JOURNAL OF DIFFERENTIAL GEOMETRY, vol.35, no.3, pp.511-519, 1992 (SCI-Expanded)
Let GL(n) be the group of n x n invertible complex matrices, and P a parabolic subgroup of GL(n). In this paper we give a geometric description of the cohomology ring of a Schubert subvariety Y of GL(n)/P. Our main result (Theorem 3.1) states that the coordinate ring A(Y intersect Z) of the scheme-theoretic intersection of Y and the zero scheme Z of the vector field V associated to a principal regular nilpotent element n of gl(n) is isomorphic to the cohomology algebra H*(Y ; C) of Y. This theorem was conjectured for any reductive algebraic group G in [4], and it was proved for the Grassmannian manifolds in [2]. We were recently informed that Professor D. H. Peterson has just proved that GL(n) is exactly the algebraic group G where the cohomology ring of any Schubert subvariety Y of the space G/B is isomorphic to A(Y intersect Z). Here B stands for a Borel subgroup of G. It is also interesting to note that the cohomology ring of the union of two Schubert subvarieties in GL(n)/P may not admit such a description. This result is due to Professor J. B. Carrell.