Witt Groups of Algebraic Groups

Abstract

Let G_k be a split reductive group over a subfield k of \mathbb C corresponding to a Lie group G . Let T be a maximal torus of G . We show the isomorphism W^*(G_k)\cong W^*(G_k/T_k) of Balmer–Witt groups. When k is algebraically closed, we prove that W^*(G_k) is isomorphic to the topological K -theory KO^{2*-1}(G/T) of the flag manifold G/T . Then we compute it explicitly by using the fact that W^*(G_k) is a Hopf algebra.