Lie Groups, Geometry, and Representation Theory pp 481-526 | Cite as

# 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}⊙_{0}σ_{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*.

## Keywords

Belkale-Kumar Schubert calculus Kostant’s harmonic forms## Mathematics Subject Classification (2010):

14N15 14M15## Preview

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