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Home , Tannakian formalism

GROUP ALGEBRA

Not for distribution This is a sketch of the construction of the dual group over integers, the part not covered by our (Mirkovi´c-Vilonen) announcement. A new element is the explicit construction of the group algebra of the dual group. The ingredients that allow the arithmetic result: (0) the decomposes into a sum of local cohomologies on semi- infinite orbits. (1) a canonical basis of the cohomology of the standard sheaves, (2) the coincidence of the standard !-sheaf with the IC-sheaf over integers, (3) convolution of standard !-sheaves has a filtration by such sheaves (over integers). These all flow from the basic observation (0). Another consequence of (0) is a construction of Unˇ [Feigin, Finkelberg, Kuznetsov, Mirkovi´c].

Contents

1. Equality of compactly supported and local cohomology on dual strata 3 2. Integrals over the semi-infinite strata 4 3. Consequences 6 4. Construction of projectives that represent the fiber 7 4.1. Bernstein’s induction 8

4.2. The explicit construction of PZ (ν) 9 5. The structure of the projectives 9 6. The algebras Uˆ(G˜) and O(G˜) given by the Tannakian formalism 11 6.1. Tannakian formalism for an abelian with a fiber functor 11 6.2. Tannakiancategorywithafiberfunctor 11 6.3. Equivalences of categories 12

6.4. G˜k is flat over k 12 7. The calculation of O(G˜) 13

Date: Long Long Time Ago in a Land Far Away . 1 2

7.1. Lemma 14 7.2. Lemma 14 7.3. Lemma 14 7.4. Lemma 14 7.5. Lemma 14 7.6. Lemma 14 8. Duality 15 9. Appendix A. Semi-small maps 15 9.1. Semi-small maps 15 9.2. Remarks 18

9.3. Functor π!∗ 18 9.4. f-semi-small maps 19 9.5. Small stratified maps 20

We will consider the categories of sheaves on the loop Grassmannian G for a G, with coefficients in modules over a commutative ring k, noetherian and of finite global dimension. 3

1. Equality of compactly supported and local cohomology on dual strata

Here we prove the following topological lemma.

1.0.1. Lemma. Let V be finite dimensional representation of a torus T = R×S with R = Gm = S, such that the weights of R are positive. Let V = V+⊕V− be a decomposition into T -invariant subspaces such that the weights of S are positive in V+ and negative in V−. Then for any T -monodromic A∈ D(V ), canonical map ∗ ∗ HV− (V, A) → Hc (V+, A), is an isomorphism

k± p± k i± . Canonical map. Denote 0֒→V±։0, 0֒→V , and V±֒→V .1.0.2

We use functors on T -monodromic sheaves: compactly supported cohomology on V+ ∗ ! ∗ C =(p+)!(i+) =(k+) (i+) , and local cohomology along V− ! ∗ ! L =(p−)∗(i−) =(k−) (i−) , (equalities come from say, the R-contraction of V ± to 0). The canonical map L→C comes from

Base Change ∗ ! ∗ ∼ L→L◦(i+)∗(i+) =(p−)∗ (i−) ◦(i+)∗ (i+) =

! ∗ ! ∗ =(p−)∗ (k+)∗(k+) (i+) =(k−) (i+) = C, ∼ ! and also symmetrically, L = C◦(i−)!(i−) →C.

! 1.0.3. Reduction to the case (i−) A =0. For A supported on either of V±, the claim follows from base change. So for j : V − V−֒→V , the claim is true for the first term in the exact triangle ! ∗ (i−)!(i−) A→A→j∗j A, hence it is equivalent for the last two. ∼ ∗ = So we can suppose that j∗j A←−A. Then LHS is zero and we need to kill the RHS. To calculate the RHS we use restriction ρ : V+ − 0֒→V+ of j and the exact triangle ! ∗ ρ!ρ (A|V+) −→A|V+ −→(k−)∗(k−) (A|V+), i.e.,

ρ!(A|V+ − 0) −→A|V+ −→(k−)∗(A|0), which we write as α A|V+ −→(k−)∗(A|0) −→ ρ!(A|V+ − 0)[1]. 4

def ∗ It remains to show that the map a=Hc (V+,α), is an isomorphism: ∗ ∗ α ∗ Hc [V+, A|V+] −−−→ Hc [V+, (k−)∗(A|0)] −−−→ Hc [V+,ρ!(A|V+ − 0)] [1]    = = = y y y . ∗ α ∗ Hc (V+, A) −−−→ A|0 −−−→ Hc (V+ − 0, A) )[1]

∗ ∗ 1.0.4. Passage to a sphere. Now let R+ act on V via R+⊆Gm = R. Choose a hermitian inner product on V , invariant under the compact part of T . Then for the sphere S of ∗ radius one around 0, map S→(V −0)/R+ is an isomorphism. Moreover, T action descends ∗ ∗ to (V − 0)/R+ and A descends to a T -meromorphic complex of sheaves on (V − 0)/R+ which one can identify with A|S. Now we interpret a in terms of the restriction of A to the sphere S, ∗ =∼ ∗ ∗ ∗ A|0 ∼= H (V, A)−→H [V, j∗(A|V − V−0)] = H (V − V−, A) ∼= H (S− V−, A) and ∗ ∼ ∗ Hc (V+ − 0, A)[1] = Hc (S∩ V+, A). In this way a is identified with the canonical map ∗ b ∗ H (S− V−, A) −→Hc (S∩ V+, A) (restriction of cohomology to a closed subspace).

1.0.5. Homotopy. Sheaf B = A|S − V− is an S-monodromic sheaf on X = S − V− and Y = S ∩ V+ is an S-invariant closed subspace with the property that for each x ∈ X, lim z·x exists and lies in Y . This implies that the restriction of cohomology map z→0 H∗(X, B)→H∗(Y, B) is an isomorphism.

1.0.6. Remarks. (1) This would be a most standard result if the sheaf would be com- patible with the projection to one of V±. (2) One would like to say that a proof in the algebraic category should work the same with S replaced by a weighted projective space (V − 0)//R or a stack (V − 0)/R.

2. Integrals over the semi-infinite strata

2.0.7. The opposite semi-infinite stratifications. Let N and N− be opposite unipo- tent radicals in G and = ·Lν and Tν = N−(K) · Lν. These are intuitively opposite stratifications ofG.

