Integral Homology of Loop Groups Via Langlands Dual Groups

REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 15, Pages 347–369 (April 20, 2011) S 1088-4165(2011)00399-X

INTEGRAL HOMOLOGY OF LOOP GROUPS VIA LANGLANDS DUAL GROUPS

ZHIWEI YUN AND XINWEN ZHU

Abstract. Let K be a connected compact Lie group, and G its complexiﬁ- cation. The homology of the based loop group ΩK with integer coeﬃcients is naturally a Z-Hopf algebra. After possibly inverting 2 or 3, we identify Z ∨ ∨ H∗(ΩK, ) with the Hopf algebra of algebraic functions on Be ,whereB is ∨ ∨ a Borel subgroup of the Langlands dual group scheme G of G and Be is the centralizer in B∨ of a regular nilpotent element e ∈ Lie B∨. We also give a similar interpretation for the equivariant homology of ΩK under the maximal torus action.

1. Introduction Let K be a connected compact Lie group and G its complexification. Let ΩK be the based loop space of K. Then the homology H∗(ΩK, Z) is a Hopf algebra over Z. The goal of this paper is to describe this Hopf algebra canonically in terms of the Langlands dual group of G. The rational (co)homology of ΩK is easy since rationally K is the product of spheres. However, the integral (co)homology is much more subtle: H∗(ΩK, Z) is not a finitely generated Z-algebra, and the simplest example H∗(Ω SU(2), Z) involves divided power structures. The integral (co)homology of ΩK has been studied extensively by different methods. The first method was pioneered by Bott (cf. [B58]). He developed an algorithm, which theoretically determines the Hopf algebra structure of integral (co)homology of all based loop groups. By applying this algorithm, he gave explicit descriptions of the (co)homology of special unitary groups, orthogonal groups, and the exceptional group G2. However, Bott’s description depends on the choice of a “generating circle” as defined in loc. cit. and is therefore not canonical. The second method to study the (co)homology of based loop groups comes from the theory of Kac-Moody groups. Namely, let G be the complexification of K.This is a reductive algebraic group over C.LetF = C(( t)) a n d O = C[[t]]. Then the quotient GrG = G(F )/G(O), known as the affine Grassmannian of G, is a maximal partial flag variety of the Kac-Moody group G(F ) (a central extension of G(F )). 1 Let ΩpolK be the space of polynomial maps (S , 1) → (K, 1K ). More precisely, let us parametrize S1 by eiφ,φ ∈ R, and let K ⊂ SO(n, R) be an embedding. Then 1 ΩpolK is the space of maps from (S , 1) → (K, 1K ) such that when composed with K ⊂ SO(n, R), the matrix entries of the maps are given by Laurent polynomials of

Received by the editors September 29, 2009 and, in revised form, October 24, 2010. 2010 Mathematics Subject Classiﬁcation. Primary 57T10, 20G07.

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iφ e .ItisknownthatΩpolK is homotopic to ΩK. On the other hand, the obvious map

ι :ΩpolK → G(F ) →GrG

is a homeomorphism between ΩpolK and GrG,whichisK-equivariant (the action of K on ΩpolK is the pointwise conjugation and on GrG is the left multiplication). See [PS86] for details about these facts. Therefore, GrG and ΩK have the same (co)homology groups. Kostant and Kumar (cf. [KK86]) studied the topology of the (partial) flag varieties of arbitrary Kac-Moody groups. In particular, they determined the (equivariant) cohomology rings of these partial flag varieties, using an algebraic construction called the nil-Hecke ring. D. Peterson (unpublished work) realized that the nil-Hecke ring can also be used to study the homology ring of the affine Grassmannian. The affine variety Spec H∗(GrG) is usually called the Peterson variety. In this paper, we will proceed from yet another perspective of this story, which is pioneered by Ginzburg (cf. [G95]). We first recall the geometric Satake isomorphism developed by Lusztig ([L81]), Ginzburg ([G95]), and Mirković and Vilonen ([MV07]). We refer the details to the paper [MV07]. Recall that GrG is a union of projective varieties. Let Pk be the abelian category of G(O)-equivariant perverse sheaves with k-coefficients on GrG whose supports are finite dimensional, where k is some commutative noetherian ring of finite global dimension. The convolution ∗ product makes Pk into a tensor category, equipped with the functor H (GrG, −)of taking (hyper)cohomology as a fiber functor. The geometric Satake isomorphism ∗ claims that H (GrG, −) gives an equivalence of tensor categories ∗ G − P ∼ ∨ (1.1) H ( rG, ): k = Rep(Gk ,k), where G∨ is the Langlands dual group scheme of G, defined over Z. Ginzburg observed that taking cohomology also gives rise to a tensor functor ∗ ∗ H (GrG, −):Pk → Mod(H (GrG,k)). Assume that G is simple and simply-connected and k = C. By Tannakian formalism, one obtains a map of Hopf algebras ∗ ∨ (1.2) H (GrG, C) → U(gC), ∨ ∨ where gC is the Lie algebra of GC. Ginzburg proved that this map is injective, and the first Chern class c1(Ldet) of the determinant line bundle Ldet on GrG gets ∨ mapped to a regular nilpotent element e in gC under (1.2). Hence the map (1.2) ∨ ⊂ ∨ ∨ ∨ factors through U(gC,e) U(gC), where gC,e is the centralizer of e in gC,e. Ginzburg concluded that this gives an isomorphism ∗ G C ∼ ∨ (1.3) H ( rG, ) = U(gC,e).

This paper is an extension of the above result to the (co)homology of GrG with integer coefficients. According to [MV07], the geometric Satake isomorphism holds for k = Z. Therefore, it is natural to expect that (1.3) should hold over Z (hence should hold modulo every prime p). However, this is not the case, and it fails for two reasons: ∗ First, as we already remarked, the integral cohomology ring H (GrG, Z)isnot Z ∨ finitely generated over whereas U(gZ,e) is. For this reason, we prefer to work with homology because it is a finitely generated Z-algebra (under the Pontryagin

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product) by a result of Bott in [B58]. The Tannakian formalism (cf. §3) gives us a natural homomorphism of group schemes ∨ (1.4) Spec H∗(GrG, Z) → GZ .

Second, although the first Chern class c1(Ldet) still gives rise to an element ∨ e ∈ gZ , it is not a regular nilpotent element modulo every prime p.Whene mod p ∨ ⊗F ∗ G F is not regular, U(ge ) p is “too large” to be isomorphic to H ( rG, p). We must carefully exclude these p’s. Now the main theorem of this note reads as 1.1. Theorem (See Theorem 6.1(1)). Let G be a reductive connected group over C such that its derived group Gder is almost simple. Let K be a maximal compact subgroup of G.LetG be the square of the ratio of the lengths of long roots and the short roots of G (so G=1, 2 or 3). Then there is a canonical isomorphism of group schemes over Z[1/G]: Z −∼→ ∨ Spec H∗(ΩK, [1/G]) Be [1/G], where B∨ is a fixed Borel subgroup of G∨, e ∈ Lie B∨ is a regular nilpotent element ∨ ∨ given by (5.3),andBe is the centralizer of e in B . This theorem describes the Hopf algebra structure of H∗(ΩK)intermsofa ∨1 purely algebraic object Be , at least after inverting G. In fact, one can also see ∨ the grading of H∗(ΩK)intermsofBe : the grading on H∗(ΩK) is the same thing as a Gm-actiononSpecH∗(ΩK), and this Gm action can be identified with the Gm ∨ G → ∨ action on Be given by the adjoint action through the cocharcter 2ρ : m T . We will also prove an equivariant version of the above theorem. Let T be a maximal torus of G such that T ∩ K is a maximal torus of K.ThenT ∩ K ∗ Z ∩ acts on ΩK via conjugation. Let RT = HT ∩K (pt, ). The T K-equivariant T ∩K homology H∗ (ΩK, Z) (for precise definition, see (2.10)) is an RT -Hopf algebra, T ∩K hence Spec H∗ (ΩK, Z) is a group scheme over RT . On the other hand, the T - T ∨ equivariant Chern classes of line bundles on GrG give an element e ∈ gZ ⊗RT [1/nG] (where nG is an integer explicitly given in Remark 5.8). The equivariant version of the main theorem reads as 1.2. Theorem (See Theorem 6.1(2)). Notations are the same as in Theorem 1.1. 1 There is a canonical isomorphism of group schemes over RT [ ]: GnG T ∩K Z 1 −∼→ ∨ Spec H∗ (ΩK, [ ]) BeT [1/G], GnG ∨ T ∨ × where BeT is the centralizer of e in B Spec RT [1/nG]. Finally, we also have descriptions of the cohomology H∗(ΩK)andtheK-equi- K ∨ variant homology H∗ (ΩK) in terms of group-theoretic data of G . For details, see §6.3 and §6.5. § ∗ Plan of the paper. In 2, we prove that the equivariant cohomology functor HT is P ∗ a fiber functor from to RT := HT (pt)-modules that is canonically isomorphic to ∗ ∗ H ⊗ RT . We also describe how the cup product by H (GrG) interacts with the ∗ § tensor structure of the functor HT .In 3, we construct a canonical homomorphism T ∨ of groups schemes over Z from Spec H∗ (GrG)toBZ .In§4, we digress to discussing

1S. Kumar informed us that D. Peterson also obtained a similar result (unpublished).

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line bundles on the aﬃne Grassmannian and their equivariant structures. In §5, we T L ∨ ⊗ determine the equivariant Chern class c1 ( det) explicitly as an element in gZ RT . Finally, in §6, we prove our main theorem.

