Schubert cells and representation theory
type: Item
instance of: scholarly article
title: Schubert cells and representation theory
zbMATH Open document ID: 0954.22011
author: Robert J. Stanton
author: Luis Casian
published in: Inventiones Mathematicae
publication date: 2001-01-30T00:00:00Z
review text: Let \(G\) be a connected real reductive group and \(P\) its parabolic subgroup. The goal of the paper is to give procedures for calculating the integer homology and cohomology groups of the real flag manifold \(G/P\) in terms of its Schubert cells. Let \(W_G\) and \(W_P\) denote the Weyl groups of the complexifications of \(G\) and of the Levi subgroup of \(P\) and let \(\widetilde W^P\) be the set of the minimal length representatives of left cosets of \(W_G\) modulo \(W_P\). The Bruhat decomposition of \(G\) gives rise to the decomposition of \(G/P\) into the Schubert cells \(C_w\) parametrized by \(w\in\widetilde W^P\), such that \(\dim C_w = l(w)\). Chain and cochain complexes for calculating \(H_*(G/P,\mathbf Z)\) and \(H^*(G/P,\mathbf Z)\) are constructed. E.g., to calculate \(H^*(G/P,\mathbf Z)\), one may use the complex \(S^*\) with \[ S^k = \bigoplus_{\substack{ w\in\widetilde W^P\\ l(w)=k}}\mathbf Z_w, \] whose coboundary operators are given by matrices with entries in \(\{0,\pm 2\}\) expressed in terms of the Weyl groups. Topologically, these entries contain information concerning non-orientability of certain submanifolds of \(G/P\). The simplest case is when \(G\) possesses the following property: for every real root \(\alpha\) of \(G\) with respect to its maximally split Cartan subgroup \(H\), the element \(\Phi_{\alpha}(-I)\) is contained in the identity component of \(H\), where \(\Phi_{\alpha}: SL(2,\mathbf R)\to G\) is the homomorphism associated with \(\alpha\). Then, for all parabolic subgroups \(P\subset G\), all the coboundaries are 0, and hence \(H^k(G/P,\mathbf Z)\simeq\mathbf Z^{s_k}\), where \(s_k\) is the number of elements of \(\widetilde W^P\) of length \(k\). Another result is a necessary and sufficient condition for a Schubert cell \(C_w\) to be a cycle; in this case, it always determines a non-zero element of \(H_{l(w)}(G/P,\mathbf Z)\). The proofs exploit perverse sheaves and principal series representations.
Mathematics Subject Classification ID: 22E47
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 57T15
Mathematics Subject Classification ID: 55N30
zbMATH DE Number: 1336938
zbMATH Keywords: real reductive group
zbMATH Keywords: parabolic subgroup
zbMATH Keywords: real flag manifold
zbMATH Keywords: integral cohomology
zbMATH Keywords: Schubert cell
zbMATH Keywords: perverse sheaf
zbMATH Keywords: Harish-Chandra module
MaRDI profile type: Publication
full work available at URL: s002220050334
OpenAlex ID: W2004196826
DOI: 10.1007/S002220050334
description: scientific article; zbMATH DE number 1336938