$T$-equivariant $K$-theory of generalized flag varieties

Abstract

Let G be a Kac-Moody group with Borel subgroup B and compact maximal torus T. Analogous to Kostant and Kumar [Kostant, B. & Kumar, S. (1986) Proc. Nati. Acad. Sci. USA 83, 1543-1545], we define a certain ring Y, purely in terms of the Weyl group W (associated to G) and its action on T. By dualizing Y we get another ring 1, which, we prove, is canonically isomorphic with the T-equivariant Ktheory KT(G/B) of GIB. Now KT(G/B), apart from being an algebra over KT(pt.) A(T), also has a Weyl group action and, moreover, KT(G/B) admits certain operators {DW},,w similar to the Demazure operators defined on A(T). We prove that these structures on KT(G/B) come naturally from the ring Y. By evaluating the A(T)-module W at 1, we recover K(G/B) together with the above-mentioned structures. We believe that many of the results of this paper are new in the finite case (i.e., G is a finite-dimensional semisimple group over C)