2.0.8. Lemma. Intersections with the G(O)-orbits. dim(∩)= ht(λ + ν) and the intersection is of pure dimension. 5

2.0.9. Lemma. Estimates based on perversity. For A∈PG(O)(G)

∗ ≤2ht(ν) ∗ ≥−2ht(ν) Hc (, A) ∈ D and H (G, A) ∈ D .

2.0.10. Lemma. Integrals over the dual strata. Let T be a Cartan subgroup of G def and Ta =T ×Gm where Gm is the group of rotations. For any Ta-monodromic sheaf A on the ind-variety G, one has a canonical isomorphism

∗ =∼ ∗ H (G, A) −→ Hc (Tν, A).

Proof. Observe that X∗(T )⊆G(K) commutes with T and the action of the rotation group Gm on T ·X∗(T ) becomes trivial modulo the subgroup T . Therefore we can use a transla- tion by X∗(T ) to reduce the claim to the case ν = 0. 7→∞ Let K be the the congruence subgroup Ker[G(C[z−1]) −−−→z G] of G(K). It is an ind-group subscheme of the ind-group scheme G(K). It lies in a group scheme 7→∞ Kˆ def=Ker[G(C[[z−1]]) −−−→z G]. Moreover, ind-scheme G lies in a scheme Gˆ which

has an open subscheme U isomorphic to Kˆ via Kˆ ∋ k7→k·L0 ∈ U for the origin ∼ −1 −1 = L0 ∈ G. Now S0 = [N(C[z ]) ∩ K]·L0 ⊆U, and N(C[z ]) ∩ K −→ S0 via −1 N(C[z ]) ∩ K ∋ k7→k·L0 ∈ S0. Also,

−1 −1 S0 =[N(C[[z ]]) ∩ Kˆ ]·L0 ∩ G =[B(C[[z ]]) ∩ Kˆ ]·L0 ∩ G,

and similarly for T0 and N−. ˆ ˆ ˆdef =∼ ˆ Let us replace U by K and then also with its Lie algebra k via exp : k=kz−1C[[−1]]−→K. Our sheaf A will then be supported in a finite dimensional vector subspace V ⊆k⊆ˆk, invariant under Ta. Let R = Gm be the rotation group, and let S = Gm act via the co-character 2ˇρ ∈ X∗(T ), so that it acts on bz−1C[[−1]] with nonnegative weights and on (n−)z−1C[[−1]] with negative weights. The problem now reduces to a case of the above lemma with V+ = V ∩ bz−1C[[−1]] and V− = V ∩ (n−)z−1C[[−1]].

2.0.11. Lemma. Two kinds of integrals over N(K)-orbits. For A∈PG(O)(G) and the longest element w0 in the Weyl group of G, one has ∗ ∗ ∗ ∗ ∗ H (G, A)= Hc (Tν, A)= Hc (Sw0·ν,w0A)= Hc (Sw0·ν, A).

2.0.12. Theorem. Integrals over N(K)-orbits are concentrated in one degree. For A∈PG(O)(G), ∗ Hc (, A) is concentrated in the degree 2ht(ν), H∗(G, A) is concentrated in the degree −2ht(ν). 6

3. Consequences

def ∗ def ∗ Denote the “fiber functor” by F =H (G, −) and for any coweight ν let Fν = Hc (, A).

∼ 3.0.13. Lemma. (a) F = ⊕ν∈X∗(T ) Fν, i.e., ∗ ∼ ∗ H (G, A) = ⊕ν∈X∗(T ) Hc (, A), A∈PG(O)(G). The same is true for the T -equivariant cohomology.

(b) Fν and F appear in one parity (on each connected component of G). In particular, they are exact on PG(O)(G). (c) Dual Tˇ of the Cartan subgroup embeds into the group G˜ given by the Tannakian formalism.

3.0.14. Lemma. [Canonical Basis ] The cohomology of the standard perverse sheaves on a G(O)-orbit Gλ has canonical basis ∼ ∼ F [I!(λ)] = k[Irr(Gλ ∩ Sν)] = F [I∗(λ)].

3.0.15. Remarks. (a) The bases here are not the ones constructed by Lusztig (irreducibles asabasisofa K-group, good algebraic properties and characterization), conjecturally they coincide with the bases Nakajima found. (b) Over Q we get two (dual) construction of bases of irreducibles, lower and upper bases.

∼ L ∼ L 3.0.16. Corollary. (a) I!(λ, k) = I!(λ, Z)⊗k and I∗(λ, k) = I∗(λ, Z)⊗k. Z Z ∼ (b) D I!(λ) = I∗(λ).

3.0.17. Theorem. The absence of torsion in Z gives

=∼ I!(λ, Z)−→I!∗(λ, Z).

∼ ∼ Proof. We give a convoluted argument: D I!(λ, Z) = I∗(λ, Z) = D I!∗(λ, Z) ! The first isomorphism comes from the canonical bases, and the second from the fact that all three sheaves coincide over C [Lusztig, Ginzburg]. The second ingredient uses the decomposition theorem which we hope to avoid some day. The second isomorphism follows from a general fact: 7

j Lemma. Let U֒→X be open and A∈P(U, Z). Suppose that .3.0.18 (a) DA is perverse, p (b) map j!A→j!∗A becomes an isomorphism over C, then ∼ = p D(j!∗A) −→ j∗A.

p Alternatively, if (b) is replaced by: (b’) map j!∗A→ j∗A is an isomorphism over C, the ∼ = p conclusion is: D j!∗(DA)−→ j!A. Proof. (1) DA is perverse iff A has no torsion subsheaves. So if DA is perverse, so are p also D(j!∗A) and D( j∗A). p (2) Since D(j!∗A)|U = A, there is a canonical map D(j!∗A)→ j∗(DA). Over complex p numbers this is the dual of the map j!A→j!∗A, hence an isomorphism. Therefore the and the are torsion sheaves supported off U: p 0→K→ D(j!∗A)→ j∗(DA)→C→0.

