The Bruhat order on symmetric varieties
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.
connected reductive linear algebraic group
scientific article; zbMATH DE number 4154733