
Definition 1.1.The L-constant term functor is Our main result is the following, cf.6: Theorem 1.2.The functor CT G L induces an exact faithful tensor functor CT G L : (P L + G (Gr G , F p ), * ) / / (P L + L (Gr L , F p ), * ).
Let us start by explaining why CT G L preserves perversity.Let X * (T ) be the group of cocharacters of T and X * (T ) + (resp.X * (T ) − ) be the monoid of dominant (resp.antidominant) cocharacters.For λ ∈ X * (T ) + let Gr ≤λ G be the reduced closure of the L + G-orbit of λ(t) in Gr G .By [Cas19, 1.5] the simple objects in P L + G (Gr G , F p ) are the shifted constant sheaves: . Let w 0 be the longest element of the Weyl group of (G, T ).In what follows we will use a letter L as a subscript or superscript to denote the corresponding objects for L.
The connected components of Gr P and Gr L are in bijection via the map q.If c ∈ π 0 (Gr L ) corresponds to Gr c L then we denote the corresponding reduced connected component of Gr P by S c .By restricting CT G L to S c we get a decomposition by weight functors: Then the fact that CT G L preserves perversity is a consequence of the following theorem, which is unique to F p -sheaves, cf.6.2.1.
Equivalently, Theorem 1.3 computes the relative F p -cohomology with compact support of the so-called Mirković-Vilonen cycles for the Levi L. The proof relies on the dynamics of G m -schemes of Bialynicki-Birula and Drinfeld, together with the existence of F p -acyclic G m -equivariant resolutions of singularities of Gr ≤λ G ; see 1.7 below for more details.Let us now comment on the tensor property of the functor CT G L .The general strategy of proof is similar to the one of Baumann-Riche for Q -coefficients [BR18,§15].It involves the Beilinson-Drinfeld global convolution Grassmannian, cf.5.3, and the key step is to show that a certain complex of sheaves is a perverse intermediate extension, cf.6.5.1.We achieve it by appealing to the main results regarding perverse F p -sheaves on F -rational varieties [Cas19, 1.6-1.7].In contrast, the analogue of the ingredient used for Q -coefficients fails; see 1.6 below.
1.3.Tannakian interpretation.By [Cas19] the functor of tensor endomorphisms of the fiber functor ⊕ i R i Γ : (P L + G (Gr G , F p ), * ) → (Vect Fp , ⊗) is represented by an affine monoid scheme M G over F p .Via the Tannakian formalism this results in an equivalence This construction is analogous to the geometric Satake equivalence [MV07].The monoid M G is pro-solvable, but beyond this little is known.We will apply the functor CT G L to deduce more information about M G .
By Theorem 1.3, the functor CT G L takes values in the symmetric monoidal subcategory P L + L (Gr L,w L 0 X * (T ) − , F p ) ⊂ P L + L (Gr L , F p ) associated to the submonoid w L 0 X * (T ) − ⊂ X * (T ) +/L in the sense of 6.2.2, and by 6.3.2 it intertwines the fiber functors.Thus denoting by M L,w L 0 X * (T ) − the Tannakian monoid of P L + L (Gr L,w L 0 X * (T ) − , F p ), the Tannaka dual to CT G L is a morphism of F p -monoid schemes M L → M G which factors as We currently have a limited understanding of the morphisms in (1.2).This is related to our lack of information on the structure of the Ext groups in the corresponding categories of representations.However, if L = T then we can say more.In this case, the category P L + T (Gr T , F p ) is semi-simple, M T = Spec(F p [X * (T )]), M T,X * (T ) − = Spec(F p [X * (T ) − ]), and the following holds, cf.7.4.5: Theorem 1.4.The Tannaka dual of CT G T induces a morphism of monoids M T → M G which factors as an open immersion followed by a closed immersion: Note that M T is the torus over F p with root datum dual to that of T .Thus, the morphism M T → M G in Theorem 1.4 is analogous to the reconstruction of the dual maximal torus in the dual group of G in [MV07].
There is another perspective on the morphism M T,X * (T ) − → M G in Theorem 1.4 as follows.By [Cas19, 1.2], the subcategory of semi-simple objects is a symmetric monoidal subcategory.Then the Tannakian monoid M ss G of P L + G (Gr G , F p ) ss identifies canonically with M T,X * (T ) − by 7.4.1.
Definition 1.5.The Tannaka dual of the above inclusion of semi-simple objects is called the eigenvalues homomorphism The morphism

By construction these morphisms satisfy
The Tannaka dual of the weight section can be viewed as a semi-simplification functor We refer to 7.4 for more discussion on this perspective.
1.4.Relation to mod p Hecke algebras.In this subsection alone we view G as a split connected reductive group over F q .We assume that all relevant subgroups are also defined over F q .Let E = F q ((t)) and O = F q [[t]], and consider the unramified mod p Hecke algebra Let U P be the unipotent radical of the parabolic subgroup P .Herzig [H11a, §2.3] defined the mod p Satake transform As ind-schemes over F q , for c ∈ π 0 (Gr L ) we have In contrast, for Q -coefficients the two transforms differ by the modulus character of P .The isomorphisms in Theorem 1.3 hold over F q , so by using that the IC-sheaves are constant we obtain a geometric proof of the following result due to Herzig.
Note that Corollaries 1.6 and 1.7 are ultimately statements about counting F q -points mod p on the Mirković-Vilonen cycles.From this point of view, the resolutions of singularities which go into the proof of Theorem 1.3 allow us to reduce this point counting to one on affine spaces.
Remark 1.8.In [Cas20, 4.5] a particular isomorphism ϕ : H G ∼ = F p [X * (T ) − ] is constructed using the function-sheaf dictionary and the formula [Cas19, 1.2] for the convolution product in P L + G (Gr G , F p ). Herzig's explicit formula [H11a, Prop.5.1] is then used to check that ϕ = S G T .Here Theorem 1.3 gives a purely geometric proof of the fact that ϕ = S G T .1.5.Relation to mod p Satake parameters.As a consequence of Corollary 1.7, the F p -algebra H G is commutative and the corresponding affine F p -scheme is identified with the space of Satake parameters From the geometric theory 1.3, this is the underlying scheme of the semi-simple monoid M ss G .Now for each standard Levi L as above, the functor CT G L preserves the subcategories of semi-simple objects by Theorem 1.3, hence by duality the morphism (1.2) admits a semi-simplification M ss L → M ss G .Then we have the following, cf.8.3.1, 8.4.1.
