Distributions on homogeneous spaces and applications

International audienceLet $G$ be a complex semisimple algebraic group. In 2006, Belkale-Kumar defined a new product $\odot_0$ on thecohomology group $H^*(G/P,{\mathbb C})$ 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 $H^*(G/P,{\mathbb C})$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$[Y]_{\odot_0}\in(H^*(G/P,{\mathbb C}),\odot_0)$ for any irreducible subvariety $Y$ of $G/P$.Given two Schubert classes $\sigma_u$ and $\sigma_v$ in$H^*(G/P,{\mathbb C})$, 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 $G/P$