(3) Since the dual of D(j!∗A) is perverse, torsion subsheaf K is zero: p 0→D(j!∗A)→ j∗(DA)→C→0.

p (4) Since DC[1] is perverse, turning the exact triangle D(C)→ D[ j∗(DA)]→ j!∗A gives an exact sequence of perverse sheaves p 0→ D[ j∗(DA)]→ j!∗A→ D(C)[1]→0.

However, j!∗A has no quotients supported off U, so C = 0.

4. Construction of projectives that represent the fiber functor

4.0.19. Lemma. Let Z⊆G be a finite closed union of G(O)-orbits. Functor Fν restricted to PG(O)(Z) is represented by a projective object PZ (ν) of PG(O)(Z).

Proof. For A in PG(O)(Z) one has def ∗ ∼ ∗ ∗ ∗ Fν(A)= Hc (Sν, A)[2ht(ν)] = HTν (G, A)[2ht(ν)]= Ext (kTν , A)[2ht(ν)]= Ext (kTν ∩Z [−2ht(ν)], A).

Now choose n >> 0 so that the G(O)-action on Z factors thru the action of G(On) for n+1 G(On) On = O/z . Then the forgetful functor FB(On) : DG(On)(Z)→DB(On)(Z) has a left G(On) adjoint γB(On) (see the construction bellow). ∗ Since ExtD(Z)(kTν ∩Z [−2ht(ν)], A) lives in one degree one has (for the obvious B(On)- structures) ∗ ∼ ∗ ∗ ∼ ∗ F A⊗H B(pt, k) Ext (k ν ∩ [−2ht(ν)], A) ⊗ H B(pt, k) Ext (k ν ∩ [−2ht(ν)], A) ν = D(Z) T Z = DB(On)(Z) T Z ∗ G(On) ∼ ∗ G(On) = Ext (k ν ∩ [−2ht(ν)], F A) Ext (γ k ν ∩ [−2ht(ν)], A). DB(On)(Z) T Z B(On) = DG(On)(Z) B(On) T Z 8

Therefore, ∼ ∼ FνA = HomD(Z)(kTν ∩Z [−2ht(ν)], A) = HomDB(On)(Z)(kTν ∩Z [−2ht(ν)], A)

∼ G(On) = HomDG(On)(Z)(γB(On)kTν ∩Z [−2ht(ν)], A). G(On) Since F = γB(On)kTν ∩Z [−2ht(ν)] ∈ DG(On)(Z) represents the Fν, we will p ≤0 find that it lies in DG(On)(Z). Let d be the highest non-vanishing degree of F (i.e. with pHd(F) =6 0). Then p d −d p d 0 =6 HomDG O (Z)(F, H F[−d]) = Ext (F, H F ) ( n) DG(On)(Z) ∼ −d p d ∗ ∼ p d −d = H [Fν( H F)⊗HB(pt, k)] = Fν( H F)⊗HB (pt, k), gives d ≤ 0.

def p 0 Therefore, PZ (ν)= H (F) represents Fν on PG(On)(Z) = PG(O)(Z). Since Fν is exact, PZ (ν) is projective.

4.0.20. Corollary. PG(O)(Z) has enough projectives.

Proof. For A∈PG(O)(Z) choose finitely generated k-projective covers fν→Fν(A). Then ∼ ∼ Hom(fν⊗PZ (ν), A) = Homk[fν, Hom(PZ (ν), A)] = Homk[fν,Fν(A)] contains a canon- ical map pν such that Fν(pν) is surjective. Therefore, ⊕ν fν⊗PZ (ν) is a projective cover of A.

4.1. Bernstein’s induction functors. For a subgroup B of a group A, the left adjoint A A def ˜ of the forgetful functor FB : DA(Z)→DB(Z) is given by γB A= a!A. Here we use the diagram p Z ←−A×Z −→ν A×Z −→a Z B with A×B acting on A×Z by (a,b)·(α,z)def=(a·α·b−1,b·z), in order to characterize A˜ ∈ ! ∼ ! DA(A×Z) by: ν A˜ = p A in DA(A×Z). B A Since the forgetful functor is exact, its left adjoint γB is right exact, i.e., A ≤0 ≤0 p Adef p 0 A γB : DB (Z)→DA (Z), and its perverse version γB = H [γB −] : PB(Z)→PA(Z) is A the left adjoint of the functor between abelian categories FB : PA(Z)→PB(Z).

4.1.1. Example. For a B-invariant Y ⊆Z denote by α : A×Y →Z the action, then B A A γB (kY ) = α! kA×Y [2dim A/B] and ΓB (kY ) = α∗ kA×Y . B B

! The first part uses p kY = kA[2dim A]⊠kY = kA×Y [2dim B][2 dim A/B] = ! ∗ ∗ ν kA×Y [2dim A/B], and the second is simpler p kY = kA×Y = ν kA×Y . B B 9

4.2. The explicit construction of PZ (ν). By construction, PZ (ν) = p 0 G(On) H [ γB(On)(kTν ∩Z [2ht(ν)]) ] with

G(On) α γB(On) kTν ∩Z = [G(On) × Tν ∩ Z −→ Z]! kG(On) × Tν ∩Z [2dim G(On)/B(On)]. B(On) B(On) T The fiber of α at η ∈ Z is B\{a ∈A, aλ ∈ Y ∩ A·η = Tν ∩Gη}. The dimension of the fiber of α at a dominant η is dA/B − ht(ν + η). th So PZ (ν) is the zero perverse cohomology of the !-image of a (shift of) a constant sheaf under an “essentially semi-small” map. Here “essentially semi-small” means that the the dimensions of fibers have the correct increment but the generic fiber is not finite.

5. The structure of the projectives

Let PZ = ⊕ν PZ (ν) be the projective that represents the fiber functor F on PG(O)(Z).

5.0.1. Lemma. (a) PZ is a projective generator of PG(O)(Z). It carries a canonical G˜- action.

(b) If Gλ is open in Z and Y = Z −Gλ, then p 0 PY = H (PZ |Y ).

(c) There is a canonical exact sequence ∗ 0→ I!(λ) ⊗ F (I∗(λ)) → PZ → PY →0.

Here I?(λ) is the standard perverse ?-extension from Gλ.