Theorem 1.9.The morphism P L equipped with its reduced structure, the space of Langlands parameters P is stratified as: The stratum S L is isomorphic to (A 1 \ {0}) rank π 0 (Gr L ) and the closure relation among the strata is given by S L = ∪ L ⊃L S L .
The underlying decomposition of the set P(F p ) was originally defined by Herzig in [H11b, §1.5, §2.4].The construction above makes the link with the Satake category P L + G (Gr G , F p ).
1.6.Obstructions to adapting proofs for Q -coefficients.Let us now explain why the known proofs that CT G L, preserves perversity and is a tensor functor fail for F p -sheaves.So that we can deal with F p and Q -coefficients simultaneously let us set IC λ, to be the -adic intersection cohomology sheaf of Gr ≤λ G .Then IC λ, is either an F p -sheaf or a Q -sheaf depending on the value of ∈ {∅, }.
For both Q -sheaves and F p -sheaves, there is a homological argument which reduces us to the case L = T .Then π 0 (Gr B ) = X * (T ) and (Gr T ) red is a disjoint union of points indexed by X * (T ), so that the weight functors are where ρ is half the sum of the positive roots.The fact that F ν preserves perversity is equivalent to the statement that By dimension estimates we have For the other inequality, one observes that there is a G m -action on Gr G such that S ν (k) is the set of k-points of the ν-component of the attractor in the sense of 2.3.1.Then Braden's hyperbolic localization theorem [Bra03] provides a comparison with the cohomology supported in the ν-component of the repeller (i.e. the attractor for the opposite G m -action), which leads to the other half of the desired vanishing (1.3) for Q -coefficients, cf.[MV07, Th. 3.5].However, we show in Appendix A that Braden's hyperbolic localization theorem fails for F p -sheaves.Braden's theorem is also the key tool from the proof of the compatibility of CT G L, with convolution [BR18, 1.15.2] that we lack in the case of F p -coefficients.
There is another approach to proving (1.3) due to Ngô-Polo [NP01].Let M ⊂ X * (T ) + be the subset of cocharacters that are either minuscule or quasi-minuscule.If λ is quasiminuscule then Ngô-Polo construct a resolution of Gr ≤λ G and explicitly stratify the fiber over S ν ∩ Gr ≤λ G by affine spaces.These stratifications allow one to estimate the dimension of H i c (S ν , IC λ, ) for (ν, λ) ∈ X * (T ) × M. If λ ∈ X * (T ) + can be decomposed as a sum of elements of M, then by considering the corresponding convolution Grassmannian m : Gr ≤λ• G → Gr ≤λ G the previous estimates allow one to prove (1.3) for any direct summand of Rm !(IC λ•, ), where IC λ•, is the IC-sheaf of Gr ≤λ• G .This is sufficient to complete the argument for Q -sheaves.However, for F p -sheaves we have Rm !(IC λ• ) = IC λ by [Cas19,6.5].Thus in our situation Ngô-Polo's approach allows us to conclude for groups of type A n only, since this is the only case where the fundamental coweights freely generating X * (T ad ) + belong to the subset M ad ⊂ X * (T ad ) + .1.7.Proof strategy for preservation of perversity.Our approach to proving Theorem 1.3 combines ideas from both [MV07] and [NP01], and works directly for L not necessarily equal to T .We start with the observation that there is a G m -action on Gr G such that Gr L (k) = Gr G (k) Gm(k) and such that the S c (k) for c ∈ π 0 (Gr L ) are the sets of k-points of the components of the attractor: Then the (unshifted) weight functor F c identifies with the hyperbolic localization functor of relative cohomology with compact support flowing in the direction of the fixed points Gr c L .Let B be the Iwahori group scheme equal to the dilation of G k[[t]] along B k .The affine flag variety F := LG/L + B is a G m -equivariant G/B-fibration over Gr G .Unlike the case of Q -coefficients, the flag variety G/B is acyclic for F p -coefficients in the sense that RΓ(G/B, F p ) = F p [0].This allows us to compare F c (IC λ ) with hyperbolic localizations on the preimage of S c ∩ Gr ≤λ G in F .Next we note that any Schubert variety in F admits a so-called Demazure resolution, which is both G m -equivariant and F p -acyclic.
Then we can appeal to a general result of Bialynicki-Birula on the structure of smooth proper G m -varieties: on the resolution, there is a unique closed attractor component, while the other components are positive-dimensional affine bundles over their fixed points.Such bundles have no relative F p -cohomology with compact support, so only the closed component contributes.
The final complete determination of F c (IC λ ) relies on the affineness of Drinfeld's attractor of a not necessarily smooth G m -scheme.
1.8.Outline.In Section 2 we recall results of Bialynicki-Birula and Drinfeld on the structure of schemes with a G m -action.The main result is 2.3.5 on F p -cohomology with compact support in the attractors on a general class of G m -schemes.In Section 3 we apply this result on the affine Grassmannian to prove 3.7.1,which is the main input in the proof of Theorem 1.3.In Sections 4 and 5 we prove Theorems 1.2 and 1.3 in the case L = T .We treat the case of general L in Section 6.In Section 7 we investigate the Tannakian consequences of Theorems 1.2 and 1.3 for the monoid M G .In Section 8 we study the stratification of P induced by the morphisms M ss L → M ss G .Finally, in Appendix A we show that Braden's hyperbolic localization theorem is false for F p -coefficients.
Notation.Let k be an algebraically closed field of characteristic p > 0 and let G be a connected reductive group over k.Fix a maximal torus and a Borel subgroup T ⊂ B ⊂ G, and let U ⊂ B be the unipotent radical of B. Let W be the Weyl group of G and let w 0 ∈ W be the longest element.
Let X * (T ) and X * (T ) be the lattices of characters and cocharacters of T , and X * (T ) + (resp.X * (T ) − ) the monoid of dominant (resp.antidominant) cocharacters determined by B. Let Φ and Φ ∨ be the sets of roots and coroots, Φ + and (Φ + ) ∨ the subsets of positive roots and positive coroots, and ∆ and ∆ ∨ the subsets of simple roots and simple coroots.For ν, ν ∈ X * (T ) we write ν ≤ ν if ν − ν is a sum of positive coroots with non-negative integer coefficients.Let ρ and ρ be respectively half the sum of the positive roots and coroots.For ν ∈ X * (T ) let ρ(ν) ∈ Z be the pairing of ρ and ν.
