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

Toshihiko Matsuki

Abstract: 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 our paper ``Double coset decompositions of reductive L ie groups arising from two involutions, Journal of 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.

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