Convention and Notation. In this paper, we always assume that G is a connected reductive algebraic group over C.From§4, we will assume that Gder is almost simple. We will fix a Borel subgroup B ⊂ G and a maximal torus T ⊂ B.LetU • be the unipotent radical of B.LetX (T )andX•(T ) be the character group and cocharacher group of T .LetK be the maximal compact subgroup of G containing the maximal compact torus in T . We shall identify ΩK with ΩpolK, and denote them simply by ΩK. The Langlands dual group G∨ is the Chevalley group scheme over Z whose root system is identified with the coroot system of G. Later a Borel subgroup B∨ ⊂ G∨ and a maximal torus T ∨ ⊂ G∨ will be fixed by the geometric Satake isomorphism. Let U ∨ be the unipotent radical of B∨. The Lie algebras of these group schemes will be denoted by g∨, b∨, t∨ and u∨,whicharefreeZ-modules. With dual groups in mind, our notation concerning the root system of G is opposite to the usual one. Let Φ∨, Φ be the set of roots and coroots of G.Coweights and coroots of G will be denoted by λ,μ,... and α,β,..., while weights and roots are denoted by λ∨,μ∨,...and α∨,β∨,....Let2ρ∨ denote the sum of positive roots. We will give X•(T ) a partial order such that λ ≤ μ if and only if μ − λ is a positive integral combination of the simple coroots. Let (−, −)Kil be the Killing form on X •(T ) given by: ∨ ∨ (x, y)Kil = α ,x α ,y . α∨∈Φ∨ ∗ The (co)homology groups H (−),H∗(−) are taken with Z-coefficients unless otherwise specified. All tensor products ⊗ with no base ring specified are understood to be taken over Z. For simplicity, we will write P for PZ. For objects F∈P,we ∗ G F ∗ G F ∗ F ∗ F will abbreviate H ( rG, ),HT ( rG, )byH ( )andHT ( ).

2. Equivariant cohomology functor Let G be a connected reductive algebraic group over C.LetF = C(( t)) a n d O = C[[t]]. Let GrG = G(F )/G(O) be the aﬃne Grassmannian of G. Each coweight λ λ ∈ X•(T ) determines a point t ∈ T (F ), and hence a point in GrG, which we still λ λ denote by t .Forλ ∈ X•(T ), let Grλ = G(O)t G(O)/G(O)betheG(O)-orbit λ λ through t .EachG(O)-orbit of GrG contains a unique point t for some dominant coweight λ.Forλ dominant, we denote the closure of Grλ by Gr≤λ.Theneach Gr≤λ is a projective variety and GrG is their union.

2.1. Weight functors and MV-ﬁltration. In [MV07], Mirkovi´c and Vilonen introduce the weight functors ∗ − P→ Z (2.1) Hc (Sμ, ): Mod( ), μ where Sμ is the U(F )-orbit through t ,μ ∈ X•(T ) (notice that in [MV07], U is denoted by N). They show in [MV07, Theorem 3.6] that there is a natural isomorphism of tensor functors: ∗ − ∼ ∗ − P→ Z (2.2) H ( ) = Hc (Sμ, ): Mod( ). μ∈X•(T )

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Moreover, under the geometric Satake isomorphism (1.1), the weight functors correspond to the weight spaces of a maximal torus T ∨ ⊂ G∨. ∗ For each F∈P, consider the ﬁltration {Fil≥μ H (F)} indexed by the partially ordered set X•(T ): ∗ ∗ ∗ Fil≥μ H (F)=ker(H (F) → H (S<μ, F)),

where S<μ = Sμ − Sμ. We call this filtration the MV-filtration. It is functorial for F∈P. Clearly, the weight functors are the associated graded pieces of the MV-filtration. Under the geometric Satake isomorphism (1.1), there is a unique ∨ ∨ ∨ Borel subgroup B ⊂ G containing T such that the natural X•(T )-filtration on the B∨-module H∗(F) coincides with the MV-filtration. Now consider the equivariant cohomology functor: ∗ − P→ gr (2.3) HT ( ): Mod (RT ). ∗ Z −∼→ X• gr Here RT = HT (pt, ) Sym( (T )), and Mod (RT ) is the category of graded RT -modules. 2.2. Lemma. There is a natural isomorphism of functors ∗ G − −∼→ ∗ G − ⊗ P→ gr HT ( r, ) H ( r, ) RT : Mod (RT ). Proof. The construction of weight functors (2.1) extends to the T -equivariant set- ∈ X ∗ F ting. For each μ •(T ), the cohomology Hc (Sμ, ) is concentrated in degree ∨ ∗ F 2ρ ,μ ([MV07, Theorem 3.5]), hence the equivariant cohomology HT,c(Sμ, )is free over RT with a canonical isomorphism ∗ F ∗ F ⊗ (2.4) HT,c(Sμ, )=Hc (Sμ, ) RT . ∗ F ∗ F There is a spectral sequence calculating HT ( ) with E1-terms HT,c(Sμ, ). Over each component of GrG, the nonzero terms of this E1 page are all in degrees of the same parity, hence the spectral sequence degenerates. In other words, we get a { ∗ F } T -equivariant version of the MV-filtration Fil≥μ HT ( ) with associated graded ∗ F pieces HT,c(Sμ, ). ∗ F ∨ Again, by [MV07, Theorem 3.5], Fil>μ HT ( ) is in degrees > 2ρ ,μ , hence we have a canonical splitting of the exact sequence → → → ∗ F → 0 Fil>μ Fil≥μ HT,c(Sμ, ) 0 given by

∨ ∨ ∗ F 2ρ ,μ F ⊗ → 2ρ ,μ ⊗ → HT,c(Sμ, )=Hc (Sμ, ) RT (H Fil≥μ) RT Fil≥μ, k where H Fil≥μ means the degree k part of Fil≥μ. Therefore, the T -equivariant ∗ F MV-ﬁltration on HT ( ) also has a canonical splitting: ∗ − ∼ ∗ − P→ gr (2.5) HT ( ) = HT,c(Sμ, ): Mod (RT ). μ∈X•(T ) Combining (2.4), (2.5) and (2.2), we get isomorphisms of functors P→ gr Mod (RT ): ∗ − ∼ ∗ − ∼ ∗ − ⊗ ∼ ∗ − ⊗ HT ( ) = HT,c(Sμ, ) = Hc (Sμ, ) RT = H ( ) RT . μ∈X•(T ) μ∈X•(T )

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∗ − 2.3. Monoidal structure on HT ( ). Although the isomorphism in Lemma 2.2 ∗ − equips HT ( ) with a monoidal structure, this monoidal structure is, apriori,not ∗ − what we want to use later. We recall here another monoidal structure on HT ( ), following [MV07, §5,§6]. First recall the deﬁnition of the convolution product ∗ on P, following [MV07, §4]. Consider the diagram

G(O) p q mG GrG ×GrG ←− G(F ) ×GrG −→ G(F ) ×GrG −−→GrG. Here p is the natural projection morphism, q is the quotient map by the right G(O)- −1 action on G(F ) ×GrG given by (x, y) · g =(xg, g y)andmG is induced by the multiplication on G(F ). By definition, F1 ∗F2 = mG,∗F, G(O) where F is the unique perverse sheaf on G(F ) ×GrG such that L ∗ ∗p 0 q (F)=p H (F1 F2). Hence G(O) ∗ F ∗F ∼ ∗ ×G F (2.6) HT ( 1 2) = HT (G(F ) rG, ). 1 Let X = AC. Recall that the global counterpart GrX of GrG classifies triples (x, E,τ)wherex ∈ X, E is a G-torsor over X and τ is a trivialization of E over O G( ) X −{x}. The global counterpart of G(F ) ×GrG is the space GrX ×GrX which classifies tuples (x1,x2, E1, E2,τ1,τ2)wherex1,x2 ∈ X, E1, E2 are G-torsors over X, τ1 is a trivialization of E1 on X −{x1} and τ2 is an isomorphism of G-torsors E | −∼→E| G ×G 1 X−{x2} 2 X−{x2}.Forx1 = x2, the fiber ( rX rX )x1,x2 is isomorphic to G(O) G ×G G ×G × rG rG; while for x1 = x2, the fiber ( rX rX )x1,x2 is isomorphic to G(F ) GrG. Given objects F1, F2 ∈P, we can “spread them over the curve X”togetper- verse sheaves K1, K2 on GrX ; we can also mimic the construction of F to get a perverse sheaf K on GrX ×GrX (see [MV07, §5] for details). The restriction of K L G ×G p 0 F F on ( rX rX )x1,x2 can be identified with H ( 1 2)whenx1 = x2 and with F when x1 = x2, under the above identification of fibers. In [MV07, Lemma 6.1, i eqn.(6.4)], it is shown that the direct image sheaves R π∗(K) are constant (as T - 2 equivariant sheaves, as we can easily see), where π : GrX ×GrX → X is the natural projection. In other words, we have a co-specialization isomorphism from the stalks i 2 2 of R π∗K along the diagonal Δ(X) ⊂ X to its stalks over X − Δ(X). Using the above identifications, the co-specialization map takes the form: G(O) L ∗ ∗ ×G F −∼→ ∗ G ×G p 0 F F (2.7) SpK : HT (G(F ) rG, ) HT ( rG rG, H ( 1 2)). The argument of [MV07, Lemma 6.1, eqn.(6.3)] has an obvious T -equivariant version (using Lemma 2.2), hence L ∗ G ×G p 0 F F ∼ ∗ F ⊗ ∗ F (2.8) HT ( rG rG, H ( 1 2)) = HT ( 1) RT HT ( 2). Combining (2.7), (2.6) and (2.8), we get the monoidal structure ∗ F ∗F −∼→ ∗ F ⊗ ∗ F HT ( 1 2) HT ( 1) RT HT ( 2).

