Classification of two Involutions on Compact Semisimple Lie Groups and Root Systems

T. Matsuki

Journal of Lie theory, Tome 12 (2002) no. 1, pp. 41-68

Résumé

Let ${\frak g}$ be a compact semisimple Lie algebra. Then we first classify pairs of involutions $(\sigma,\tau)$ of ${\frak g}$ with respect to the corresponding double coset decompositions $H\backslash G/L$. (Note that we don't assume $\sigma\tau=\tau\sigma$.) In a previous paper ["Double coset decompositions of reductive Lie groups arising from two involutions", J. Algebra 197 (1997) 49--91], we defined a maximal torus $A$, a (restricted) root system $\Sigma$ and a ``generalized'' Weyl group $J$ and then we proved $$J\backslash A\cong H\backslash G/L$$ when $G$ is connected. In this paper, we also compute $\Sigma$ and $J$ for some representatives of all the pairs of involutions when $G$ is simply connected. By these data, we can compute $\Sigma$ and $J$ for ``all'' the pairs of involutions.