Journal of Lie Theory

no. 1
Classification of two Involutions on Compact Semisimple Lie Groups and Root Systems
Journal of Lie Theory, Volume 12 (2002) no. 1, pp. 41-68
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.
DOI: 10.5802/jolt.252
Classification: 22E46, 17B20
Keywords: semisimple Lie group, involutions, semisimple Lie algebra, double coset decomposition, root system, Weyl group 
@article{JOLT_2002_12_1_a3,
     author = {T. Matsuki},
     title = {Classification of two {Involutions} on {Compact} {Semisimple} {Lie} {Groups} and {Root} {Systems}},
     journal = {Journal of Lie Theory},
     pages = {41--68},
     year = {2002},
     volume = {12},
     number = {1},
     doi = {10.5802/jolt.252},
     zbl = {0998.22004},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.252/}
}
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T. Matsuki. Classification of two Involutions on Compact Semisimple Lie Groups and Root Systems. Journal of Lie Theory, Volume 12 (2002) no. 1, pp. 41-68. doi: 10.5802/jolt.252

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