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∗ − § 2.4. Lemma. Under the monoidal structure of HT ( ) introduced in 2.3 and the ∗ ∗ monoidal structure of H (−) ⊗ RT induced from that of H (−),thenaturaliso- morphism in Lemma 2.2 is a monoidal isomorphism. In particular, the monoidal ∗ − § structure on HT ( ) introduced in 2.3 is also a tensor structure (i.e., it is compat- ible with the commutativity constraints2). Proof. Using the same argument of [MV07, §6], one can show that the sum of the T -equivariant weight functors has a natural monoidal structure. Moreover, the isomorphism in (2.5) is a monoidal isomorphism. Using (2.4) and the argument of [MV07, Proposition 6.4], it is easy to see that ∗ − ∼ ∗ − ⊗ P→ X•(T ) (2.9) HT,c(Sμ, ) = Hc (Sμ, ) RT : Mod (RT ) μ∈X•(T ) μ∈X•(T ) ∗ − ∼ ∗ − ⊗ as monoidal functors. The natural isomorphism HT ( ) = H ( ) RT in Lemma 2.2 is given by the composition of monoidal isomorphisms (2.5), (2.9) and (2.2), hence it is also a monoidal isomorphism.

2.5. Remark. Similarly, we can consider ∗ − ∗ − P→ gr HG(O)( )=HG( ): Mod (RG). ∗ Z Here RG = HG(pt, ). Using the argument of [MV07, Proposition 6.1], one can ∗ − show that HG( ) is also a tensor functor. Further argument shows that it is actually a fiber functor (i.e., it is exact and faithful). By [Sa72], this fiber functor defines a ∨ ∨ G -torsor E over Spec RG (here we use the fact that G is flat over Z). Moreover, by Lemma 2.2, the pull back of E to Spec RT admits a canonical trivialization. However, E itself does not have a canonical trivialization. G ∗ G ∗ 2.6. Equivariant (co)homology of rG. Since HT ( rG)=HT ∩K (ΩK)andΩK ∗ G is a homotopy commutative H-space, HT ( rG) is naturally a commutative and co- G ∗ G commutative Hopf algebra over RT . By the cell-decomposition of rG, HT ( rG) is a free RT -module concentrated in even degrees. We denote the coproduct on ∗ G HT ( rG)byΔ. We define the T -equivariant homology T G ∗ G gr (2.10) H∗ ( rG):=HomRT (HT ( rG),RT ) ∗ G T G i G to be the graded dual of HT ( rG)asanRT -module, i.e., Hn ( rG) sends HT ( rG) i−n T G ∼ T G T G −∗ G D to RT .NotethatH∗ ( rG)=−lim→ H∗ ( r≤λ)whereH∗ ( r≤λ)=HT ( r≤λ, ≤λ) (D≤λ is the dualizing complex of Gr≤λ with the canonical T -equivariant structure). ∗ G T G As the RT -dual of HT ( rG), the T -equivariant homology H∗ ( rG)isalsoa commutative and co-commutative Hopf algebra over RT . The multiplication on T T ∩K H∗ (GrG) is identified with the Pontryagin product on H∗ (ΩK), and is denoted by ∧. G Similarly, we can define the G-equivariant homology H∗ (GrG)ofGrG either as ∗ G −∗ G D the RG-dual of HG( rG) or the direct limit of HG ( r≤λ, ≤λ). This is a Hopf algebra over RG. ∗ G ∗ G − Next, we consider the action of HT ( rG) on the fiber functor HT ( rG, )via cup product.

2See [MV07, Section 6] for the deﬁnition of the commutativity constraints on P.

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F F ∈P ∈ ∗ G 2.7. Proposition. For any 1, 2 ,andanyh HT ( rG), we have a commutative diagram ∗ F ⊗ ∗ F ∼ / ∗ F ∗F HT ( 1) RT HT ( 2) HT ( 1 2) Δ(h) h ∗ F ⊗ ∗ F ∼ / ∗ F ∗F HT ( 1) RT HT ( 2) HT ( 1 2) ∗ − where the horizontal maps are given by the tensor structure of the functor HT ( ) (see the proof of Lemma 2.2). ∗ A similar statement also holds for G-equivariant cohomology HG. ∈ ∗ G ∗ F ∗F Proof. From (2.6) we see that the action of h HT ( rG)onHT ( 1 2)isthe G(O) G(O) ∗ ∈ ∗ ×G ∗ ×G F same as the action of mG(h) HT (G(F ) rG)onHT (G(F ) rG, ). Let us keep track of the cup product by the cohomology of the relevant spaces in the construction of the tensor structure in §2.3. Since the morphism π : GrX ×GrX → X2 is stratified by affine space bundles (see [MV07, Lemma 6.1, argument for i (6.4)]), an easy spectral sequence argument shows that the sheaves R π∗Z are locally constant, hence constant. In other words, we also have a co-specialization i isomorphism from the stalks of R π∗Z along Δ(X) to stalks elsewhere, i.e., G(O) ∗ ∗ ×G −∼→ ∗ G ×G (2.11) Sp : HT (G(F ) rG) HT ( rG rG). Moreover, by the naturality of co-specialization maps, (2.7) and (2.11) are compat- ible under the cup product: ∗ ∪ ∗ ∪ ∗ SpK(h v)=Sp (h) SpK(v), G(O) G(O) ∈ ∗ ×G ∈ ∗ ×G F for any h HT (G(F ) rG),v HT (G(F ) rG, ). ∈ ∗ G ∗ F ∗ From the above discussion, the cup product action of h HT ( rG)onHT ( 1 F ∗ F ⊗ ∗ F 2), when transported to HT ( 1) RT HT ( 2) under the isomorphism given by ∗ − ∗ ∗ ∈ the tensor structure of HT ( ), is the cup product action of the class Sp (mG(h)) ∗ G ×G ∗ G ⊗ ∗ G HT ( rG rG)=HT ( rG) RT HT ( rG). Therefore, to prove the proposition, it suffices to show that ∗ ∗ ∗ G → ∗ G ×G (2.12) Δ = Sp mG : HT ( rG) HT ( rG rG).

Recall that the coproduct Δ is induced from the multiplication mK :ΩK × ΩK → ΩK, hence × ∗ ∗ ∗ ∗ G → ∗ × (2.13) (ι ι) Δ=mK ι : HT ( rG) HT ∩K (ΩK ΩK). 2 Now we construct a map ι :ΩK × ΩK × X →GrX ×GrX . For a point (γ1,γ2 : 1 → ∈ × × 2 C C S K, x1,x2) ΩK ΩK X , we consider the complexifications γ1 ,γ2 : 1 A −{0}→G (recall that γi are polynomial maps so that the complexifications make sense). Let γC : A1 −{x }→G be the composition of the translation i,xi i A1 −{ } −∼→ A1 −{ } C xi 0 with γi . We define triv triv ι(γ1,γ2,x1,x2)=(x1,x2, E1 = E , E2 = E ,τ1,τ2) triv where E is the trivial G-torsor on X, and the isomorphism τi (viewed as an automorphism of E triv over X −{x })isgivenbyγC : X −{x }→G for i =1, 2. i i,xi i The restriction of ι to points x1 = x2 can be identified with the map ι(2) : O ι×ι q G( ) ΩK × ΩK −−→ G(F ) ×GrG −→ G(F ) ×GrG; the restriction of ι to x1 = x2 is

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2 exactly the map ι × ι :ΩK × ΩK →GrG ×GrG.WeviewΩK × ΩK × X as a constant family over X2,thenι commutes with the co-specialization maps, i.e., we have a commutative diagram

G(O) ∗ ∗ ×G Sp / ∗ G ×G HT (G(F ) rG) HT ( rG rG)

∗ ∗ ι(2) (ι×ι) ∗ × ∗ × HT ∩K (ΩK ΩK) HT ∩K (ΩK ΩK). On the other hand, we also have a commutative diagram

m ΩK × ΩK K /ΩK

ι(2) ι G(O) mG / G(F ) ×GrG GrG.

Combining the two commutative diagrams, we get × ∗ ∗ ∗ ∗ ∗ ∗ ∗ (ι ι) Sp mG = ι(2)mG = mK ι . In view of (2.13), we get × ∗ × ∗ ∗ ∗ (ι ι) Δ=(ι ι) Sp mG. Since (ι × ι)∗ is an isomorphism, the equality (2.12) follows. This ﬁnishes the proof of the proposition.

T 3. Relating H∗ (GrG) to the Langlands dual group Consider the ﬁber functor ∗ − ⊗ T G P→ gr T G HT ( ) RT H∗ ( rG): Mod (H∗ ( rG)).

We now construct a canonical tensor automorphism σcan of this functor. For any F∈P ∗ F ⊗ T G , the action of σcan on HT ( ) RT H∗ ( rG)isgivenby i σcan(v ⊗ h)= (h ∪ v) ⊗ (hi ∧ h), i ∈ ∗ F ∈ T G { i} { } where v HT ( ),h H∗ ( rG)and h , hi are dual bases of the free RT - ∗ G T G modules HT ( rG)andH∗ ( rG). One readily checks that σcan does not depend i on the choice of the dual bases {h }, {hi}.

