Schubert cells and representation theory

Abstract

Let G be a linear, real, reductive group, and let P be a parabolic subgroup. The Bruhat decomposition of G gives a cellular decomposition of the generalized flag manifold X = G/P. Originating in the work of Schubert on Grassmann manifolds, this cellular decomposition was used by Ehresmann [Eh] to give a proof of the basis theorem for the integral cohomology of Grassmannians. Since Ehresmann it has been clear that, for a generalized flag manifold X, a more detailed understanding of this decomposition by generalized Schubert cells would provide a determination of the integral (co-)homology of X. The aim of this article is to give a representation theoretic determination of the differentials in the Schubert cell decomposition of X and thereby obtain a combinatorial description of the integral (co-)homology of X. Our primary tool will be the infinite-dimensional representation theory of G. If G and P are complex groups, elementary considerations show that all Schubert cells define non-zero integral homology classes and that none are torsion. Moreover, the cohomology has the well-known connection to finitedimensional representation theory ([Ct], [Ko1], [Ko2], [Bt]). For real groups G and P, even in low dimensions, torsion can be present in the homology of real flag manifolds. Thus the major new development in this paper is that the infinite-dimensional representation theory of G detects which real Schubert cells define integral, in particular torsion, classes in the (co-)homology. The innovation on the representation-theoretic side that makes possible the consideration of (co-)homology with integer coefficients is the geometric formulation of representation theory introduced by Beilinson-Bernstein. To compute homology we use a topological technique that applies to spaces with filtration. Let X p be the union of Schubert cells whose dimension