Distributions on Homogeneous Spaces and Applications
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 σu and σv 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 = σu⊙0σv. 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.
KeywordsBelkale-Kumar Schubert calculus Kostant’s harmonic forms
Mathematics Subject Classification (2010):14N15 14M15
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