Some results on algebraic groups with involutions. (English) Zbl 0628.20036

Springer, Tony A.

Algebraic groups and related topics, Proc. Symp., Kyoto and Nagoya/Jap. 1983, Adv. Stud. Pure Math. 6, 525-543 (1985).

[For the entire collection see Zbl 0561.00006.]
Let G be a reductive linear algebraic group over an algebraically closed field of characteristic not 2, and let \(\theta\) be an involution of G. Denote by K the fixed point group of \(\theta\) and by B a Borel subgroup. Then K has finitely many orbits in the flag manifold G/B. The author establishes some basic facts about these orbits. He gives a rather explicit description of these orbits as algebraic varieties, studies K- fixed vectors in G-modules, and describes orbit closures.

MSC:

20G05	Representation theory for linear algebraic groups
14M15	Grassmannians, Schubert varieties, flag manifolds
20G15	Linear algebraic groups over arbitrary fields

Keywords:

reductive linear algebraic group; involution; fixed point group; Borel subgroup; orbits; flag manifold

Citations:

Zbl 0561.00006