2 5 N ov 2 00 3 EQUIVARIANT ( K - ) HOMOLOGY OF AFFINE GRASSMANNIAN AND TODA LATTICE

1.1. Let G be an almost simple complex algebraic group, and let GrG be its affine Grassmannian. Recall that if we set O = C[[t]], F = C((t)), then GrG = G(F)/G(O). It is well-known that the subgroup ΩK of polynomial loops into a maximal compact subgroup K ⊂ G projects isomorphically to GrG; thus GrG acquires the structure of a topological group. An algebro-geometric counterpart of this structure is provided by the convolution diagram G(F)×G(O) GrG → GrG. It allows one to define the convolution of two G(O) equivariant geometric objects (such as sheaves, or constrictible functions) on GrG. A famous example of such a structure is the category of G(O) equivariant perverse sheaves on Gr (“Satake category” in the terminology of Beilinson and Drinfeld); this is a semi-simple abelian category, and convolution provides it with a symmetric monoidal structure. By results of [10], [19], [2] this category is identified with the category of (algebraic) representations of the Langlands dual group. The starting point for the present work was the observation that a similar definition works in another setting, yielding a monoidal structure on the category of G(O) equivariant perverse coherent sheaves on Gr (the “coherent Satake category”). The latter is a non-semisimple artinian abelian category, the heart of the middle perversity t-structure on the derived category of G(O) equivariant coherent sheaves on GrG; existence of this t-structure is due to the fact that dimensions of all G(O)-orbits inside a given component of GrG are of the same parity, cf. [3]. The resulting monoidal category turns out to be non-symmetric, though its Grothendieck ring K(GrG) is commutative. One of the results of this paper is a computation of this ring. Along with K(GrG) we compute its “graded version”, the ring H G(O)(Gr) of equivariant homology of Gr, where the algebra structure is again provided by convolution. (The ring H G(O) • (GrG) was essentially computed by Dale Peterson [20], cf. also [15].) To describe the answer, let Ǧ be the Langlands dual group to G, and let ǧ be its Lie algebra. Consider the universal centralizers Zǧ and Z Ǧ Ǧ : if we denote by CǦ,ǧ ⊂ Ǧ× ǧ (resp. CǦ,Ǧ ⊂ Ǧ× Ǧ) the locally closed subvariety formed by all the pairs (g, x) such that Adg(x) = x and x is regular (resp. all the pairs (g1, g2) such that Adg1g2 = g2 and