Distributions on homogeneous spaces and applications

Abstract

Let $G$ be a complex semisimple algebraic group. In 2006, Belkale-Kumar defined a new product $odot\_0$ on thecohomology group $H^{\ast }(G/P,\mathbb{C})$ of any projective $G$-homogeneousspace $G/P$.Their definition uses the notion of Levi-movability for triples ofSchubert varieties in $G/P$.In this article, we introduce a family of $G$-equivariant subbundlesof the tangent bundle of $G/P$ and the associated filtration of the DeRham complex of $G/P$ viewed as a manifold. As a consequence one gets a filtration of the ring $H^{\ast }(G/P,\mathbb{C})$and proves that $\odot \_0$ is the associated graded product.One of the aim of this more intrinsic construction of $\odot \_0$ isthat there is a natural notion of fundamental class$[Y]\_\odot \_0\in (H^{\ast }(G/P),\odot \_0)$ for any irreducible subvariety $Y$ of $G/P$.Given two Schubert classes $\sigma \_u$ and $\sigma \_v$ in$H^{\ast }(G/P)$, we define a subvariety $\Sigma \_u^v$ of $G/P$. This variety should play the role of the Richardson variety; moreprecisely, we conjecture that$[\Sigma \_u^v]\_\odot \_0=\sigma \_u\odot \_0\sigma \_v$.We give some evidence for this conjecture, and prove special cases.Finally, we use the subbundles of $TG/P$ to give a geometriccharacterization of the $G$-homogeneous locus of any Schubertsubvariety of $G/P$.