Schubert cells and cohomology of the spaces G/P (RMS 28:3 (1973) 1–26)

Summary

We study the homological properties of the factor space G/P, where G is a complex semisimple Lie group and P a parabolic subgroup of G. To this end we compare two descriptions of the cohomology of such spaces. One of these makes use of the partition of G/P into cells (Schubert cells), while the other consists in identifying the cohomology of G/P with certain polynomials on the Lie algebra of the Cartan subgroup H of G. The results obtained are used to describe the algebraic action of the Weyl group W of G on the cohomology of G/P.

Introduction

Let G be a linear semisimple algebraic group over the field C of complex numbers and assume that G is connected and simply-connected. Let B be a Borel subgroup of G and X = G/B the fundamental projective space of G.

The study of the topology of X occurs, explicitly or otherwise, in a large number of different situations. Among these are the representation theory of semisimple complex and real groups, integral geometry and a number of problems in algebraic topology and algebraic geometry, in which analogous spaces figure as important and useful examples. The study of the homological properties of G/P can be carried out by two well-known methods. The first of these methods is due to A. Borel [1] and involves the identification of the cohomology ring of X with the quotient ring of the ring of polynomials on the Lie algebra h of the Cartan subgroup H ⊂ G by the ideal generated by the w-invariant polynomials (where W is the Weyl group of G).