Notation 2.2.1.Given a scheme X, we denote by |X| its underlying topological space.
• A decomposition of X is a family of subschemes X i ⊂ X, i ∈ I, such that • A filtration of X is a finite decreasing sequence of closed subschemes The subschemes Z n \ Z n+1 , n = 0, . . ., N − 1, are the cells of the filtration.
Corollary 2.2.3.Let X be a k-scheme.Assume that X admits a filtration whose cells are positive dimensional affine spaces.Then Proof.This follows from 2.1.1 and the long exact sequence of F p -cohomology with compact support associated to the decomposition of a scheme into an open and a complementary closed subscheme.
2.3.Some G m -schemes.Let X be a scheme of finite type over k, equipped with a G maction.Recall from [Dri13] the following definitions and results.
• The space of fixed points is the fppf sheaf where Spec(k) is equipped with the trivial G m -action.
• The attractor is the fppf sheaf X + := Hom Gm k ((A 1 ) + , X) where (A 1 ) + is the affine line over k equipped with the G m -action by dilations.
Evaluating at 1 and 0 defines maps p and q: The space of fixed points is representable by a closed subscheme X 0 ⊂ X.The attractor is representable by a k-scheme.The morphism q is affine, and the section X 0 ⊂ X + obtained by precomposing with the structural morphism (A 1 ) + → Spec(k) induces an identification (X + ) 0 = X 0 ; the morphism p restricts to the identity between X 0 ⊂ X + and X 0 ⊂ X.Moreover, the morphism q has geometrically connected fibers, cf.[Ric19, Cor.1.12], so that the decomposition of X + as a disjoint union of its connected components is the preimage by q of the corresponding decomposition of X 0 : For i ∈ π 0 (X 0 ) we will denote by q i : X i → X 0 i the induced retraction.Remark 2.3.2.Suppose that X is separated over k.Then p : X + → X is a monomorphism, which induces the following identifications of sets: and for each i ∈ π 0 (X 0 ), Now consider the following hypothesis: (H) for each i ∈ π 0 (X 0 ), the restriction p| X i : X i → X is an immersion.
(1) Suppose that (H) is satisfied, and that X is proper over k.Then the family of subschemes (X i ) i∈π 0 (X 0 ) is a decomposition of X.
(2) Suppose that there exists a G m -equivariant immersion of X into some projective space P(V ) where G m acts linearly on V .Then (H) is satisfied, and if moreover X is proper, there exists a filtration as its family of cells. Proof.
In particular When (H) is satisfied, then for each i there exists a unique subscheme p(X i ) ⊂ X such that p| X i decomposes as an isomorphism X i ∼ − → p(X i ) followed by the canonical immersion p(X i ) ⊂ X.Thus, identifying X i with p(X i ), we get that the family (X i ) i∈π 0 (X 0 ) is a decomposition of X.
(2) When X admits a G m -equivariant immersion into some projective space P(V ) where G m -acts linearly on V , then, as noted in [Dri13, B.0.3 (iii)], the fact that (H) is satisfied follows from the case X = P(V ).If the immersion is closed, the fact that the decomposition (X i ) i∈π 0 (X 0 ) of X can be realized as the cells of a filtration follows again from the case X = P(V ), as proved in [ByB76, Th. 3]1 .Theorem 2.3.4.
(1) Suppose that X is smooth and separated over k.Then (H) is satisfied, X 0 and X + are smooth over k, and for each i ∈ π 0 (X 0 ), there exists an integer d i ≥ 0 such that i , and there exists exactly one such X i lying in each connected component of X.
(2) Suppose that X is normal and projective over k.Then there exists a G m -equivariant closed immersion of X into some projective space P(V ) where G m -acts linearly on V . Proof.
Corollary 2.3.5.Let X be a proper k-scheme equipped with a G m -action satisfying (H).
Suppose that there exists a connected smooth projective k-scheme X equipped with a G maction, and a surjective G m -equivariant morphism of k-schemes f : X −→ X.
Then there exists at most one i =: . Then for i ∈ π 0 (X 0 ), we have: Remark 2.3.6.If X can be embedded equivariantly into some P(V ) where G m acts linearly on V , then by 2.3.3 (2) there exists at least one i ∈ π 0 (X 0 ) such that X i ⊂ X is closed, hence then there is exactly one such i.
Proof of Corollary 2.3.5.Let i ∈ π 0 (X 0 ).Define Y i and f i by the fiber product diagram Since p| X i is an immersion by hypothesis, so is the canonical map Y i → X, and we write X i ⊂ X and Y i ⊂ X for the corresponding subschemes.Also by 2.3.4 (1) the schemes X j , j ∈ π 0 ( X 0 ), are realized as subschemes of X, and they form a decomposition of the latter, cf.2.3.3 (1).Then we have the following identity of subspaces of | X|: indeed this can be checked on k-points, where it follows from the definitions, cf.2.3.2.Thus the immersions X j → X, for f (j) = i, factor through Y i ⊂ X (note that the schemes X j are reduced, cf.2.3.4 (1)), and the family ( X j ) f (j)=i is a decomposition of the scheme Y i .Further, by 2.3.4 (2) and 2.3.3 (2), one may form a filtration of X, But since X is connected, there is exactly one X j ⊂ X which is closed, say X j 0 , by 2.3.4 (1).Thus i = f (j 0 ) =: i 0 is uniquely determined.
Finally, suppose moreover that Then recall the filtration of Y i constructed above.For every 0 ), and we have the commutative diagram Descending in this way along the filtration of Y i , we obtain which concludes the proof.
Lemma 2.3.7.Let X be a proper k-scheme equipped with a G m -action satisfying (H).
Then for each i ∈ π 0 (X 0 ) such that X i ⊂ X is closed, the retraction q i : X i → X 0 i is a universal homeomorphism and the section X 0 i ⊂ X i induces the identity of reduced schemes Proof.As we have recalled, the retraction q : X + → X 0 is always affine, [Dri13, Th. 1.4.2(ii)], with geometrically connected fibers, cf.[Ric19, Cor.1.12].In particular its restrictions q i : X i → X 0 i above each X 0 i have the same properties.Now let i ∈ π 0 (X 0 ) such that X i ⊂ X is closed.Then X i is proper over k, so that the morphism q i is proper.Consequently, in this case q i is a universal homeomorphism.So its canonical section X 0 i ⊂ X i identifies (X 0 i ) red and (X i ) red .