(d) PZ has a G˜-equivariant filtration with canonical G˜-isomorphisms ∼ ∗ ∼ Gr(PZ ) = ⊕ I!(λ)⊗F [I∗(λ)] = ⊕ I!(λ)⊗F [I!(−w0·λ)]. Gλ⊆Z Gλ⊆Z In particular, F [P(ν)] is a free k-module and ∼ Gr[F (PZ )] = F [Gr(PZ )] = ⊕ F [I!(λ)]⊗F [I!(−w0·λ)] Gλ⊆Z has a canonical basis - the union of products of canonical basis. ∼ L (e) PZ (ν, k) = PZ (ν, Z)⊗k. Z

Proof. (a) G˜ = Aut(F ) acts on F |PG(O)(Z), hence also on PZ . p ≤0 (b) Clearly, on PG(O)(Y ), complex PZ |Y ∈ DG(O)(Y ), still represents F , and then so p 0 does H (PZ |Y ) ∈PG(O)(Y ).

(c1) Restriction PZ (ν)|Gλ is up to a shift a constant sheaf kGλ [2ht(λ)] ⊗ ?. Moreover, ? is a free k-module - the basis is given by irreducible components of the fiber at λ ∈ Gλ 10

of of the semi-small map that occurs in the construction of PZ (ν) (actually, the union of such over all co-weights ν). This gives a map I!(λ)⊗? → PZ .

Let k = Z for a moment. Then I!(λ) coincides with I!∗(λ) hence the map I!(λ) ⊗ ?→ PZ p <0 is an embedding. The quotient Q is supported on Y . Since I!(λ)|Y ∈ D (Y ), exact p 0 p 0 triangle I!(λ)|Y ⊗ ? → PZ |Y → Q , gives Q = H (Q)= H (PZ (ν)|Y )= PY (ν).

So for k = Z we find by the induction in the number of G(O)-orbits in Z that PZ (ν) has a filtration whose quotients are standard !-sheaves. ∼ L L (e) Therefore, I!(λ, k) = I!(λ, Z) ⊗ k implies that PZ (Z)⊗ k is perverse. Since Z Z

L ∼ L ∼ L ∼ Hom[PZ (k), PZ (Z)⊗ k] = F [PZ (Z)⊗ k] = F [PZ (Z)]⊗ k = F [PZ (Z)]⊗ k = [PZ (Z)]⊗ k, Z Z Z Z Z

L there is a canonical map PZ (k)→PZ (Z)⊗ k. Again, one can see that this is an isomorphism Z by the induction since it is clearly an isomorphism over Gλ. One can see the same from our construction of projectives. For F(k) = L L G(On) ∼ γ kTν ∩Z [−2ht(ν)], one clearly has F(k) = F(Z)⊗k. So PZ (ν, k) = PZ (ν, Z)⊗k B(On) Z Z L L L p 0 p follows by applying H to the triangle τ<0F(Z)⊗k→ F(Z)⊗k → PZ (ν, Z)⊗k (since Z Z Z the first term in is in pD<0).

(c2) For a general k, exact sequence 0→ I!(λ) ⊗ ? → PZ → PY →0 is now obtained by tensoring the sequence for Z. The multiplicity ? is given by ∼ ∗ ∼ ∗ ? = Hom[PZ , I∗(λ)] = F [I∗(λ)] .

To obtain the first isomorphism we apply Hom[−, I∗(λ)] to the exact sequence

0→ I!(λ) ⊗ ? → PZ → PY →0. p Recall that for a A supported in ∂Gλ one has Ext [A, I∗(λ)] = 0, for ∗ p = 0, 1. (Since A|Gλ = 0 gives Ext [A, (Gλ֒→Z)∗kGλ ] = 0, for the cone C of the p ∼ p−1 canonical map I∗(λ)→(Gλ֒→Z)∗kGλ [dim Gλ] one has Ext [A, I∗(λ)] = Ext (A, C). The p >0 claim follows from C ∈ D (Z).) For A = PY this vanishing gives the first isomorphism in =∼ ∼ ∗ Hom[PZ , I∗(λ)]−→Hom[I!(λ) ⊗ ?, I∗(λ)] = ? , ∼ the second comes from the identification Hom[I!(λ), I∗(λ)] = k. p 0 (d) The canonical map PZ → H (PZ |Y )= PY comes from F |PG(O)(Y ) being a restriction of F |PG(O)(Z), so it is G˜-equivariant. Therefore the filtration on PZ is G˜-equivariant. Moreover, G˜-action on the kernel I!(λ)⊗?⊆ PZ is given by the corresponding action on ∗ ∼ ? = Hom[I!(λ), PZ ]. Finally, isomorphism ? = F [I∗(λ)] is equivariant since G˜ = AutF . 11

5.0.2. Remark. The proof of (c) over a filed is abstract, but in general we seem to need the fact that the multiplicity of I!(λ) is Z-free.

def p 5.0.3. Injectives. The Verdier dual IZ =D(PZ ) is perverse since PZ ∈ PG(O)(Z), i.e., p F [P(ν)] is a free k-module. IZ is clearly an injective object in the category PG(O)(Z) which is closed under the Verdier duality.

6. The algebras Uˆ(G˜) and O(G˜) given by the Tannakian formalism

6.1. Tannakian formalism for an with a fiber functor. Consider an abelian category C with a “fiber functor” ω : C→m(k), i.e., an exact and fully faithful k-functor. The Tannakian formalism [P. Deligne, Categories Tannakiens] says that ω can be viewed as an equivalence of C and the category of k-modules with additional data of an action of the k-algebra End(ω). If End(ω) is k-projective one can instead describe C as the category of comodules for the coalgebra End(ω)∗. Moreover, using a projective generator P of C one can describe this coalgebra explicitly, End(ω)∗ ∼= ω(P )∗ ⊗ ω(P ). EndC(P ) All of this is clear if we know a representing object P for ω. It is a projective generator o of C since the functor is exact and fully faithful. Then End(ω) ∼= EndC(P ) ∼= ω(P ), hence End(ω)∗ ∼= ω(P )∗ and the coalgebra structure is the dual of the algebra structure on ω(P ) ∼= EndC(P ).

6.1.1. Lemma. F : PG(O)(Z)→m(k) gives an equivalence of PG(O)(Z) with the right ∼ ∼ 0 modules for the algebra F (PZ ) = End(PZ ) = End(ω) . We can also view these as the comodules for the coalgebra ∗ ∼ ∗ ∼ ∗ ∼ End(ω) = End(PZ ) = F (PZ ) = F (IZ ).