3.1. Lemma. The natural transformation σcan is a tensor automorphism of the ∗ − ⊗ T G ﬁber functor HT ( ) RT H∗ ( rG). F F ∈P ∈ ∗ F Proof. Let 1, 2 and vi HT ( i)fori =1, 2. We have to show that

σcan(v1 ⊗ v2)=σcan(v1) ⊗ σcan(v2). ⊗ ∗ F ∗F Here we view v1 v2 as an element in HT ( 1 2) under the identiﬁcation ∗ F ⊗ ∗ F ∼ ∗ F ∗F HT ( 1) RT HT ( 2) = HT ( 1 2).

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On one hand, we have i j (3.1) σcan(v1) ⊗ σcan(v2)= (h ∪ v1 ⊗ hi) ⊗ (h ∪ v2) ⊗ hj i,j i j = (h ∪ v1) ⊗ (h ∪ v2) ⊗ (hi ∧ hj ). i,j On the other hand, by Proposition 2.7, k (3.2) σcan(v1 ⊗ v2)= Δ(h )(v1 ⊗ v2) ⊗ hk. k ∗ G ∧ Since the coproduct Δ on HT ( rG) is adjoint to the Pontryagin product on T H∗ (GrG), the two expressions (3.1) and (3.2) are the same.

T By this lemma and the Tannakian formalism, σcan deﬁnes an H∗ (GrG)-valued ⊗ ∗ ∨ × point of the group scheme Aut (HT ), which is canonically isomorphic to G Spec RT by Lemma 2.2: T T ∨ (3.3) σ :SpecH∗ (GrG) → G × Spec RT . We also have the non-equivariant counterpart: ∨ (3.4) σ :SpecH∗(GrG) → G . 3.2. Lemma. The morphisms σT and σ as in (3.3) and (3.4) are homomorphisms of group schemes over RT and Z. Proof. We give the argument for the non-equivariant version, and the equivariant ∗ ∗ version is similar. We abbreviate H∗(GrG)byH∗ and H (GrG)byH . We need to check that the following diagram is commutative

∨ τ˜ / O(G ) H∗

∗ μ Δ∗ ∨ ∨ τ˜⊗τ˜ / O(G ) ⊗O(G ) H∗ ⊗ H∗ ∗ ∨ ∗ where μ and Δ∗ are the coproducts on O(G )andH∗. The composition (˜τ ⊗τ˜)◦μ ∗ corresponds to the following automorphism of the tensor functor H (−)⊗H∗ ⊗H∗: ∗ ∗ H (F) ⊗ H∗ ⊗ H∗ → H (F) ⊗ H∗ ⊗ H∗, i j (3.5) v ⊗ 1 ⊗ 1 → (h ∪ h ∪ v) ⊗ hi ⊗ hj , i,j ∗ i ∗ for v ∈ H (F)and{h }, {hi} dual bases of H and H∗ as before. On the other hand, the composition Δ∗ ◦ τ˜ is given by the automorphism ∗ ∗ H (F) ⊗ H∗ ⊗ H∗ → H (F) ⊗ H∗ ⊗ H∗, k (3.6) v ⊗ 1 ⊗ 1 → (h ∪ v) ⊗ Δ∗(hk). i ∗ ∗ ∗ Since Δ∗ : H∗ → H∗ ⊗ H∗ is adjoint to ∪ : H ⊗ H → H , the two expressions (3.5) and (3.6) are the same. 3.3. Proposition. T ∨ (1) The homomorphism σ factors through B × Spec RT ;

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∼ ∨ (2) We have a natural isomorphism of group schemes Spec H0(GrG) −→ Z , the center of G∨; (3) If G is simply-connected, σ factors through U ∨; (4) In general, let Gsc be the simply-connected cover of Gder, and we identify the G 0 G neutral component rG with rGsc . Then we have a natural isomorphism of Hopf algebras: ∼ H∗(GrG) = H0(GrG) ⊗ H∗(GrGsc ). ∨ The morphism σ :SpecH∗(GrG) → B factors as ∗ ∨ ∨ ∨ Spec H∗(GrG) → Spec H0(GrG) × Spec H∗(GrGsc ) −→ Z × U → B , where the starred arrow is the product of the morphisms in (2) and (3). Here we have identified unipotent radicals U ∨ of B∨ for G∨ and (Gsc)∨ = (G∨)ad. ∗ − ⊗ T G Proof. (1) The action of σcan preserves the MV-filtration on HT ( ) RT H∗ ( rG), which implies that σT factors through B∨ × Spec R . ∼ ∼ T (2) The component group π0(GrG) = π1(K) = X•(T )/ZΦ(hereZΦistheco- root lattice of G). Therefore, H0(GrG) is the group algebra of X•(T )/ZΦ, hence isomorphic to O(Z∨)asHopfalgebras. (3) and (4) Since the tensor automorphism σcan preserves the MV-filtration, it induces a tensor automorphismσ ¯can on the associated graded pieces, i.e., on the tensor functor ∗ − ⊗ G P→ X•(T ) G Hc (Sμ, ) H∗( rG): Mod (H∗( rG)). μ∈X•(T ) ∨ The tensor automorphismσ ¯can gives a homomorphism Spec H∗(GrG) → T which σ ∨ π ∨ is clearly the same as the composition Spec H∗(GrG) → B → T . On the other ∗ F hand, since Hc (Sμ, ) is concentrated in one degree, the action ofσ ¯can on it is given by → ∗ ∪ ⊗ ∈ ∗ F ⊗ G v (¯μ v) μ¯ Hc (Sμ, ) H0( rG). ∈ ∗ F ∈ X X Z ∨ ⊂ Z X Z ∨ Here v Hc (Sμ, ),μ ¯ is the image of μ •(T )in •(T )/ Φ [ •(T )/ Φ ]= ∗ 0 H0(GrG), andμ ¯ is the element in H (GrG) that takes value 1 on the component corresponding toμ ¯ and 0 on other components. This means that the tensor auto- morphismσ ¯can in fact comes from a tensor automorphism of the functor ∗ − ⊗ G P→ X•(T ) G Hc (Sμ, ) H0( rG): Mod (H0( rG)) μ∈X•(T )

X•(T ) X•(T ) followed by the functor Mod (H0(GrG)) → Mod (H∗(GrG)) given by ⊗ G H0 H∗( rG). In other words, there is a commutative diagram

σ / ∨ (3.7) Spec H∗(GrG) B

π ∼ / ∨ / ∨ Spec H0(GrG) Z T

which implies that σ factors through π−1(Z∨)=Z∨ × U ∨ ⊂ B∨.

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sc ∨ ∨ Now if G = G , Z is trivial and the map H∗(GrG) → B factors through U ∨. This proves (3). In general, by the functoriality of the geometric Satake isomorphism for the homomorphism Gsc → G, we have a commutative diagram / ∨ ∨ (3.8) Spec H∗(GrG) Z × U

/ ∨ Spec H∗(GrGsc ) U . Combining (3.7) and (3.8), the assertion is proved. 3.4. Remark. The composition

T T σ ∨ ∨ (3.9) Spec H∗ (GrG) −−→ B × Spec RT → T × Spec RT can be interpreted via equivariant localization. In fact, as we remarked in the proof T ∨ of Proposition 3.3(3),(4), the above morphism Spec H∗ (GrG) → T × Spec RT comes from the tensor automorphismσ ¯can of the T -equivariant weight functors ∗ G ∗ − (2.9). On the other hand, the cup product action of HT ( rG)onHT,c(Sλ, ) ∗ ∗ G → ∗ −∼→ ∗ { λ} factors through the restriction map iλ : HT ( rG) HT (Sλ) HT ( t ) (the λ second arrow is an isomorphism because Sλ contracts to t under a Gm-action). ∈ ∗ F Therefore, for vλ HT,c(Sλ, ), we have ∗ i ∪ ⊗ ⊗ σ¯can(vλ)= (iλ(h ) vλ) hi = vλ iλ,∗(1), i T { λ} → T G ∗ ∗ where iλ,∗ : RT = H∗ ( t ) H∗ ( rG) is the adjoint of iλ. The product of iλ is the equivariant localization map: ∗ ∗ ∗ G → ∗ { λ} Loc = iλ : HT ( rG) HT ( t ). λ∈X•(T ) Let T (3.10) Loc∗ = iλ,∗ : RT [X•(T )] → H∗ (GrG)

λ∈X•(T ) ∗ be the adjoint of Loc ,whereRT [X•(T )] is the RT -valued group ring of the abelian group X•(T ). The above discussion shows that the morphism (3.9) coincides with T ∨ Spec(Loc∗):SpecH∗ (GrG) → Spec RT [X•(T )] = T × Spec RT .