F p -cohomology with compact support of the MV-cycles
3.1.The affine Grassmannian.For an affine group scheme H over k (or more generally, over k[[t]]) we have the loop group functor and the non-negative loop group functor The affine Grassmannian of G is the fpqc-quotient Gr G := LG/L + G.It is represented by an ind-scheme over k.
3.2.The Cartan decomposition.The set X * (T ) + embeds in Gr G (k) via the identification λ → λ(t).For λ ∈ X * (T ) + , denote by Gr λ G the reduced L + G-orbit of λ(t) in Gr G .Then we have the decomposition of the reduced ind-closed subscheme (Gr G ) red ⊂ Gr G : Let Gr λ G be the closure of Gr λ G in Gr G .Then Gr λ G is an integral projective k-scheme, of dimension 2ρ(λ), which is the union of the Gr µ G with µ ≤ λ; it will also be denoted by Gr ≤λ G .Moreover (Gr G ) red is the limit of the Gr λ G : Then by functoriality we get a diagram Passing to the reductions, we get the decomposition of (Gr B ) red into its connected components and a decomposition of (Gr G ) red by ind-subschemes 3.4.The Mirković-Vilonen cycles.
The MV-cycles can be reconstructed from the theory of G m -schemes, as follows.
The adjoint action of the torus T on LG normalizes L + G and hence induces an action on Gr G .Fixing a regular dominant cocharacter G m → T , we equip Gr G with the resulting G m -action.
Let λ ∈ X * (T ) + .Then Gr λ G and Gr λ G are stable under the G m -action.Thus X := Gr λ G is a projective G m -scheme over k.Moreover, it can be embedded equivariantly in some P(V ) where G m acts linearly on V : indeed, one can construct on the affine Grassmannian Gr G some G-equivariant very ample line bundle, cf.[Zhu17, §1.5].Consequently by 2.3.3 (2) the connected components of the attractor X + are realized as subschemes of X.Then, it follows from (2.3.2 and) the Iwasawa decomposition of G(k((t))) that Thus the MV-cycles indexed by (ν, λ) for varying ν are precisely the (X ν ) red ⊂ X, which decompose X as X = ν∈X * (T )∩X (X ν ) red .
3.5.Generalization to the standard Levi subgroups.Let P = U P L ⊂ G be a parabolic subgroup of G containing B with unipotent radical U P and Levi factor L. Then the decomposition of (Gr P ) red into its connected components and a decomposition of (Gr G ) red by ind-subschemes Fix a dominant cocharacter G m → T whose centralizer in G is equal to L, and equip Gr G with the restriction to G m of the adjoint action of T along this cocharacter.
Let λ ∈ X * (T ) + and X := Gr λ G .The connected components of the attractor X + are realized as subschemes of X, and and so we compute using 2.3.2 that ∀c ∈ π 0 (X 0 ), (X c ) red = S c ∩ Gr λ G .Thus the MV-cycles indexed by (c, λ) for varying c are precisely the (X c ) red ⊂ X, and they decompose X as 3.6.Equivariant resolutions of Schubert varieties.Let be the affine Weyl group and the Iwahori-Weyl group.Consider the length function Let S a be the set of elements of length 1 which are contained in W a .Then (W a , S a ) is a Coxeter system.Let Ω ⊂ W be the set of elements of length 0. This is a subgroup and is representable by a connected smooth projective scheme over k, and it is equipped with a T -action by multiplication on the left on the factor L + P s 1 .The morphism Proof.The morphism f 1 : X → F λw 0 G spelled out in the proposition is nothing but the well-known affine Demazure resolution of the Schubert variety . Indeed, decompose it as considering the Artin-Schreier short exact sequences on X and on On the other hand, the morphism f 2 : by proper base change and the Bruhat decomposition of the flag variety G/B (which can be filtered), cf.2.2.3. Thus Remark 3.6.2.The morphism X → F λw 0 G in 3.6.1 is moreover birational, so that it is a resolution of singularities of the Schubert variety F λw 0 G , and X → X in 3.6.1 is the composition of the latter with the G/B-fibration Instead, we could also have used a T -equivariant resolution of singularities of the variety Gr λ G itself, e.g. the affine Demazure resolution of F λ G followed by the birational projection F λ G → Gr λ G .In fact, this resolution of Gr λ G is a very particular case of the equivariant resolutions of singularities of Schubert varieties in the twisted affine flag variety associated to any connected reductive group over k((t)) constructed in [Ric13]; precisely it is a particular case of [Ric13, 3.2 (i)] 2 .If the reductive group over k((t)) splits over a tamely ramified extension and the order of the fundamental group of its derived subgroup is prime-to-p, then any Schubert variety has rational singularities by [PR08,8.4];since 'having rational singularities' is an intrinsic notion by [CR11, Th. 1] (see also [Kov20]), then in this case all the resolutions f from [Ric13] satisfy Rf * F p = F p [0] (using Artin-Schreier).
3.7.F p -direct images with compact support of the MV-cycles.
for the normalization of the Kottwitz map as in [PR08], which is opposite to the one in [Ric13].
be the morphism of k-schemes defined by the diagram Proof.Let η T : G m → T be a regular dominant cocharacter.We start by applying 2.3.5 to X := Gr λ G equipped the G m -action η T (G m ) obtained by restriction of the adjoint T -action along η T ; it does apply thanks to 2.3.3 (2) combined with [Zhu17, §1.5], and 3.6.1.
