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The Bruhat order on symmetric varieties. (English) Zbl 0704.20039
Let G be a connected reductive linear algebraic group over an algebraically closed field of characteristic $$\neq 2$$. Let $$\theta$$ be an automorphism of G of order 2. Let K be the fixed point subgroup of $$\theta$$. Then $$X(=G/K)$$ is called the symmetric variety defined by (G,$$\theta$$). Let B be a $$\theta$$-stable Borel subgroup of G. It is known that the number of double cosets BgK is finite. We have a natural partial order on the set of B-orbits in X (for the action of B on X by left translations), namely, given two B-orbits O, $$O'$$, we define $$O'\leq O$$, if $$O'$$ is contained in the Zariski closure of O. (The authors refer to this partial order as the Bruhat order on the symmetric variety X.) In this paper, the authors give a combinatorial description of the Bruhat order on the symmetric variety X.
Reviewer: V.Lakshmibai

MSC:
 20G15 Linear algebraic groups over arbitrary fields 20G05 Representation theory for linear algebraic groups 14L30 Group actions on varieties or schemes (quotients)
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