Intersections of double cosets in algebraic groups
In this note we prove several results concerning the intersection of double cosets in algebraic groups. Let \(G\) be a connected (affine) algebraic group and let \(H\), \(K\) and \(L\) be closed connected subgroups of \(G\) satisfying the following conditions: (i) \(H \cap K\) is connected and \(\text{Lie}(H) + \text{Lie}(K) = \text{Lie}(G)\); (ii) the sets \(H \setminus G/L\) of \((H,L)\) double cosets and \(K \setminus G/L\) of \((K,L)\) double cosets are finite. We consider double coset intersections \(H x L \cap K x L\). We show that each such double coset intersection is a smooth irreducible variety and that the closure of the intersection is the intersection of the closures. In particular, let \(G\) be a reductive group, let \(B\) be a Borel subgroup of \(G\) and let \(T\) be a maximal torus of \(B\). Let \(P\), \(Q\) and \(R\) be parabolic subgroups of \(G\) containing \(B\) and let \(R^-\) be the parabolic subgroup containing \(T\) which is opposite to \(R\). We consider double coset intersections of the form \(P x Q \cap R^- x Q\). We give a nice parametrization of the set of these double coset intersections and we describe the partial order on the set of these intersections given by inclusion of the closures of the intersections. Although all of our proofs are elementary, the results seem to be new and they are quite useful (see the author, \textit{G. Röhrle} and \textit{R. Steinberg} [Invent. Math. 110, 649-671 (1992; Zbl 0786.20029)]).
scientific article; zbMATH DE number 57233