Next let L be a standard Levi.We have the canonical commutative diagram It shows that for each c ∈ π 0 (Gr L ), Intersecting with Consequently, the subscheme (X c ) red ⊂ X is η T (G m )-stable, and the reduced connected components of its attractor are realized by the subschemes (X ν ) red , ν ∈ X * (T ) ∩ (Gr c L ∩X).In particular, by 2.3.3 (2), there exists at least one nonempty closed (X ν ) red ⊂ (X c ) red .Now let η L : G m → T be a dominant cocharacter whose centralizer in G is L, and equip X := Gr λ G with the G m -action η L (G m ) obtained by restriction of the adjoint T -action along η L .Thanks to 2.3.3 (2) combined with [Zhu17, §1.5], there exists at least one nonempty (X c 0 ) red := (X c ) red ⊂ X which is closed.Choosing (X ν 0 ) red ⊂ (X c 0 ) red nonempty and closed, then we get (X ν 0 ) red ⊂ X red nonempty and closed, so that ν 0 = w 0 (λ) by the torus case.Hence c 0 = c(w 0 (λ)).And by 3.7.2(2) below, The theorem in the case of the standard Levi L follows by 2.3.5, which applies thanks to 3.6.1.Lemma 3.7.2.Let c ∈ π 0 (Gr L ) and λ ∈ X * (T ) + .
Proof.Let ∆ ∨ ⊂ Φ ∨ be the set of simple coroots of G with respect to the pair (B, T ), and let ∆ ∨ L ⊂ ∆ ∨ be the subset of simple coroots of the Levi L with respect to (B ∩ L, T ).By the Cartan decomposition As Gr c L ∩ Gr ≤λ G ⊂ Gr L is closed and L + L-stable, to prove (i) it suffices to show that, for λ ∈ c and µ as above, Gr c L ∩ Gr µ L = ∅ unless µ ≤ L λ.To prove this, suppose Gr c L ∩ Gr µ L = ∅.Then λ − µ ∈ Z∆ ∨ L , and moreover since µ Thus µ ≤ L λ and hence the claim follows.Finally, (ii) can be proved similarly, since then Gr Finally, we record from the proof of 3.7.1 (and 2.3.7):

Hyperbolic localization on the affine Grassmannian
4.1.Perverse F p -sheaves on the affine Grassmannian.For a separated scheme X of finite type over k let P b c (X, F p ) be the abelian category of perverse F p -sheaves on X as defined in [Cas19,§2].This is an abelian subcategory of D b c (X, F p ) in which all objects have finite length.The definition of perverse sheaves extends to ind-schemes of ind-finite type as in [Cas19,3.13].
Let P L + G (Gr G , F p ) ⊂ D b c (Gr G , F p ) be the full abelian subcategory of L + G-equivariant perverse F p -sheaves on Gr G .By [Cas19, 1.1], the category P L + G (Gr G , F p ) is symmetric monoidal and the functor is an exact faithful tensor functor.The definition of the convolution product * will be reviewed in Subsection 5.3.By [Cas19, 1.5], the simple objects in P L + G (Gr G , F p ) are the shifted constant sheaves: and ∀i ∈ Z, We have Proof.Since IC λ is the shifted constant sheaf F p [2ρ(λ)] supported on Gr ≤λ G then parts (i) and (ii) follow immediately from 3.7.1.To prove part (iii), by dévissage we can assume that F • = IC λ for some λ ∈ X * (T ) + .Then part (iii) follows from (i) and (ii).
Notation 5.1.3.Given an abstract abelian monoid A, we will denote by (Vect Fp (A), ⊗) the symmetric monoidal category of finite dimensional A-graded F p -vector spaces equipped with the tensor product F p (a) ⊗ F p (b) := F p (a + b), where F p (a) denotes the vector space F p placed in 'degree' a ∈ A. Definition 5.1.4.The total weight functor is Remark 5.2.1.In the case of characteristic 0 coefficients, Baumann and Riche construct an isomorphism between H and ν∈X * (T ) F ν in the proof of [BR18, 1.5.9].In our proof of 5.2.2 below we use 4.3.3,which is unique to F p -sheaves, to compare the functors H and F − .
By [Cas19, 6.9], R i Γ(F • ) = 0 for all F • ∈ P L + G (Gr G , F p ) and i > 0. Set Z − := Z ≤0 .For all i ∈ Z − , the adjunction between Ri * ν and Ri ν * induces a natural transformation of functors Hence there is a natural transformation of functors Theorem 5.2.2.The natural transformation of functors is an isomorphism.In particular, for all i ∈ Z it restricts to an isomorphism Proof.Let λ ∈ X * (T ) + .Combining [Cas19, 6.9] and 4.2.2 (i), taking the stalk at {w 0 (λ)} defines an isomorphism in Vect Fp [Cas19,6.11]and each F ν is exact by 5.1.2.Hence it follows by induction on the length of F • that the above map is an isomorphism in general.By 5.2.2, composing F − with the forgetful functor Vect Fp (X * (T ) − ) → Vect Fp gives H.
Remark 5.2.3.Using the method in [MV07, 3.6] one can show that the decomposition H ∼ = ⊕ ν∈X * (T ) − F ν is independent of the choice of the pair (T, B).

Recollections on convolution. We first recall the definition of the convolution product in P L
(5.1) Here p is the quotient map on the first factor, q is the quotient by the diagonal action of L + G, and m is induced by multiplication in LG.We set there exists a unique perverse sheaf

1]).
There is a convolution morphism m I : Gr G,X I → Gr G,X I and a projection f I : Gr G,X I → X I .Since X = A 1 , for I = { * } there are canonical isomorphisms So in the sequel we keep the notation I for the set {1, 2} only.Let U ⊂ X 2 be the complement of the image of the diagonal embedding ∆ : X → X 2 .Then we have the following commutative diagram with Cartesian squares: Let τ : Gr G,X = Gr G ×X → Gr G be the projection and let . By [Cas19, 7.6, 7.10] there is a perverse sheaf (5. 3) The sheaf H n−2 (Rf I,! (Rm I,! F • 1,2 )) is constant by [Cas19,7.9].Therefore, by summing (5.3) over n we get an isomorphism ).This gives H the structure of a monoidal functor.
We finally recall that the associativity constraint in (P L + G (Gr G , F p ), * ) is constructed using the one of the bifunctor L and proper base change [Cas19, 6.8], and the commutativity constraint as follows.There is a morphism Gr G,X 2 → Gr G,X 2 which swaps the factors in X 2 .Using that this morphism restricts to the identity map over ∆(X), it is shown in the proof of [Cas19,7.11]that there is a canonical isomorphism .
