
Journal of Lie Theory 12 (2002), No. 1, 041068 Copyright Heldermann Verlag 2002 Classification of two Involutions on Compact Semisimple Lie Groups and Root Systems Toshihiko Matsuki Faculty of Integrated Human Studies, Kyoto University, Kyoto 6068501, Japan [Abstractpdf] 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) 4991], 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. [ Fulltextpdf (256 KB)] 