6.1.2. Passage to PG(O)(G). Since PG(O)(G) is a union of abelian sub-categories PG(O)(Z), we get a pro-projective object P = lim PZ that pro-represents the fiber functor F , and an ← Z ind-injective object I = lim IZ . Now F gives an equivalence between PG(O)(G) and the → Z def of the pro-algebra F (P )= lim F (PZ ) (a complete topological algebra), ← def which can be seen also as the category of comodules for the ind-coalgebra F (I)= lim F (IZ ) → (a union of sub-coalgebras of finite rank).

p 6.2. Tannakian category with a fiber functor. The PG(O)(G)⊆PG(O)(G) consisting of sheaves A with F (A) a projective k-module, is a Tannakian category with a p def p fiber functor F . Filtration by sub-categories PG(O)(λ)=PG(O)(Gλ) indexed by dominant p p p co-weights λ is compatible with convolution: PG(O)(λ)∗PG(O)(µ) ⊆PG(O)(λ+µ), and with p p duality (it sends PG(O)(λ) to PG(O)(−w0·λ)). This makes I into an ind-Hopf object I = 12

lim I(λ), with the multiplication given by the corresponding maps I(λ) ∗ I(µ)→I(λ + µ). → In consequence, F (I) is a Hopf algebra. The Tannakian formalism says that this is the group algebra O(G˜) of the group scheme G˜def= Aut(F ) of automorphisms of the fiber functor. def As an algebra it has an increasing filtration by O(G˜)λ = F (Iλ) (λ a dominant co-character), with ∼ Grλ(O(G˜)) = F [I∗(λ)]⊗F [I∗(−w0·λ)].

(This is dual of the above formula for Gr[F (PZ )].) The dual Hopf algebra is the topological Hopf algebra Uˆ(G˜)= F (P ).

6.3. Equivalences of categories. By the definition of G˜, we can view the fiber functor p p ˜ F as a functor between the Tannakian categories PG(O)(G) and m (G), the category of algebraic G˜-modules which are k-projective. The Tannakian formalism guarantees that this is an equivalence.

p ∼ p ˜ 6.3.1. Lemma. The equivalence of Tannakian categories PG(O)(G) = m (Gk) extends to a ∼ canonical equivalence of abelian categories PG(O)(G) = m(G˜k).

Proof. The main point is that any object A in PG(O)(G) is a quotient of an object P p in PG(O)(G), but for A ∈ PG(O)(Z) we saw that we can choose such P which is even projective in PG(O)(Z).

6.4. G˜k is flat over k.

6.4.1. Lemma. (a) O(G˜Z) is a free Z-module and U(G˜Z) has no torsion.

(b) O(G˜Z) has no zero divisors.

(c) O(G˜k) ∼= O(G˜Z)⊗k. Z

(d) Lie algebra g˜Z has no torsion and it is a Z-form of gˇ. Proof. (a) follows from the description of Gr[O(G˜)], as a free module with a canonical basis.

(b) follows since now O(G˜Z)⊆O(G,˜ Z)⊗C = O(G,˜ C) ∼= O(G,ˇ C) and the last algebra Z is integral. (c) The base change claim follows from such claim for the basic projectives - canonical ∼ L isomorphisms PZ (ν, k) = PZ (ν, Z)⊗k. Z

(d) In particular, the primitive part g˜Z of the topological Hopf algebra U(G˜Z) has no torsion, while U(G˜Z)⊆U(G˜C) gives g˜Z⊗C ∼= g˜C ∼= gˇ. Z 13

7. The calculation of O(G˜)

∼ 7.0.2. Lemma. (a) The algebra structure on Gr(O(G˜)) = ⊕ F [I∗(λ)]⊗F [I∗(−w0·λ)], is λ the quotient of the algebra ⊕ F [I∗(λ)] ⊗ ⊕ F [I∗(µ)] = A⊗A. Here λ µ

def A= ⊕ F [I∗(λ)] λ is given the algebra structure thru the canonical maps F [I∗(λ)]⊗F [I∗(µ)]→F [I∗(λ + µ)]. (b) Filtration on O(G˜) is compatible with the coalgebra structure and each summand ∼ ∗ F [I∗(λ)]⊗F [I∗(−w0·λ)] = F [I∗(λ)]⊗F [I!(λ)] of Gr[O(G˜)] is a sub-coalgebra. Its dual ∗ ∼ is the associative algebra structure (possibly without a unit) on F [I∗(λ)] ⊗F [I!(λ)] = Hom[F I∗(λ),F I!(λ)], defined via the canonical map F [I!(λ)]→F [I∗(λ)]. Proof. (a) One checks that the maps I(λ) ∗ I(µ)→I(λ + µ) defined by p p ∗ p PG(O)(λ)×PG(O)(µ) −→ PG(O)(λ + µ), induce on the graded pieces GrλIλ = I∗(λ)]⊗F [I∗(−w0·λ)] the maps GrλI(λ) ∗ GrµI(µ)→Grλ+µI(λ + µ) given in each factor by the canonical maps I∗(λ) ∗I∗(µ)→I∗(λ + µ).

7.0.3. Part (b) will not be needed.

7.0.4. Corollary. O(G˜) is finitely generated, hence G˜ is noetherian.

Proof. The maps F [I∗(λ)]⊗F [I∗(µ)]→F [I∗(λ + µ)] are surjective, hence A is generated by the sum of fundamental representations F [I∗(ωi)].

7.0.5. Lemma. Let k be a closed field.

(a) G˜k is connected.

(b) If G˜k is reduced then G˜k ∼= Gˇk.

“ Proof. ” (a) For any field k, noetherian group scheme G˜k is connected. Otherwise it would have a finite group scheme quotient and its representations would form a tensor subcategory of PG(O)(G) supported in a compact subvariety. Since dimsupp(A∗B) = dim supp(A) + dimsupp(B) such quotient is trivial.

(b) G˜k = U ⋉ R for a reductive group R and a unipotent group U (?).

