Distributions on homogeneous spaces and applications
N Ressayre (ICJ)

TL;DR
This paper introduces a new geometric framework for understanding a specialized product on the cohomology of homogeneous spaces, linking algebraic and geometric structures and providing evidence for a conjectured correspondence with Richardson varieties.
Contribution
It develops a natural, intrinsic construction of the Belkale-Kumar product using filtrations of the DeRham complex, and introduces subvarieties that generalize Richardson varieties within this context.
Findings
Established a filtration of the cohomology ring compatible with the new product.
Conjectured and provided evidence that certain subvarieties represent the product of Schubert classes.
Provided a geometric characterization of the G-homogeneous locus of Schubert subvarieties.
Abstract
Let be a complex semisimple algebraic group. In 2006, Belkale-Kumar defined a new product on thecohomology group of any projective -homogeneousspace .Their definition uses the notion of Levi-movability for triples ofSchubert varieties in .In this article, we introduce a family of -equivariant subbundlesof the tangent bundle of and the associated filtration of the DeRham complex of viewed as a manifold. As a consequence one gets a filtration of the ring and proves that is the associated graded product.One of the aim of this more intrinsic construction of isthat there is a natural notion of fundamental class for any irreducible subvariety of .Given two Schubert classes and in, we define a subvariety…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Geometry and complex manifolds
