. R T / 0 30 92 07 v 2 3 J un 2 00 4 AFFINE WEYL GROUPS IN K - THEORY AND REPRESENTATION THEORY

We give an explicit combinatorial Chevalley-type formula for the equivariant K-theory of generalized flag varieties G/P. The formula implies a simple combinatorial model for the characters of the irreducible representations of G and, more generally, for the Demazure characters. The construction is given in terms of a certain R-matrix, that is, a collection of operators satisfying the Yang-Baxter equation. It reduces to combinatorics of decompositions in the affine Weyl group and enumeration of saturated chains in the Bruhat order on the (nonaffine) Weyl group. The formula implies several symmetries of coefficients in the equivariant K-theory. We derive a Pieri-type formula and a dual Chevalley-type formula for this ring. The paper contains some other applications and examples. Finally, we conjecture a Pieri-type formula for the quantum K-theory of G/B. The proofs are completely combinatorial.