Algebraic groups with a commuting pair of involutions and semisimple symmetric spaces
Let F be an algebraically closed field of char\(\neq 2\), and let G be a connected reductive algebraic group defined over F. In this paper the author classifies the isomorphism classes (under both inner and outer automorphisms) of (ordered) pairs (\(\theta\),\(\sigma)\) of commuting involutions of G. Given such a pair there exists a torus T of G which is maximally (\(\theta\),\(\sigma)\)-split: the classification goes via Araki type diagrams describing the simple roots of T, and their restrictions to the (\(\theta\),\(\sigma)\)-split part of T. If \(F={\mathbb{C}}\), and if G is semisimple, then given a pair (\(\theta\),\(\sigma)\) there exists a \(\theta\)- and \(\sigma\)-stable compact real form \({\mathfrak u}\) of G's Lie algebra \({\mathfrak g}\), with conjugation \(\tau\). Let \({\mathfrak g}_ 0\) be the real form of \({\mathfrak g}\) determined by the conjugation \(\theta\) \(\tau\). Then (\({\mathfrak g}_ 0,\sigma | {\mathfrak g}_ 0)\) is an infinitesimal semisimple symmetric pair; all such pairs are obtained in this way, and the above classification yields in particular a classification of semisimple symmetric pairs. The subclass of pairs (\(\theta\),\(\theta)\) gives rise to the Riemannian symmetric pairs in this way, and one obtains the old classification of Riemannian symmetric pairs by means of Araki diagrams. The classification of semisimple symmetric pairs for a real semisimple Lie algebra was achieved already by \textit{M. Berger} [Ann. Sci. Éc. Norm. Supér., III. Sér. 74, 85-177 (1957; Zbl 0093.356)], by means of a different method. The present method has the advantage that it clarifies much more of the fine structure of semisimple symmetric pairs.
scientific article; zbMATH DE number 4122249
scientific article