On the other hand, we have the following: Proposition 5.3.1.There is a canonical isomorphism Proof.By the arguments in the proof of [Cas19, 7.10 (ii)], there is a canonical isomorphism ).On the other hand, by [Cas19,7.8]we have (5.4) Consequently, we get a commutativity isomorphism In order to make this commutativity isomorphism compatible with that of ⊗ it must be modified by certain sign changes which depend on the parities of the dimensions of the strata occurring in the support of the F • i ; see the proof of [Cas19,7.11]for more details.5.4.Compatibility with convolution.
Remark 5.4.1.In this subsection we use 4.3.3 in order to take H 2ρ(ν) ({ν}, •) as our definition of F ν .This allows us to give a proof that F − is a tensor functor which is unique to F p -sheaves and simpler than that in [MV07,6.4].In particular, we need only globalize the points {ν} relative to a curve instead of the S ν .In Subsection 6.6 we globalize the S ν to give a proof of the compatibility between convolution and the constant term functor CT G L with respect to a general Levi subgroup L ⊂ G.By taking L = T this provides an alternative proof of Theorem 5.4.2 below which is analogous to that in [MV07,6.4].
For ν ∈ X * (T ) − let {ν}(X 2 ) ⊂ Gr G,X 2 be the reduced closure of The reduced fiber of {ν}(X 2 ) over ∆(X) is isomorphic to {ν} × X ⊂ Gr G ×X. Denote by i ν,X 2 : {ν}(X 2 ) → Gr G,X 2 the inclusion.For ν ∈ X * (T ) − and The total weight functor is a tensor functor Proof.By the same considerations as in the proof of (5.3) in [Cas19, 7.10], we have (5.5) From the adjunction between Ri * ν,X 2 and Ri ν,X 2 , * we get a natural map (5.6)By 5.2.2 and the description of the stalks in (5.3), (5.5) the above map (5.6) is an isomorphism over closed points in X 2 .Since each of the sheaves in (5.6) is constructible then this is an isomorphism of sheaves on X 2 .As H n−2 (Rf I,! (Rm I,! F • 1,2 )) is constant by [Cas19, 7.9], then each of the sheaves H n−2 ( Fν (Rm I,! F • 1,2 )) is also constant.Hence by (5.5) we get a natural isomorphism By summing over ν ∈ X * (T ) − we get an isomorphism ).The associativity isomorphism in P L + G (Gr G , F p ) is constructed from the associativity of the operation (see the proof of [Cas19, 7.11]), so the above isomorphism is compatible with the usual associativity isomorphism in Vect Fp (X * (T ) − ).Moreover, using (5.4) and (5.5) one can verify directly from the construction in [Cas19,7.11]that the commutativity isomorphism in P L + G (Gr G , F p ) is compatible with the commutativity isomorphism in Vect Fp (X * (T ) − ).Thus F − is a tensor functor.
We denote by P L + G (Gr G , F p ) ss the full subcategory of P L + G (Gr G , F p ) consisting of semisimple objects.By [Cas19, 1.2] it is a Tannakian subcategory with fiber functor given by the restriction of H.
Remark 5.4.4.We can summarize this section as follows.Let 2ρ − : X * (T ) − → Z − be the additive map induced by the group homomorphism 2ρ : X * (T ) → Z, and let 2ρ − : Vect Fp (X * (T ) − ) → Vect Fp (Z − ) be the induced functor.Then the exact faithful symmetric monoidal functor factors as a composition of exact faithful symmetric monoidal functors 6.The constant term functor 6.1.The definition of CT G L .We return to the setup in Subsection 3.5 following the geometric setting explained in [BD,§5.3.27];see also [BR18,§1.15.1].In particular, P ⊂ G is a parabolic subgroup containing B, and L ⊂ P is the Levi factor containing T .We may consider for L all the objects that we consider for G; we will denote them using a letter L as a subscript or a superscript.There is a diagram The connected components of Gr L are parametrized by L is the set of coroots of L with respect to T .For c ∈ π 0 (Gr L ) let Gr c L and Gr c P be the corresponding connected components of Gr L and Gr P .
Let ρ L be half the sum of the positive roots of L. Then 2(ρ − ρ L )(c) is a well-defined integer for c ∈ π 0 (Gr L ) since ρ = ρ L on Φ ∨ L .Define the locally constant function where Gr P → π 0 (Gr P ) sends Gr c P to c. Definition 6.1.1.The L-constant term functor is Let c ∈ π 0 (Gr L ).Since (Gr c P ) red = S c , then by restricting (6.1) to S c we get a diagram Definition 6.1.2.The weight functor associated to c is Proof.This follows from the definitions and the topological invariance of the étale site.6.2.Preservation of perversity.Theorem 6.2.1.Let c ∈ π 0 (Gr L ) and Furthermore, for λ ∈ X * (T ) + we have Proof.The description of F c (IC λ ) follows from 3.7.
. Then the perversity of F c (F • ) for general F • follows by induction on the length of F • .For equivariance, we observe that F • is L + L-equivariant, and that S c is L + L-stable and σ c : S c → Gr c L is L + Lequivariant.As pullback along a smooth morphism is t-exact (up to a shift) for the perverse t-structure by [Cas19, 2.15], then it follows that F c (F • ) ∈ P L + L (Gr c L , F p ) by the proper base change theorem (cf.[Cas19, 3.2]).Notation 6.2.2.Given a subset A ⊂ X * (T ) + , we set This is an ind-closed subscheme of Gr G , which is stable under the L + G-action.There is a natural embedding which identifies P L + G (Gr G,A , F p ) with the full subcategory of P L + G (Gr G , F p ) whose objects are supported on Gr G,A .Let Then the simple objects in P L + G (Gr G,A , F p ) are the IC λ for λ ∈ A. Moreover, if A ⊂ X * (T ) + is a submonoid, then so is A and it follows from [Cas19, 1.2, 6.7] that the full subcategory P L + G (Gr G,A , F p ) inherits from P L + G (Gr G , F p ) the structure of a symmetric monoidal category.
simple this follows from 6.2.1.The general case follows by induction on the length of F • .Corollary 6.2.4.The L-constant term functor is an exact functor Proof.This follows from 6.2.3 and 6.1.3.Note that for L = T , we recover the functor F − , i.e.