4. Remarks on line bundles From this section on, we assume that Gder is almost simple. In this case, it is well known that the Picard group of each component of GrG is isomorphic to Z. Let Ldet denote the line bundle on GrG whose restriction to each component is the positive generator of the Picard group. 4.1. Equivariant line bundles. Fix a representation V of G. The homomorphism φ : G → GL(V ) induces a morphism of ind-schemes Grφ : GrG →GrGL(V ) over C. We identify GrGL(V ) with the set of O-lattices in V ⊗ F in the usual way. Fix a ∈G L ∈G lattice Λ1 rGL(V ). Deﬁne a line bundle Λ1 whose value at a point Λ rGL(V ) is the line ⊗−1 det(Λ : Λ1) := det(Λ/Λ ∩ Λ1) ⊗ det(Λ1/Λ ∩ Λ1)

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where det(−) means taking the top exterior power of a ﬁnite dimensional C-vector space. Let Lφ = Gr∗L be the pull-back line bundle on Gr . Λ1 φ Λ1 G

Let G(F )Λ1 be the stabilizer of the lattice Λ1 in G(F )(via the action φ). Then Lφ has a natural G(F ) -equivariant structure: for each g ∈ G(F ) , we identify Λ1 Λ1 Λ1 det(Grφ(x):Λ1) and det(Grφ(gx):Λ1)=det(Grφ(gx):gΛ1) via the action of g. ⊂ ∈G Let H G(F )Λ1 be an algebraic subgroup. Let x rG and Hx be the stabilizer of x in H, so that we can identify the H-orbit through x with the homogeneous space H/Hx. Via the induction functor, we have an identiﬁcation H H • Pic (H · x)=Pic (H/Hx)=X (Hx). φ H • It is easy to see that the image of L | · ∈ Pic (H · x)inX (H ) is the character Λ1 H x x given by the action of Hx on the line det(Grφ(x):Λ1) (notice that Hx stabilizes both Grφ(x)andΛ1). We consider the special case where Λ0 = V ⊗C O. Since Λ0 is stabilized by G(O), Lφ has a natural G(O)-equivariant structure. Take H = G ⊂ G(O)andx = tλ. Λ0 Then Hx = Pλ ⊂ G is a parabolic subgroup containing the maximal torus T .We can identify the image of Lφ in X•(P ) ⊂ X•(T ) as the character of the action of Λ0 λ λ T on det(t Λ0 :Λ0), which is ∨ ∨ (4.1) − dim Vχ∨ · χ ,λ χ . χ∨∈weight(V )

We can further specialize to the case H = Hx = T , then the element in (4.1) is also the restriction of cT (Lφ ) to the point tλ, if we identify X•(T ) with H2 (pt). 1 Λ0 T 4.2. Lemma (See also [KNR94, Section 5] and [So00, Lemma 10.6.1]). We have ⊗ Lφ = L dV ∈ Pic(Gr ) where Λ1 det G 1 ∨ 2 d = dim V ∨ · χ ,θ . V 2 χ χ∨∈weight(V ) Here θ stands for the coroot corresponding to the highest root θ∨ of G. ∈ G G i Proof. For i π0( rG), let rG denote the corresponding connected component. We ﬁrst claim that the degree of Lφ on the generating cycle of H (Gri )(∼ Z) Λ1 2 G = G i ∈ is independent of the component rG.Infact,foranyg G(F ), by choosing a trivialization of the line det(gΛ1 :Λ1), we get an isomorphism of line bundles g∗Lφ ∼ Lφ .SinceG(F ) transitively permutes the components of Gr , our asser- Λ1 = Λ1 G G 0 tion follows. In the following we concentrate on the neutral component rG. Let I ⊂ G(O) be the Iwahori subgroup of G(F ) containing B. The only 1- G 0 −θ dimensional I-orbit in rG is It , and the fundamental class of its closure generates H (Gr0 ). Therefore the number d in question is the degree of Lφ on the closure 2 G V Λ1 of It−θ. Fix an sl(2)-triple (xθ∨ ,θ,x−θ∨ )wherex±θ∨ belongs to the root space gθ∨ .Let φ0 :SL(2)→ G(F ) be the homomorphism which sends the standard triple (e, h, f) −1 −θ to (t xθ∨ ,θ,tx−θ∨ ). Then the closure of It in GrG is the φ0(SL(2))-orbit through −θ −θ t , and the stabilizer of t is the upper triangular matrices B2 ⊂ SL(2). To compute the degree of Lφ on this orbit, we will choose a lattice Λ which is stable Λ1 1 −θ under the SL(2)-action and calculate the character of B2 acting on det(t Λ0 :Λ1).

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 360 ZHIWEI YUN AND XINWEN ZHU Let V = n Vn be the grading according to the action of θ: Vn is the di- ∨ ∨ rect sum of weight spaces Vχ such that χ ,θ = n. Then the lattice Λ1 = −[n/2]O⊗ n t C Vn is stable under the SL(2)-action. On the other hand, the lattice −θ −nO⊗ −θ t Λ0 = n t C Vn. Therefore, the action of B2 on det(t Λ0 :Λ1)isviaa character whose pairing with θ is:

2 n n 1 ∨ 2 (4.2) n(n − [ ]) dim V = dim V = dim V ∨ · χ ,θ . 2 n 2 n 2 χ n n χ∈weight(V )

Here the ﬁrst equality follows from the fact that dim Vn =dimV−n. As a well- 1 known fact for line bundles on P =SL(2)/B2, the number in (4.2) is the degree of restriction of the line bundle Lφ to the SL(2)-orbit through t−θ, hence the degree Λ1 of Lφ on the generating cycle of H (Gr0 )(hereΛ can be any reference lattice, Λ1 2 G 1 because diﬀerent choices of Λ1 give rise to isomorphic line bundles).

4.3. Remark. ThelemmashowsthatsomepowerofLdet admits a natural G(O)- equivariant structure (the G(O)-equivariant structure is necessarily unique if G is almost simple). However, Ldet itself may not have a G(O)-equivariant structure. G 0 ∼ Z Assume that under the isomorphism Pic( rG) = , the image of the natural for- T G 0 → G 0 Z ⊂ Z getful map Pic ( rG) Pic( rG)isn . Then the number n coincides with the number nG which will be introduced in Remark 5.8.

5. The regular element eT We continue using the notation introduced in the previous section. Since Lφ ∼ Λ0 = ⊗ L dV det admits a canonical G-equivariant structure (hence a canonical T -equivariant structure), we can take its equivariant Chern class cT (Lφ ) ∈ H2 (Gr ). 1 Λ0 T G 5.1. Lemma. The element cT (Lφ ) is primitive in the Hopf algebra H∗ (Gr ). 1 Λ0 T G

Proof. Let mK :ΩK × ΩK → ΩK be the multiplication. Then the T ∩ K- equivariant line bundle m∗ (Lφ ) restricts to Lφ on both ΩK × {∗} and {∗} × ΩK K Λ0 Λ0 ({∗} is the unit element in ΩK). This forces cT (Lφ ) to be primitive. Therefore, 1 Λ0 T c1 is also primitive.

5.2. Corollary. The element 1 cT = cT (Lφ ) ∈ H2 (Gr , Q) 1 1 Λ0 T G dV is independent of the choice of the representation φ : G → GL(V ).

Proof. For diﬀerent choices of (φ, V ), the classes 1 cT (Lφ ) have the same image dV 1 Λ0 2 c1(Ldet)inH (GrG). However, there can be at most one primitive element in 2 G Q 2 G Q HT ( rG, ) with a given image in H ( rG, ).

T F∈P T 5.3. The element e . For each object Q, cup product with c1 induces a functorial map ∪ T ∗ F → ∗+2 F c1 : HT ( ) HT ( ).

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T F F ∈P By Proposition 2.7, the fact that c1 is primitive implies that for any 1, 2 Q, there is a functorial commutative diagram ∗ F ⊗ ∗ F ∼ / ∗ F ∗F HT ( 1) RT HT ( 2) HT ( 1 2)

cT ⊗id + id ⊗cT cT 1 1 1 ∗ F ⊗ ∗ F ∼ / ∗ F ∗F HT ( 1) RT HT ( 2)[2] HT ( 1 2)[2]. ∨ In general, let c is an endomorphism of the functor ωR :Rep(G ) → Mod(R) (the composition of the forgetful functor with the tensor product ⊗R) satisfying ⊗ ⊗ ∈ ∨ ⊗ cV1⊗V2 = cV1 idV2 +idV1 cV2 ,thenc determines an element e g R.In ∨ 2 fact, Consider the ﬁber functor ωR, :Rep(G ) → Mod(R[]/ ) sending V → V ⊗ R[]/2 = V ⊗ R ⊕ V ⊗ R. The maps

id 0 (5.1) V : V ⊗ R ⊕ V ⊗ R → V ⊗ R ⊕ V ⊗ R cV idV ∨ 2 give a tensor automorphism of ωR, , hence an element e ∈ G (R[]/ ). Since the 2 induced tensor automorphism of ωR (via R[]/ → R) is the identity, e lies in the kernel of the reduction map G∨(R[]/2) → G∨(R), which is g∨ ⊗ R. ∗ P→ ⊗ Q Applying the above discussion to the functor HT : Mod(RT ) (which ∨ → ⊗ Q can be identiﬁed with the functor ωRT ⊗Q :Rep(G ) Mod(RT ) by Lemma T T T ∈ ∨ ⊗ ⊗ Q 2.2) and c = c1 , we conclude that c1 gives an element e g RT . 5.4. Remark. More generally, suppose ω :Rep(G∨) → Mod(R) is a ﬁber functor

and c is an endomorphism of ω satisfying c = c ⊗ id +id ⊗c . V1⊗V2 V1 ω (V2) ω (V1) V2 Then ω determines a G∨-torsor E over Spec R.Letδ :SpecR[]/2 → Spec R be the structure map and let δ∗E be the pull-back G∨-torsor to Spec R[]/2. Then the same construction as in (5.1) determines an automorphism of δ∗E whose restriction to the closed subscheme Spec R → Spec R[]/2 is the identity. Spelling these out, we see that c determines a Higgs ﬁeld of the G∨-torsor E, i.e., a global section G∨ e ∈ Γ(Spec R, Ad(E)) where Ad(E)=E × g∨ is the adjoint bundle. In other words, the pair (ω ,c ) determines an R-point (E,e )ofthestack[g∨/G∨]. ∗ − Applying the above discussion to the functor HG( ) (which determines the torsor E G L L G by Remark 2.5) and c1 ( ) for any G-equivariant line bundle on rG,wegeta commutative diagram:

cT (L) 1 /∨ Spec RT g

∗ (H (−),cG(L)) G 1 / ∨ ∨ Spec RG [g /G ].

