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 cohomology 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 functor 7 4.1. Bernstein’s induction functors 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 category 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 reductive group 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.