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


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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