Now irreducible representations of G˜k and R are the same (?). Since the parameterization of irreducibles is given by the same cone in the same lattice as in Gˇk we see that T˜k is a maximal torus of G˜k and R, and the root systems of R and G˜k coincide.

Since Gr[O(G˜k)] ∼= Gr[O(Gˇk)] as a module for R ∼= Gˇk we see that U = 1. 14

7.0.6. Lemma. I!(λ) ∗I!(µ) has a canonical filtration with ∼ η Gr[I!(λ) ∗I!(µ)] = ⊕η Mλ,µ⊗I!(η) η for some free modules Mλ,µ with a canonical basis. The same is true for standard ∗-sheaves. Proof. The general case follows from the case k = Z for !-sheaves. The proof of this case is very similar to the one for the structure of the projectives (

7.1. Lemma. filt). The convolution is given by the direct image under a stratified semi- small map and one peels one by one the layers corresponding to open orbits in the re- mainder. This is formalized bellow in the notion of “f-semi-small maps” (

7.2. Lemma. f).

7.2.1. Lemma. (a) Over an integral ring k, the pairings given by the canonical maps F [I∗(λ)]⊗F [I∗(µ)]→F [I∗(λ + µ)] have no zero divisors.

(b) For any closed field k, G˜k is a connected . Proof. Let us start with k = C. Over C any noetherian group scheme is always reduced, so G˜C ∼= G˜C by

7.3. Lemma. ident. Now (a) is clear over C since for a reductive group these maps are given by tensoring the sections of line bundles on a (connected) flag variety. This implies (a) for k = Z. The general case follows since the kernel of F [I∗(λ)]⊗F [I∗(µ)]→F [I∗(λ + µ)] is defined over integers by

7.4. Lemma. filt.

Applying (a) to G×G shows that Gr[O(G˜k)] has no zero divisors, and then the same is true for O(G˜k). So over a field k, G˜k is an algebraic group.

7.4.1. Lemma. G˜Z is smooth.

Proof. G˜Z is flat over Spec(Z) since O(G˜Z) is a free Z-module. So smoothness is equivalent to the smoothness of the fibers over closed points, but these are algebraic groups.

7.4.2. Theorem. G˜Z ∼= GˇZ.

Proof. G˜Z is a reductive group since G˜k is for any closed field k by

7.5. Lemma. reduced and

7.6. Lemma. ident. G˜Z is split since for all dominant λ the Weyl module has a G˜Z-form. 15

8. Duality

Verdier duality corresponds to the composition of the duality for G˜-modules with the involution −w0 ∈ Aut(G˜) (more precisely, a representative of this involution is chosen canonically by our fixed choice of the pinning b˜, h˜, e˜ - the last one is the regular nilpotent given by the hyperplane section.

9. Appendix A. Semi-small maps

A will denote some coefficient ring (commutative, with a unit, noetherian and of finite global dimension).

def 9.1. Semi-small maps. A map π : X→Y defines constructible subsets Yk(X) = {y ∈ −1 def −1 Y, dim(X ∩ π y)= k} and Xk =X ∩ π Yk(X), k ∈ N. For any real number k we denote def def X≥k = ∪ Xp, and Y≥k(X)= ∪ Yp(X). We use the same notation for any subvariety S⊆X p≥k p≥k and the restriction of the map to S. Map π : X→Y is said to be dimensionally semi-small if X is irreducible and codimX X≥k ≥ k, k ∈ N. An irreducible subvariety S of X is said to be dimensionally semi-small relative to the map π : X→Y if π|S is dimensionally semi-small. An A-Stratification S of X is said to be dimensionally semi-small relative to the map π if all strata satisfy this property. We say that S is semi-small relative to π if it is dimensionally semi-small and for any stratum S ∈S, restriction π|S¯ is proper.

9.1.1. Lemma. (a) If X has a stratification dimensionally semi-small for π : X→Y , then π is dimensionally semi-small. (b) Let π : X→Y be dimensionally semi-small. Any irreducible subvariety S of X which is “transversal” to the filtration X≥k of X in the sense that

codimSS ∩ X≥k ≥ k, k ∈ N;

is dimensionally semi-small for π and meets X0.

(c) For a dimensionally semi-small S⊆X one has dim Y≥k(S) ≤ dim S − 2k, k ∈ R. Proof. (a) If S is a stratification dimensionally semi-small for π, then

dim Xk = max dim S ∩ Xk ≤ max dim S − k ≤ dim X − k. S∈S S∈S

(b) Since Sk⊆ ∪ S ∩ Xp, transversality condition above implies that for k ∈ N, p≥k

dim Sk ≤ max dim S ∩ Xp ≤ max dim S − p = dim S − k; p≥k p≥k 16

hence π|S is dimensionally semi-small. S meets X0 since S−X0 = S∩X≥1 has codimension ≥ 1.

(c) For k ∈ N, dim Yk(S) ≤ dim Sk − k ≤ dim S − 2k = dim π(S) − 2k, hence also dim Y≥k(S) ≤ dim π(S) − 2k. The same claim for real numbers k ≥ 0 follows.

9.1.2. Lemma. (a) Let S be a stratification of X, dimensionally semi-small for the map π : X→Y . Then the functor π! : DS (X,A)→D(Y,A) is right exact (i.e. it preserves p ≤0 p ≥0 D ), and π∗ : DS (X,A)→D(Y,A) is left exact (preserves D ). (b) If S is semi-small for π : X→Y , direct image preserves perversity, i.e.

π∗ : PS (X,A)→P(Y,A).

Proof. (b) is a consequence of (a). Part (a) is the claim that for any perverse sheaf F ≤0 ≥0 in PS (X,A), one has π!F is in D (Y ) and π∗F is in D (Y ), i.e. for any integer d, constructible sets ∗ i Yd (F)= {y ∈ Y, H (π!F)y =6 0 for some i ≥−d} and ! i ! Yd (F)= {y ∈ Y, H (π∗F)y =6 0 for some i ≤ d}, have dimension ≤ d. Let y be a point in Y and S a stratum in S. Denote

def −1 ρ k Sy =p y −−−→ S −−−→ X    q p  y y πy y −−−→i Y Y.