T In the remaining part of this section, we determine the element e explicitly. ∨ ∨ G We give an (even) grading on g = n∈Z g2n according to the action of m via the conjugation of 2ρ∨ : G → T ∨: m ∨ ∨ g2n = gα. α∈Φ, 2ρ∨,α =2n ∨ ⊗ ∪ T We give g RT the tensor product (even) grading. The fact that c1 increases the T ∨ cohomological degree by 2 implies that e is homogeneous of degree 2 in g ⊗ RT ,

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i.e., T ∈ ∨ ⊗ X• ⊗ Q e g2(1−n) Symn( (T )) . n∈Z 5.5. Lemma. T ∈ ∨ ⊗ Q ⊕ ∨ ⊗ X• ⊗ Q (5.2) e g2 t (T ) . ∪ T ∗ F Proof. The action of c1 preserves the T -equivariant MV-ﬁltration on HT ( ) ∨ ∨ E| which gives the canonical B -reduction of the trivial G -torsor Spec RT (see Re- T ∨ T mark 2.5). Hence, e is an RT ⊗ Q-valued element of the Lie algebra b .Sincee ∨ is homogeneous of degree 2 in b ⊗ RT ⊗ Q, it must take the form (5.2). T ∈ ∨ ⊗ Q ∈ ∨ ⊗ X• ⊗ Q Write e = e + h where e g2 and h t (T ) . We ﬁrst calculate e. Clearly, e is obtained from the endomorphism of the functor ∗ H : P→Mod(Z)givenby∪c1(Ldet), the ordinary Chern class of Ldet.Inpartic- ∨ ∨ ular, e ∈ g . For each simple root αi of g ,letxi be the element of a Chevalley ∨ ∨ basis of g in the root space gαi .Thene is an integral combination of the xi’s.

5.6. Proposition. Up to changing the Chevalley basis elements {xi} by signs, we have 2 (5.3) e = |αi| xi i 2 where || is a W -invariant quadratic form on X•(T ) normalized so that short 2 coroots (of G) have length one (hence |αi| =1, 2 or 3). λ p λ ∨ λ Proof. Let I∗ betheperversesheaf j∗ Z[ 2ρ ,λ ]wherej : Grλ →GrG is the i− 2ρ∨,λ λ i inclusion. We ﬁrst claim that the restriction map H (I∗ ) → H (Grλ)isan λ λ ∨ isomorphism for i ≤ 2. In fact, the cone F of I∗ → j∗ Z[ 2ρ ,λ ] has perverse degree ∨ ≥ 1, and is supported on Gr<λ = Gr≤λ −Grλ, which has dimension ≤ 2ρ ,λ −2. i ∨ Therefore, H (F)vanishesfori<− dim Gr<λ +1≤− 2ρ ,λ + 3. This implies our assertion. ∨ Now we prove the proposition. Without changing e and g , we can assume that G is of adjoint type. Write e = i nixi for some integers ni.Foreachsimple coroot αi of G,letλi be the corresponding fundamental coweight. By [MV07, ∨ ∗ −w0(λi) Proposition 13.1], the G -module H (I∗ ) is the Schur module with highest weight −w0(λi), where w0 stands for the element of longest length in the Weyl group of G. Therefore, the Lie algebra element xi sends the lowest weight vector vlow (with weight −λi) to a generator of the rank 1 weight space of weight −λi +αi. For j = i, clearly xj · vlow = 0. Therefore, e · vlow is ni-times a generator of the rank 1 weight space of weight −λi + αi. Translating to geometry, this means that the action of c1(Ldet) sends the fun- − ∨ − −w (λ ) G ∈ 2ρ , w0(λi) I 0 i damental class [ r−λi ] H ( ∗ )toni-times a generator of ∨ − 2− 2ρ ,−w (λ ) w0(λi) H 0 i (I∗ )(whichisafreeZ-module of rank one). By the dis- ∨ − i− 2ρ ,−w (λ ) w0(λi) cussion at the beginning of the proof, we can identify H 0 i (I∗ ) i G L ∈ 0 G with H ( r−λi ). Hence c1( det) sends 1 H ( r−λi )toni-times a generator 2 G of H ( r−λi ) (which is unique up to sign). To calculate this number ni,itsuf- L | ∈ 2 G ﬁces to calculate the class c ( ) G − H ( r− ), or its further restriction to 1 det r λi λi − − 2 · λi · λi →G H (G t ) (because the inclusion G t r−λi is a homotopy equivalence). −λi 2 −λi Now the restriction of c1(Ldet)toG·t is ni-times the generator of H (G·t ), which is a free Z-module of rank 1. Let φi :SL(2)→ G be the homomorphism

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−λi ∼ 1 corresponding to the coroot αi of G.Thenφi(SL(2)) · t =SL(2)/B2 = P is the − generating 2-cycle of the partial ﬂag variety G·t λi (by the Bruhat decomposition), −λi and ni equals the degree of the restriction of Ldet to φi(SL(2)) · t . Let Ad : G → GL(g) denote the adjoint representation of G. The degree of LAd Λ0 − ∼ − · λi P1 T LAd | − ∈ 2 { λi } on φi(SL(2)) t = is the same as the pairing of c1 ( Λ ) t λi HT ( t )= X• ∨ 0 (T ) with αi . By (4.1), this degree equals ∨ ∨ − α , −λi α ,αi =(λi,αi)Kil. α∨∈Φ∨

Ad 1 By Lemma 4.2, c1(L )=dAdc1(Ldet)= (θ, θ)Kilc1(Ldet), therefore the degree of Λ0 2 −λi Ldet on φi(SL(2)) · t is

2(λi,αi)Kil (αi,αi)Kil ni = = . (θ, θ)Kil (θ, θ)Kil 2 Since θ is a short coroot, we conclude that ni = |αi| .

Next we determine the element h ∈ t∨ ⊗ X•(T ) ⊗ Q. 5.7. Proposition. Identifying t∨ with X•(T ) and t∨ ⊗ X•(T ) with the dual of X•(T ) ⊗ X•(T ), the element h is the (Q-valued) symmetric bilinear form −2(λ, μ) h(λ, μ)= Kil . (θ, θ)Kil

∨ • Proof. Write h = hi ⊗ gi,wherehi ∈ t ,gi ∈ X (T ). Since the cup product T T with c1 preserves the T -equivariant MV-ﬁltration, there is an induced action e ∗ − on the associated graded pieces HT,c(Sλ, ). The element h is given by the action eT . More precisely, for λ ∈ X•(T ), the action of eT on the λ-weight subspace is given by the multiplication of hi,λ gi. Translating back to geometry, this T ∗ − means that the cup product of c1 on HT,c(Sλ, ) should be the multiplication ∈ X• ∼ 2 T by hi,λ gi (where as before, gi (T ) = HT (pt)). On the other hand, c1 ∗ − { λ} → acts on HT,c(Sλ, ) via its restriction to Sλ.Since t Sλ is a T -equivariant T ∗ − homotopy equivalence, the action of c1 on HT,c(Sλ, ) is given by the multiplication ∗ T ∈ 2 { λ} Q X ⊗ Q of iλ(c1 ) HT ( t , )= •(T ) . Applying (4.1) to the line bundle LAd associated with the adjoint representation Λ0 T 2 { λ} Ad of G, and using Lemma 4.2, we see that the restriction of c1 to HT ( t )is given by: − ∗ T 2 ∨ ∨ iλ(c1 )= α ,λ α . (θ, θ)Kil α∨∈Φ∨ Therefore, −2 h = α∨ ⊗ α∨. (θ, θ)Kil α∨∈Φ∨

If we view h as a bilinear form on X•(T ), then for any λ, μ ∈ X•(T ),

∨ ∨ −2 ∨∈ ∨ α ,λ α ,μ −2(λ, μ) h(λ, μ)= α Φ = Kil . (θ, θ)Kil (θ, θ)Kil

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∨ • 5.8. Remark. Let nG be the least positive integer such that nGh ∈ t ⊗ X (T )(or T ∨ equivalently, such that nGe ∈ g ⊗ RT ). Clearly, we have nG = nGder .Onthe other hand, for G is almost simple, this number can be found in [So00, Table C], in which nG is denoted by b. We reproduce these numbers explicitly in the following list: #π1(G) • nG = if G is of type A; gcd(#Z(G), #π1(G)) • ± nG =2ifG is PSp2m (for m odd), SO4m (for m odd), PSO4m,orPE7; • nG =3ifG is PE6; • nG =4ifG is PSO4m+2; • nG =1otherwise. ± Here P stands for the adjoint form, and SO4m are the two quotients of Spin4m by Z/2 which are not isomorphic to SO4m.

