The ranks of homotopy groups of Kac-Moody groups

Abstract

Let $A$ be a Cartan matrix and $G(A)$ be the Kac-Moody group associated to Cartan matrix $A$. In this paper, we discuss the computation of the rank $i_k$ of homotopy group $\pi_k(G(A))$. For a large class of Kac-Moody groups, we construct Lie algebras with grade from the Poincar\'{e} series of their flag manifolds. And we interpret $i_{2k}$ as the dimension of the degree $2k$ homogeneous component of the Lie algebra we constructed. Since the computation of $i_{2k-1}$ is trivial, this gives a combinatorics interpretation of $i_k$ for all $k>0$.