∗ Any F in PS (X,A) is an extension of the sheaves k!k F, S ∈ S; hence (π!F)y is an ∗ extension of the terms (π!(k!k F))y, corresponding to the various strata S in X. So ∗ ∗ ∗ ! ! ! Yd (F)⊆ ∪ Yd (k!k F) and similarly Yd (F)⊆ ∪ Yd (k∗k F). So it suffices to see that S∈S S∈S ∗ ∗ ! ∗ Y (k!k F) ⊆ Y≥ 1 − (S) ⊇ Y (k!k F), d 2 (dim S d) d the estimates we need will then follow from 1 dim Y≥ 1 (dim S−d)(S) ≤ dim S − 2 (dim S − d)= d. 2 2 To check the first inclusion we estimate the stalk at a point y in Y . The base change gives ∗ ∗ ∗ ∗ [π!(k!k F)]y =[p!k F]y = Hc (Sy, k F). Since F is perverse and S-constructible, k∗F is in the degrees ≤ − dim S, hence ∗ ∗ ∗ ∗ Hc (Sy, k F) is in the degrees ≤ − dim S + 2dim Sy. So y ∈ Yd (k!k F) implies 1 −d ≤− dim S + 2dim Sy, i.e. dim Sy ≥ 2 (dim S − d). 17

The costalk at y ∈ Y is

! ! def ! ! ! ! ∗ ! [π∗(k∗k F)]y =i [p∗k F]= q∗ρ (k F)= H [Sy, (kρ) F].

def ! Since F is in PS (X,A), L=k F is a smooth complex of sheaves on S concentrated in the ! degrees ≥ − dim S, and ρ L is therefore in the degrees ≥ − dim S +2 · codimSSy. [To ! ! ! check this, choose a stratification C of Sy. Then ρ L is an extension of all σ∗σ (ρ L) for σ ! the strata C ⊂ Sy in C. Since L, S and C are smooth, (ρσ) L is in the degrees of L shifted up by 2 · codimSC ≥ 2 · codimSSy, i.e. in the degrees ≥− dim S +2 · codimSSy.] ! ! For y ∈ Yd (k∗k F), we get d ≥ − dim S +2 · codimSSy = dim S − 2dim Sy, hence again 1 dim Sy ≥ 2 (dim S − d).

9.1.3. Lemma. Let S be a stratification of X and let π : X→Y be a proper map. Then the following is equivalent: (a) S is π-semi-small,

(b) π∗ : PS (X,A)→P(Y,A),

(c) π∗IC(S,A) is perverse for each stratum S ∈S. In particular, validity of (b) or (c) is independent of the the choice of the ring A. Proof. We only have to show that if for some choice of the coefficient ring A one has

(∗) π∗IC(S,A) ∈P(Y,A) for all strata S ∈S, then any stratum S in S is dimensionally semi-small for π. i j :(Partition ∂S ֒→ S¯ ←֓S of S¯, gives a Mayer-Vietoris triangle for the sheaf L = IC(S,A ∗ j!(L|S)→L→i∗i L.

We apply π∗ and get another exact triangle ∗ π∗ j!(L|S)→π∗L→π∗ i∗i L.

def Observe that the restriction πS =π|∂S : ∂S→Y is proper and with the stratification S|∂S of ∂S = S¯ − S, it satisfies (∗). Therefore, by an induction assumption S|∂S is semi-small for πS. Now, since L|∂S is in strictly negative perverse degrees, so is (by the preceding lemma) π∗(L|∂S). By the assumption (∗), π∗L is in perverse degrees zero, so from a triangle above we deduce that π∗j!(L|S) is in the non-negative perverse degrees. Therefore, for any integer d one has dim Y(d) ≤ d for the constructible set Y(d) consisting of all y ∈ Y , such that for some i ≥−d cohomology group i i −1 H [π∗ j!(L|S)]y = Hc(S ∩ π y, A[dim S])

does not vanish. However, for y ∈ π(Sk), the highest nontrivial cohomology is in the −1 degree − dim S +2· dim S ∩ π y = − dim S +2k. So π(Sk)⊆Y(dim S − 2k) and therefore dim Sk ≤ dim π(Sk)+ k ≤ dim S − 2k + k = dim S − k. 18

9.2. Remarks. (1) The proof of the lemma actually proves the following. For a stratification S of X and a map π : X→Y , the following is equivalent: (a) S is π dimensionally semi-small for π, ≤0 (b) π! is right exact on DS (X,A), i.e. π! : PS (X,A)→D (Y,A), ≤0 (c) π!IC(S,A) is in D (Y,A) for each stratum S ∈S. (2) Implications 2⇒3 and 3⇒2 are proved in lemmas 3 and 4, however, one can probably prove both at the same time? (3) The converse in lemma 2a holds, but it is not true that for a semi-small π any stratification S which is a refinement of X = ∪Xk is semi-small: S = {X0,X1} for X = N˜ →N = Y in sl2 is a counterexample.

9.2.1. Lemma. If π : X→Y is semi-small there is a semi-small stratification S of X, hence in particular, π∗IC(Y ) is perverse.

9.3. Functor π!∗.

9.3.1. Dimensionally semi-small maps. Let S be a stratification of X, dimensionally semi- small for π : X→Y . On PS (X,A), map of functors π!A→π∗A factors into a diagram (self-dual if A is a field) p 0 ι p 0 π! → H π! −→ H π∗ → π∗, ≤0 ≥0 since π! maps into D (Y,A) and π∗ into D (Y,A). We define a functor (self-dual if A is a field) def p 0 ι p 0 π!∗ : PS (X,A)→P(Y,A), π!∗ = Im[ H π! −→ H π∗].

9.3.2. Lemma. For mixed sheaves, π!∗ preserves weight, hence it satisfies the decomposi- tion theorem. If T is a stratification of Y that makes π into a stratified map, then for a local system L on a stratum S ∈S ∼ π!∗IC(S, L) = ⊕ IC(T, LT ), T ∈TS

−1 here TS⊆T consists of S-relevant strata, i.e, such that the fiber Sy = π y ∩ S has dim(S)−dim(T ) dimension 2 (maximal possible), and the stalk of the local system LT at y ∈ T has a canonical basis consisting of all irreducible components of Sy of maximal dimension.

