ar X iv : m at h / 03 06 41 3 v 1 [ m at h . A G ] 2 9 Ju n 20 03 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). The equivariant cohomology ring H• G(O)(GrG,C) = H • G(O)(GrG) was computed by V. Ginzburg [7] in terms of the Langlands dual group Ǧ. More precisely, let ǧ be the Lie algebra of Ǧ, and let Zǧǧ be the universal centralizer: if we denote by Cǧ,ǧ ⊂ ǧ× ǧ the locally closed subvariety formed by all the pairs (x1, x2) such that [x1, x2] = 0 and x2 is regular, then Z ǧ ǧ is the categorical quotient Cǧ,ǧ//Ǧ with respect to the diagonal adjoint action of Ǧ. The projection to the second (regular) factor Zǧǧ → ǧreg//Ǧ = ť/W (the quotient of the dual Cartan Lie algebra with respect to the Weyl group) makes Zǧǧ a sheaf of abelian Lie algebras. V. Ginzburg identifies H• G(O)(GrG) with the global sections of the relative universal enveloping algebra Uť/W ( Z ǧ ǧ ) . In other words, the spectrum of the commutative ring H• G(O)(GrG) is identified with the tangent space T(̌t/W ). The sheaf of sections of T(̌t/W ) in the flat topology coincides with the sheaf of W -equivariant maps from ť to ť. Finally, it is known that T(̌t/W ) can be described as an affine blow-up of (̌t × ť)/W . More precisely, let Ř be the set of roots of Ǧ; for α̌ ∈ Ř let us denote by 1α̌ (resp. 2α̌) the linear function on ť × ť obtained as a composition of α̌ with the projection on the first (resp. second) factor. Then C[T(̌t/W )] = C[̌t × ť, 1α̌ 2α̌ , α̌ ∈ Ř]W .