6. Proof of the main result T T ∨ Recall the morphism σ :SpecH∗ (GrG) → B ×Spec RT defined in Proposition ∨ 3.3(1) and its non-equivariant counterpart σ :SpecH∗(GrG) → B . It is clear that ∗ − ⊗ T G the action of σcan on the fiber functor HT ( ) RT H∗ ( rG) commutes with the cup T T product action of c1 . Therefore, σ further factors through the homomorphism T T G → ∨ σ :SpecH∗ ( rG)[1/nG] BeT , ∨ of group schemes over Spec RT [1/nG]. Here BeT is the centralizer group scheme of T ∨ ∨ ∨ e ∈ g (RT [1/nG]) in B × Spec RT [1/nG]. Since e is already an element in g , we also have the homomorphism G → ∨ × ∨ → ∨ σ :SpecH∗( rG) Z Ue Be of group schemes over Z (the first arrow above follows from Proposition 3.3(4)). 6.1. Theorem. Assume that Gder is almost simple. G → ∨ (1) The homomorphism σ :SpecH∗( rG) Be is a closed embedding, and is an isomorphism over Spec Z[1/G],whereG =1, 2 or 3 is the square of the ratio of lengths of the long and short coroots of G.Inparticular, G → ∨ × ∨ Z Spec H∗( rG) Z Ue is also an isomorphism over Spec [1/G]. T T G → ∨ (2) The homomorphism σ :SpecH∗ ( rG)[1/nG] BeT is a closed embed- 1 ding, and is an isomorphism over Spec RT [ ]. GnG Proof. Step I. We first prove that σT and σ are closed embeddings, i.e., τ T : O ∨ → T G O ∨ → G (BeT ) H∗ ( rG)[1/nG]andτ : (Be ) H∗( rG) are surjective. It suffices to prove the surjectivity of τ because any homogeneous lifting of a Z[1/nG]-basis T T of H∗(GrG)[1/nG]toH∗ (GrG)[1/nG] generates H∗ (GrG)[1/nG]asanRT [1/nG]- T O ∨ → module, and we can choose these liftings to be in the image of τ since (BeT ) O ∨ (Be ) is surjective. Fix a dominant coweight λ of G. Consider the following diagram:

β ∨ G / ∨ Iλ /O (6.1) H∗( rλ)O H∗− 2ρ ,λ ( ∗ ) (Be ) OOO OOO OOO τ OO' / H∗(Gr≤λ) H∗(GrG)

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λ ∗ λ where H∗(I∗ )istheZ-dual of H (I∗ ). In the following, we describe the various arrows in this diagram and verify its commutativity. First we concentrate on the triangle in (6.1). We have the natural morphisms between complexes on Gr≤λ: λ ∨ λ Z →I∗ [− 2ρ ,λ ] → j∗ Z. Taking duals of their cohomology, we get the triangle in (6.1), which is therefore commutative. Next we examine the square in (6.1). The map β is defined to be the matrix ∨ ∗ λ coefficient of the G -module H (I∗ ) with respect to the lowest weight vector − 2ρ∨,λ λ vlow =[Gr≤λ] ∈ H (I∗ ). λ More precisely, for u ∈ H∗(I∗ ), · ∀ ∈ ∨ β(u)(g)= u, g vlow , g Be . λ We check that commutativity of the square in (6.1). For u ∈ H∗(I∗ )andthe ∨ H∗(Gr )-valued point σ ∈ B (H∗(Gr )), we have G can e G i (6.2) τβ(u)=β(u)(σcan)= u, σcanvlow = u, h · vlow hi ∈ H∗(GrG) i i ∗ where {h }, {hi} are dual bases of H (GrG)andH∗(GrG). On the other hand, ∗ G → ∗− 2ρ∨,λ Iλ → · the map H ( r≤λ) H ( ∗ ) sends h h vlow, hence the adjoint map ∨ Iλ → G → i · H∗− 2ρ ,λ ( ∗ ) H∗( r≤λ) sends u i u, h vlow hi. Combining with (6.2), we have verified that the square in (6.1) is commutative. O ∨ → G From diagram (6.1), we conclude that the image of (Be ) H∗( rG) contains λ the image of H∗(G · t )=H∗(Grλ) → H∗(GrG) for any λ ∈ X•(T ). When G is of adjoint type, by [B58, Theorem 1], there exists a single λ (called the “gener- λ ating circle” by Bott), such that the image of H∗(G · t ) → H∗(GrG) generates G Z O ∨ → G H∗( rG)asa -algebra. This implies that (Be ) H∗( rG)issurjectivein the case G is adjoint. By Proposition 3.3(4), this means that Spec H∗(GrG)= ∨ ∨ Spec H0(GrG) ⊗ H∗(GrGsc ) → Z × U is a closed embedding when G is ad- ∼ ∨ ∨ joint. Since Spec H0(GrG) = Z , we conclude that H∗(GrGsc ) → U is also a closed embedding. For general G, using Proposition 3.3(4) again, we conclude that ∨ ∨ Spec H∗(GrG) → Z × U is always a closed embedding. 1 Step II. We prove that B∨ [1/ ]isflatoverR := R [ ]. Consider the eT G T T n G G morphism over RT : − T − T ∨ × → ∨ × φ =Ad( )e e : B Spec RT u Spec RT . ∨ −1 ∨ ∨ × Then BeT [1/G]=φ (0), i.e., BeT [1/G] is the closed subscheme of the B ∨ Spec RT (which is smooth over RT )cutoutbydimu equations. Hence the fiber ∨ → ∨ − ∨ ∨ dimensions of BeT Spec RT are at least dim B dim u = r, the rank of G . ∨ → If we can show that all the fibers of BeT [1/G] Spec RT are r-dimensional, then ∨ BeT [1/G] will be locally a complete intersection over RT . Using that all the fibers ∨ are of the same dimension again and that RT is regular, we can conclude that BeT

is flat over RT (see [M80, 20.D] with obvious modifications). Therefore, it suffices ∈ ∨ to show that for any closed point s Spec RT , the fiber (Be )s has dimension r. Since RT is graded, Spec RT admits a natural Gm-action with fixed point locus z :SpecZ → Spec RT defined by the augmentation ideal of RT .Moreover,this fixed point locus is attracting: for each point s ∈ Spec RT ,letη ∈ Spec RT be the

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generic point of the Gm-orbit of s, then the closure of η intersects the fixed point locus z(Spec Z). In other words, η specializes to a point s0 ∈ z(Spec Z)ofthesame ∨ ∨ ≤ ∨ residue characteristic as s. Hence dim(Be )s = dim(Be )η dim(Be )s0 . Therefore, ∨ → 0 in order to show that all fibers of BeT [1/G] Spec RT have dimension r, it suffices ∨ → Z to show that all (geometric) fibers of Be [1/G] Spec [1/G]havedimensionr. Let k be an algebraic closure of Fp for some prime p not dividing G.Wewould ∨ ⊗ like to show that Be Z k has dimension r over k. Webasechangethesituationfrom Z ∨ to k without changing notation. So in the rest of this paragraph Be is over k,

etc. By Proposition 5.6, the element e has the form i nixi where each ni =0ink. ∈ ∨ ∨ → × r We can choose t T (k) such that Ad(t)e = i xi (because αi : T (k) (k ) is surjective). Therefore, we only need to treat the element e1 = i xi. Itiswell ∨ ∨ × ∨ ∨ − ∨ known that Be = Z Ue ; therefore, it suﬃces to show that dim Ue = r dim Z . 1 1 1 ∨ This equality follows from the main result of [K87]. This proves that BeT is ﬂat

over RT . ∨ Z The above argument also shows that Be [1/G]isﬂatover [1/G]. Step III. Now we can ﬁnish the proof. (2) The equivariant version. In view of Step I, it remains to prove that

O ∨ → T G 1 (BeT )[1/G] H∗ ( rG)[ ] GnG 1 O ∨ T G is injective. Since both (BeT )[1/G]andH∗ ( rG)[ ]areﬂatoverRT ,it GnG ∨ T suﬃces to show that O(B )[1/ ] ⊗ Q → H∗ (Gr ) ⊗ Q is injective where e G RT G RT Q =Frac(RT ). Recall from Remark 3.4 that we have a commutative diagram

τ T Q / ∨ Spec(H∗ (GrG) ⊗R Q) B SST eQ SSSS Spec(SSS Loc∗) SSSS SS) T ∨ × Spec Q

∨ T where eQ ∈ b (Q) is the value of e over the generic point of Spec RT .Bythe equivariant localization theorem, the localization map Loc∗ in (3.10) is an isomorphism. On the other hand, by Proposition 5.7, eQ is a regular semisimple element ∨ ∨ → ∨ × in b (Q), hence the projection BeQ T Spec Q is also an isomorphism. This shows that the closed embedding τQ is in fact an isomorphism, and (2) is proved. (1) The non-equivariant version. As a consequence of the equivariant ver- G → ∨ Q T sion, σ :SpecH∗( rG) Be is an isomorphism over (webasechangeσ to Q → Z −→z ⊗ Q O ∨ ⊗ Q → G Q Spec Spec Spec RT ). Therefore, τ : (Be ) H∗( rG, )isan O ∨ Z isomorphism. Since (Be )[1/G]isﬂatover [1/G] by Step II, hence torsion-free, O ∨ → G we conclude that τ[1/G]: (Be )[1/G] H∗( rG)[1/G] is injective. Combining this with Step I, (1) is proved.