9.3.3. Lemma. If π : X→Y is dimensionally semi-small there is a semi-small stratification ≤0 ≥0 S of X. Therefore, π!IC(Y ) ∈ D (Y,A), π∗IC(Y ) ∈ D (Y,A), and π!∗IC(Y ) is defined and perverse.

∼ 9.3.4. Lemma. (πτ)!∗ = π!∗τ!∗. 19

9.4. f-semi-small maps. In this section we will weaken the above definitions to allow def for a generic fiber to have dimension fX→Y = dim X − dim π(X).

9.4.1. f-semi-small maps. Map π : X→Y is said to be f-semi-small if X is irreducible, f = fS→Y and

codimX X≥k+f ≥ k, k ∈ N, i.e.,

codimX X≥p ≥ p − f, k ∈ N.

An irreducible subvariety S of X is said to be f-semi-small relative to the map π : X→Y if π|S is f-semi-small. An A-Stratification S of X is said to be f-semi-small relative to the map π if all strata satisfy this property (so denote fS→Y by fS→Y ).

9.4.2. Lemma. (a) Let S be a stratification of X, f-semi-small for the map π : X→Y . p ≤f p ≥−f Then π! : PS (X,A)→ D (Y,A), and π∗ : PS (X,A)→ D (Y,A). (b) Suppose that moreover, T is a stratification of Y such that π : (X, S)→(Y, T ) is a stratified map. For a stratum S ∈ S consider the highest perverse cohomology p f H [π! I!(S)] of the !-direct image of the standard !-sheaf for S. It has a filtration such that the subquotients are the perverse sheaves IT , T ∈T , of the following form.

Let LT be the local system on T with stalks (LT )y = k[Irr(Sy)], (Irr(Sy) denotes the set of all irreducible components of the fiber Sy of π|S at y, that have the (maximal possible) dimension ). Then IT is between the standard !-sheaf and the IC-sheaf: I!(T, LT ) →→ IT →→ I!∗(T, LT ). p f ∼ (c) Suppose moreover, that π is proper. Then H [π! IC(S, k)] = ⊕T ∈T IC(T, LT ). p −f Similarly for H (π∗IC(S, k)).

j i Sublemma. (a) Divide X into an open and a closed part U֒→X←֓Y = X − U. For .9.4.3 A∈PS (X) one has

p p ∗ 0→ Im[ j!(A|U)→A] → A → i∗( i A) →0,

p p p and I = Im[ j!(A|U)→A] is between j!(A|U) and j!∗(A|U) : j!(A|U) →→ I →→ j!∗(A|U). − dim T (b) For A∈PT (Y ) and a stratum T ∈ T , AT = H (A|T ) is a local system on T . ∼ p Perverse sheaf A has a filtration with GrA = ⊕T ∈T IT where IT is between j!(AT ) and j!∗(AT ). p Proof. (a) Since the canonical map j!(A|U)→A is an identity over U, the cokernel C is p p ∗ supported on Y and the image I is between j!(A|U) and j!∗(A|U) . Therefore i (I) is ∼ p p p ∗ = p ∗ a quotient of i ∗ ( j!(A|U)) = 0, and this implies that i∗( i A)−→ i∗( i C)= C. 20

p 0 iT ∗ p ∗ b) Clearly, AT = H (T ֒→Y ) A = iT A. Replace Y with the support of A, let) k p ∗ T be an open stratum and Y − T ֒→Y . then (a) gives 0→ IT → A → k∗( k A) →0 for p p ∗ p ∗ = IT = Im[ (iT )!(AT )→A]. Since for a stratum S⊆Y − T one has (S֒→YT ) k A .p(S֒→Y )∗A we have reduced the number of the strata

9.4.4. Fast Proof of the lemma. (a) follows by the kind of estimates that were used in the basic lemma for semi-small maps. (b) follows from the sublemma by observing that p f for A = H [π! I!(S)] one has AT = LT - the estimate in (a) for the integral of I!(S) over the intersection of the fiber with a stratum different from S is better by two then for arbitrary perverse sheaves, hence only stratum S contributes and others do not subtract.

The proof of (c) is the same - I!∗(S) has estimate better by only 1, so only S contributes, others do not subtract since the exact sequences split by the decomposition theorem (?).

9.4.5. Remarks. (a) One way f-semi-small maps arise is by restricting a semi-small map to closed subvariety of the target. p f (b) It is not true that H [π! I!(S)] has a filtration by standard !-sheaves. Let X be a minimal resolution of the affine quadric xy = uv, and S = {X} while T consists of the singular point p and T = Y − p. Then π!IC!(X)= π!CX [3] = IC(Y ) while I!(T )= CT [3] is larger: 0→IC(Y )→I!(T )→Cp→0.

9.5. Small stratified maps. Let us consider two complex stratified spaces (Y, T ) and (X, S) and a map f : Y → X. We assume that the two stratifications are locally trivial with connected strata and that f is a stratified with respect to the stratifications T and S, i.e., that for any T ∈ T the image f(T ) is a union of strata in S and for any S ∈ S the map f|f −1(S): f −1(S) → S is locally trivial in the stratified sense. We say that f is a stratified semi-small map if

a) for any T ∈T the map f|T¯ is proper b) for any T ∈T and any S ∈S such that S ⊂ f(T¯) we have 1 dim(f −1(x) ∩ T¯) ≤ (dim f(T¯) − dim S) 2 for any (and thus all) x ∈ S . 21

Next the notion of a small stratified map. We say that f is a small stratified map if there exists a (non-trivial) open stratified subset W of Y such that a) for any T ∈T the map f|T¯ is proper b) the map f|W : W → f(W ) is proper and has finite fibers c) for any T ∈T , T ⊂ W , and any S ∈S such that S ⊂ f(T¯) − f(T ) 1 we have dim(f −1(x) ∩ T¯) ≤ (dim f(T¯) − dim S) 2 for any (and thus all) x ∈ S .

The result below follows directly from dimension counting:

9.5.1. Lemma. If f is a semi-small stratified map then Rf∗A ∈ PS (X, k) for all A ∈ PT (Y, k). If f is a small stratified map then, with any W as above, and any A ∈PT (W, k), we have Rf∗j!∗A = ˜j!∗f∗A, where j : W ֒→ Y and ˜j : f(W ) ֒→ X denote the two inclusions.