Distributions on Homogeneous Spaces and Applications

Abstract

Let G be a complex semisimple algebraic group. In 2006, Belkale-Kumar defined a new product ⊙₀ on the cohomology group H*(G/P,C) of any projective G-homogeneous space G/P. Their definition uses the notion of Levi-movability for triples of Schubert varieties in G/P.

In this article, we introduce a family of G-equivariant subbundles of the tangent bundle of G/P and the associated filtration of the de Rham complex of G/P viewed as a manifold. As a consequence one gets a filtration of the ring H*(G/P,C) and proves that ⊙₀ is the associated graded product. One of the aims of this more intrinsic construction of ⊙₀ is that there is a natural notion of a fundamental class [Y]⊙₀ ∈ (H*(G/P,C), ⊙₀) for any irreducible subvariety Y of G/P.

Given two Schubert classes σ_u and σ_v in H*(G/P,C), we define a subvariety $\sum _u^v$ of G/P. This variety should play the role of the Richardson variety; more precisely, we conjecture that [$\sum _u^v$]⊙₀ = σ_u⊙₀σ_v. We give some evidence for this conjecture, and prove special cases.

Finally, we use the subbundles of TG/P to give a geometric characterization of the G-homogeneous locus of any Schubert subvariety of G/P.