In particular, CT L T = F L − .Remark 6.2.5.Let us set L , which is a submonoid of the abelian group π 0 (Gr L ), and The latter is an equality for L = T , but it is strict in general.Indeed, for any Remark 6.2.6.There is a more general version of Theorem 4.3.3 as follows.Let c ∈ π 0 (Gr L ) and denote by i c : Gr c L → Gr G the inclusion.Then one can show that there is a natural isomorphism of functors We will only use the functor F c because it does not require a perverse truncation.
and if ν ∈ c then Hence by the proper base change theorem Now take the cohomology of both sides in degree 2ρ L (ν) .
Corollary 6.3.2.For all ν ∈ X * (T ), In particular, there is a canonical transitivity isomorphism and the functor CT G L is faithful.
Proof.The first part follows from 6.1.3and 6.3.1.Then the transitivity isomorphism is obtained by summing over ν (in X * (T ) − ).Finally the faithfulness of CT G L follows from the transitivity isomorphism and the faithfulness of H. 6.4.The ind-schemes S c (X) and S c (X 2 ).For c ∈ π 0 (Gr L ) let S c (X) ⊂ Gr G,X and S c (X 2 ) ⊂ Gr G,X 2 be the reduced ind-subschemes realizing relative versions of S c as in [BR18, §1.15.1].They can be identified with the corresponding connected components of (Gr P,X ) red and (Gr P,X 2 ) red .Let Gr c L,X and Gr c L,X 2 denote the connected components of Gr L,X and Gr L,X 2 determined by c.We denote the relative versions of the ind-immersion s c : S c → Gr G and the projection σ c : S c → Gr c L as follows: in particular we have the projection τ : Gr G,X → Gr G and the associated shifted pull-back The important facts about the geometry of these ind-schemes are summarized in the following commutative diagram from [BR18, §1.15.1] whose squares are Cartesian (up to possible non-reducedness of fiber products) and are obtained by restriction to U ⊂ X 2 or its complementary diagonal ∆(X) ⊂ X 2 : We have canonical identifications and 6.5.The key isomorphism for the compatibility with convolution.
Theorem 6.5.1.There is a canonical isomorphism Contrary to the case of characteristic 0 coefficients, we cannot appeal to Braden's theorem to compute the co-restriction of the left side of 6.5.1 over ∆(X) as in [BR18,1.15.2].This complication is the primary obstacle we must overcome in order to prove 6.5.1.We begin by reducing to the case where the F • i are simple.Reduction of 6.5.1 to the case of simple F • i .By a diagram chase involving the proper base change theorem and the Künneth formula, the two complexes in 6.5.1 are canonically identified over U .Once we show that the complex on the left is isomorphic to the one on the right, by [Cas19,2.11]there will be a unique isomorphism which restricts to our canonical identification over U .
We claim that it suffices to show the left side is the intermediate extension of its restriction to U in the case where the F • i are simple.By the properties characterizing j c L! * in [Cas19, 2.7], it follows that if the outer two terms in an exact triangle are intermediate extensions, then so is the middle term (cf. the proof of [Cas19,7.8]).While j I,! * may not be exact in general, (5.4) allows us to replace j I,! * by the triangulated functor Rm I,! .Thus, by induction on the lengths of the F • i we can assume that F • i = IC λ i for λ i ∈ X * (T ) + .The remainder of the proof will be an explicit computation of both sides of 6.5.1 in the special case F • i = IC λ i for λ i ∈ X * (T ) + .For convenience we denote λ Let Gr ≤λ• G,X 2 be the closure of Gr ≤λ 1 G × Gr ≤λ 2 G ×U ⊂ Gr G,X 2 with its reduced scheme structure.If p |π 1 (G der )| then by [Zhu17, 3.1.14]we have | this isomorphism should be modified by passing to the reduced subscheme on the left side.
Lemma 6.5.2.There is a canonical isomorphism Proof.We first observe that τ integral and F -rational by [Cas19,7.4],so j I,! * (τ G,X 2 is a universal homeomorphism (see [Cas19,7.12]for more details), so by topological invariance of the étale site it follows that j I,! * (τ ) is still a shifted constant sheaf supported on Gr ≤λ• G,X 2 .Hence in any case there is a canonical isomorphism as stated.
Proof.By the assumption of the lemma, if c 1 + c 2 = c then w 0 (λ i ) / ∈ c i for i = 1 or 2. For such i we have F c i (IC λ i ) = 0 by 6.2.1, so both sides of 6.5.1 vanish over U .Therefore the right side of 6.5.1 vanishes.On the other hand, by 6.5.2 and the proper base change theorem, This complex is also zero by 6.2.1, so the left side of 6.5.1 is zero.
Lemma 6.5.4.If F • i = IC λ i for λ i ∈ X * (T ) + and w 0 (|λ • |) ∈ c, then the right side of 6.5.1 is canonically isomorphic to the shifted constant sheaf Proof.By 6.2.1, the right side of 6.5.1 is canonically isomorphic to . Now apply 6.5.2 to L instead of G.
From now on we assume w 0 (|λ G,X 2 be its complement with the reduced scheme structure.Then Z c λ• is a locally closed subscheme of Gr ≤λ• G,X 2 such that Proof.By 3.7.3,σ2 c,λ• restricts to a universal homeomorphism over U and ∆(X), so it is universally bijective.The natural morphism (Gr c L,X 2 ) red → S c (X 2 ) coming from the morphism L → P induces a section to σ2 c,λ• , so it is a universal homeomorphism.
Lemma 6.5.6.If F • i = IC λ i for λ i ∈ X * (T ) + and w 0 (|λ • |) ∈ c, then the left side of 6.5.1 is canonically isomorphic to the shifted constant sheaf Proof.By abuse of notation, let us view σ2 c as a morphism Then by the definition of F 2 c and 6.5.2, the left side of 6.5.1 is G,X 2 be the inclusion.By 3.7.2(ii), we have hence is supported in Gr ≤w L 0 w 0 (λ•) L,X 2 by 6.5.4 since the left and right sides of 6.5.1 agree over U .It follows that Consequently, by applying Rσ 2 c! to the exact triangle associated to the decomposition of ]. Now we conclude by using 6.5.5.Proof of 6.5.1.We have reduced to the case where ∈ c both sides of 6.5.1 vanish by 6.5.3, and if w 0 (|λ • |) ∈ c both sides are canonically identified with the same complex , F p ) by 6.5.4 and 6.5.6.6.6.Compatibility with convolution.Theorem 6.6.1.The L-constant term functor is a tensor functor First, apply Fc .After unwinding the definitions and using the proper base change theorem, there is a canonical isomorphism Fc (τ A similar diagram chase yields a canonical isomorphism . Second, use the key isomorphism 6.5.1 to get Third, use 5.3.1 for L instead of G to get By taking the sum over the c ∈ π 0 (Gr L ) we obtain finally (cf.6.1.3) . By appealing to the constructions in Subsection 5.3 one can verify that this isomorphism is compatible with the associativity and commutativity constraints.The arguments are analogous to the case of characteristic 0 coefficients as in [BR18, 1.15.2];we leave the details to the reader.Corollary 6.6.2.The functor CT G L induces an equivalence of symmetric monoidal categories Proof.The last assertion follows from 6.1.3and 6.2.1.In particular, it implies that the restriction CT G L | P L + G (Gr G ,Fp) ss factors through ) is a tensor functor, which is also a consequence of 6.6.1.
