Intersections of double cosets in algebraic groups
type: Item
instance of: scholarly article
title: Intersections of double cosets in algebraic groups
zbMATH Open document ID: 0833.22001
DOI: 10.1016/0019-3577(92)90028-J
published in: Indagationes Mathematicae. New Series
publication date: 1992-09-27T00:00:00Z
review text: 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)]).
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 20G15
zbMATH DE Number: 57233
zbMATH Keywords: double cosets
zbMATH Keywords: algebraic groups
zbMATH Keywords: connected (affine) algebraic group
zbMATH Keywords: reductive group
zbMATH Keywords: Borel subgroup
zbMATH Keywords: maximal torus
zbMATH Keywords: parabolic subgroups
zbMATH Keywords: parametrization
zbMATH Keywords: partial order
author: R. W. jun. Richardson
MaRDI profile type: Publication
cites work: Q5580321
cites work: Q3260091
cites work: On Lusztig's isomorphism theorem for Hecke algebras
cites work: On some geometric aspects of Bruhat orderings. I: A finer decomposition of Bruhat cells
cites work: Q3998721
cites work: Q4128900
full work available at URL: 0019-3577(92)90028-j
OpenAlex ID: W1972025572
description: scientific article; zbMATH DE number 57233