6.2. Remark. (1) Inverting G in the statement of the above theorem is necessary because e ∨ F is not regular modulo G, hence the ﬁber of Be over G does not have the G F same dimension as Spec H∗( rG, G ). However, the proof of the above theorem implies that Spec H∗(GrG) is always isomorphic to the Zariski ∨ closure in Be of its generic ﬁber.

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(2) The theorem shows that no primes other than G need to be inverted, even for those “bad primes” of G (i.e., those dividing the coefficients of the highest root in terms of a linear combination of simple roots). For example, let G be the exceptional group G2. Then Bott (cf. [B58]) shows that ∼ 2 H∗(ΩK) = Z[u, v, w]/(2v − u ). On the other hand, one can show (using formulae in [K87]) that O ∨ ∼ Z − 2 − 3 2 − 4 (Be ) = [A, C, D, E, F ]/(2C 3A , 3(D A ), 3(E + C 3A )). F ∼ O ∨ ⊗ F We find that H∗(ΩK, 2) = (Be ) 2 although 2 is considered as a bad prime for group G2. 6.3. Description of the cohomology. Dually, we have a description of the cohomology of GrG. Recall that if H is an affine group scheme over some commutative ring k, the algebra of distributions on H (cf. [J03, §I.7])isdefinedas Dist(H)={f : O(H) → k|f is k-linear,f(mn+1)=0forsomen}. Here m is the augmented ideal of O(H). It has a natural algebra structure dual to the coalgebra structure on O(H). Dist(H) can be also regarded as the algebra of left invariant differential operators on H. If, in addition, H is infinitesimally flat (i.e. O(H)/mn+1 is flat over k for any n), then Dist(H) has a natural structure of a Hopf algebra over k. ∨ ∨ ⊗ F Now let H = Be . It is proved by Springer (cf. [Sp66]) that Be p is smooth if the prime p is good (i.e. not bad) for G∨. This fact, together with the flatness of ∨ Z ∨ B over [1/G] as shown in the course of proving Theorem 6.1, implies that Be is smooth over Z ,whereZ is obtained from Z by inverting the bad primes of G∨. ∨ Z In particular, Be is infinitesimally flat over . 6.4. Corollary. Assume that G is almost simple and simply-connected. Then there is an isomorphism of algebras over Z, ∗ ∗ G −∼→ ∨ (6.3) τ : H ( rG) Dist(Ue ), which is an isomorphism of Hopf algebras over Z . ∨ ∨ ∨ Proof. Observe that Z is trivial in this case, hence Ue = Be . In the course of ∨ proving Theorem 6.1 we have shown that τ : O(U ) → H∗(GrG) is surjective, and ∨ is an isomorphism up to torsion. Observe that Dist(Ue ) can be identified with the O ∨ O ∨ graded dual of (Ue ) because elements in the augmentation ideal of (Ue )all have positive degrees. Taking the graded dual of τ we get the desired isomorphism τ ∗ in (6.3). 6.5. Description of the G-equivariant homology. In this subsection, we assume that G is almost simple and simply-connected. We now interpret the G- G equivariant homology H∗ (GrG) in terms of the regular centralizer group scheme of G∨. Let us recall that a prime p is called a torsion prime of G if H∗(G, Z)has p-torsion. Let S1 be the multiplicative set generated by the torsion primes of G, Z and let S1 be the corresponding localization, i.e., the localization away from the torsion primes. Borel (see [B53]) proved that ⊗ Z −∼→ ⊗ Z W (6.4) RG S1 (RT S1 ) . T Since NG(T )actsonGrG, it induces an action of W = NG(T )/T on H∗ (GrG)com- G T patible with the natural W -action on RT . The natural map H∗ (GrG) → H∗ (GrG)

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T then factors through the W -invariants of H∗ (GrG). Using the isomorphism (6.4), it is easy to show that G G Z −∼→ T G Z W H∗ ( rG, S1 ) H∗ ( rG, S1 ) ⊗ Z as Hopf algebras over RG S1 . On the other hand, let S2 be the multiplicative set generated by bad primes of ∨ G and those dividing n +1 if G is of type An. Clearly, G,nG ∈ S2,andthe ⊂ Z Z result in [B60] implies S1 S2.Let S2 be the localization of with respect to ∨ ∨ ∨ S2. Consider the universal centralizer scheme I over g (i.e., the fiber of I over ∈ ∨ ∨ x g is the centralizer Gx ). After inverting primes in S2, we have the following identifications: ∨ × Z −∼→ ∨ × Z −∼→ ⊗ Z (6.5) (g //G) Spec S2 (t //W ) Spec S2 (Spec RG S2 ) where the latter isomorphism uses the element f in Proposition 5.7. It can be ∨ ∨ reg × Z shown that the restriction of I to the regular locus (g ) Spec S2 descends ∨ ⊗ Z ∼ ∨ × Z to a smooth group scheme J over Spec(RG S2 ) = (g //G) Spec S2 .Infact, this is essentially proved in [N06] with all primes dividing #W inverted; but further argument shows that only primes in S2 need to be inverted. 6.6. Proposition. There is a natural isomorphism of group schemes over ⊗ Z Spec(RG S2 ): G G Z −∼→ ∨ Spec H∗ ( rG, S2 ) J . ⊗ Z → ⊗ Z Sketch of proof. Let π : Spec(RT S2 ) Spec(RG S2 ), then by the definition ∨ → ∨ ∗ ∨ of the regular centralizer, we have a closed embedding β : BeT GeT = π J ∨ ⊗ Z (from now on we view BeT as a scheme over Spec(RT S2 )). Using the flat- ∨ ∗ ∨ ⊗ Z ness of BeT and π J over RG S2 ,itiseasytoseethatβ is an isomorphism. Therefore, the coordinate ring O(J ∨) can be identified with the W -invariants of O ∨ −∼→O ∨ G G Z (BeT ) (GeT ). Recall from (6.5) that H∗ ( rG, S2 ) is also identified with T G Z the W -invariants of H∗ ( rG, S2 ). It remains to show that the W -actions on T G Z O ∨ T H∗ ( rG, S2 )and (BeT ) correspond to each other under the isomorphism σ in Theorem 6.1(2). For this, it is enough to argue over Frac(RT ) and use equivariant localization (Remark 3.4) again. The details are left to the reader. Acknowledgment The research of Z.Y. is supported by the National Science Foundation under the agreement No. DMS-0635607. Any opinions, findings and conclusions or recom- mendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The research of X.Z. was partially conducted during the period he was employed by the Clay Mathematics Institute as a Liftoff Fellow. References

[B53] Borel, A. Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. Ann. of Math. (2) 57, (1953). 115–207. MR0051508 (14:490e) [B60] Borel, A. Commutative subgroups and torsion in compact Lie groups. Bull. amer. Math. Soc. 66 1960 285–288. MR0117299 (22:8080) [B58] Bott, R. The space of loops on a Lie group. Michigan Math. J. 5(1958), 35–61. MR0102803 (21:1589) [G95] Ginzburg, V. Perverse sheaves on a Loop group and Langlands’ duality. arXiv:math/9511007.

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use HOMOLOGY OF LOOP GROUPS 369

[J03] Jantzen, J. C. Representations of algebraic groups. Second edition. Mathematical Sur- veys and Monographs, 107. American Mathematical Society, Providence, RI, 2003. MR2015057 (2004h:20061) [K87] Keny, S. V. Existence of regular nilpotent elements in the Lie algebra of a simple algebraic group in bad characteristics. J. Algebra 108 (1987), no. 1, 195–201. MR887203 (88d:17011) [KK86] Kostant, B; Kumar, S. The nil Hecke ring and the cohomology of G/P for a Kac-Moody group G. Adv. in Math. 62 (1986), 187–237. MR866159 (88b:17025b) [KNR94] Kumar, S; Narasimhan, M.S; Ramanathan, A. Infinite Grassmannians and moduli spaces of G-bundles. Math. Annalen 300 (1994), 41–75. MR1289830 (96e:14011) [L81] Lusztig, G. Singularities, character formulas, and a q-analog of weight multiplicities. Analysis and topology on singular spaces, II, III (Luminy, 1981), 208–229, Astérisque, 101-102, Soc. Math. France, Paris, 1983. MR737932 (85m:17005) [M80] Matsumura, H. Commutative algebra. Second edition. Mathematics Lecture Note Se- ries, 56. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR575344 (82i:13003) [MV07] Mirković, I.; Vilonen, K. Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. of Math. (2) 166 (2007), no. 1, 95–143. MR2342692 (2008m:22027) [N06] Ngô, B-C. Fibration de Hitchin et endoscopie. Invent. Math. 164 (2006), no. 2, 399–453. MR2218781 (2007k:14018) [PS86] Pressley, A; Segal, G. Loop groups. The Clarendon Press Oxford University Press, New York, 1986. Oxford Science Publications. MR900587 (88i:22049) [Sa72] Saavedra Rivano, N. Catégories Tannakiennes. Lecture Notes in Mathematics, Vol. 265. Springer-Verlag, Berlin-New York, 1972. MR0338002 (49:2769) [So00] Sorger, C. Lectures on moduli of principal G-bundles over algebraic curves. School on Algebraic Geometry (Trieste, 1999), 1–57, ICTP Lect. Notes, 1, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2000. MR1795860 (2002h:14017) [Sp66] Springer, T. A. Some arithmetical results on semi-simple Lie algebras. Inst. Hautes Etudes´ Sci. Publ. Math. No. 30 (1966), 115–141. MR0206171 (34:5993)

Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540 Current address: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139 E-mail address: [email protected] Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138 E-mail address: [email protected]

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