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 $W^{\ast }(G_k)\cong W^{\ast }(G_k/T_k)$ of Balmer–Witt groups. When $k$ is algebraically closed, we prove that $W^{\ast }(G_k)$ is isomorphic to the topological $K$-theory $K{O}^{2\ast -1}(G/T)$ of the flag manifold $G/T$. Then we compute it explicitly by using the fact that $W^{\ast }(G_k)$ is a Hopf algebra.