To conclude the proof, it remains to see that CT G L induces a bijection between the sets of (isomorphism classes of) simple objects, in other words, that the inclusion We need to check that λ ∈ X * (T ) − , which means that α, λ ≤ 0 for all the simple roots α ∈ ∆ ⊂ Φ.The inequality holds if α ∈ ∆ L ⊂ ∆ since λ ∈ X * (T ) −/L .Now assume that α ∈ ∆ \ ∆ L .As λ ≤ L µ, we have µ ≤ L λ i.e.

Tannakian interpretation
7.1.The Satake equivalence.Recall from [Cas19, 1.1] the Tannaka equivalence given by the geometric Satake equivalence with F p -coefficients: In particular M G is an affine monoid scheme over F p which represents the functor of tensor endomorphisms of the fiber functor H.
• Let A ⊂ X * (T ) + be a submonoid.Then the full subcategory P L + G (Gr G,A , F p ) ⊂ P L + G (Gr G , F p ) introduced in 6.2.2 is a Tannakian subcategory with fiber functor given by the restriction of H.We denote by M G,A the corresponding F p -monoid scheme and by S G,A the resulting Tannaka equivalence.It fits into a commutative diagram We have a canonical homomorphism M G → M G,A , which for A = X * (T ) + is the identity.
• Given an arbitrary abstract abelian monoid A, the category (Vect Fp (A), ⊗) introduced in 5.1.3is Tannakian with fiber functor given by forgetting the grading.Its Tannaka monoid is the diagonalizable F p -monoid scheme Remark 7.1.2.In the case G = T , we have for all submonoids A ⊂ X * (T ).In particular M T = M T,X * (T ) = D(X * (T )) = T ∨ , the torus dual to T .
7.2.The dual of the torus embedding.As noticed in 5.4.4,we have obtained a factorization of H as Under the equivalences S G and S T it corresponds to a sequence of tensor functors Fp ), ⊗), i.e. by Tannaka duality to a sequence of morphisms of F p -monoid schemes Remark 7.2.1.We show in 7.4.5 that D(F − ), denoted there by w, is a closed immersion, and that T ∨ → D(X * (T ) − ) is an open immersion.7.3.The dual of the Levi embedding.As noticed in 6.6.3,we have obtained a factorization of F − as Under the equivalences S G , S L , and S T it corresponds to a diagram Definition 7.4.3.We call the canonical homomorphism From 5.4.3 we have the equivalence . By Tannaka duality, it corresponds to the identity • We call the composition the semi-simplification functor.
• We call its Tannaka dual The first inclusion follows from the definition of the elements λ α .For the second one, note that for λ ∈ X * (T ) − \ ∆ ⊥ we can find integers m > 0, m α ≥ 0, such that mλ − α m α λ α ∈ ∆ ⊥ .Since the elements e µ for µ ∈ ∆ ⊥ are units, the second inclusion follows.Hence P \ L∈L\{G} j L (D(X * (T ) −/L )) is equal to the subset underlying the closed subscheme s(S G ).
Remark 8.3.2.We have seen in the proof of 8.In particular, the scheme P is integral.The monoid X * (T ) + is generated by the elements By sending the indeterminates x, y, z to the corresponding generators in F p [X * (T ) + ], we get a surjection Since I is a prime ideal and F p [X * (T ) + ] is an integral domain of dimension 2 then this map is an isomorphism.In particular the ring F p [X * (T ) + ], equivalently the ring F p [X * (T ) − ], is not regular.

P
defined by the constant term functor CT G L is an open immersion.Moreover, denoting by L the finite set of standard Levi subgroups T ⊂ L ⊂ G and setting ∀L ∈ L, S L := P L \ L ∈L L L

3. 3 .
The Iwasawa decomposition.From our fixed choice B = U T ⊂ G, we have the quotient map B → T and the closed immersion B → G: hence a union of Cartan closures for the affine Grassmannian Gr L which are contained in the connected component Gr c L .Such Cartan closures are irreducible, and all contain the unique minimal L + L-orbit of Gr c L , so any union of them is connected.It follows that π 0 (X 0 ) = {| Gr c L ∩X| | c ∈ π 0 (Gr L ) and Gr c L ∩X = ∅}.Next, the bijection Gr P (k) ∼ − → Gr G (k) corresponds to the decomposition Finally, denote by B the Iwahori group scheme equal to the dilation of G k[[t]] along B k , and for each s ∈ S a , by P s the parahoric group scheme increasing B determined by s.Now let λ ∈ X * (T ) + .Choose a reduced expression of λw 0 ∈ W , i.e. an (n + 1)-tuple (s 1 , . . ., s n , ω) ∈ S n a ×Ω such that s 1 • • • s n ω = λw 0 and n = (λw 0 ).In the next proposition, we denote by F λw 0 G the Schubert variety of λw 0 in the affine flag variety F G := LG/L + B, i.e. the closure of F λw 0
ind-projective we have m != m * .We now recall the construction of the monoidal structure on H following [Cas19, §7].Let X = A 1 .The construction uses the Beilinson-Drinfeld Grassmannians Gr G,X I and the global convolution Grassmannians Gr G,X I for I = { * } and I = {1, 2} (see also [Zhu17, §3.
then up to possible non-reducedness of the fiber product we have a Cartesian diagram
. During the preparation of this article R.C. was partially supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1144152, and C.P. was partially supported by the Agence Nationale pour la Let A d be the affine space over k of dimension d.Then