Cup products of line bundles on homogeneous varieties and generalized PRV components of multiplicity one

TL;DR

This paper investigates the conditions under which cup product maps between cohomology groups of line bundles on flag varieties are nonzero and irreducible, linking these to representation theory and tensor product multiplicities.

Contribution

It provides criteria for nonzero cup product maps of line bundles on flag varieties and characterizes generalized PRV components with multiplicity one.

Findings

Criteria for nonzero cup product maps

Characterization of irreducible components in tensor products

Bounds on multiplicities related to Littlewood-Richardson cone

Abstract

Let X=G/B be a complete flag variety, and L' and L" two line bundles on X. Consider the cup product map H^{d'}(X,L') x H^{d"}(X, L") --> H^{d}(X,L), where L=L' x L" and d=d'+d". We answer two natural questions about the map above: When is it a nonzero map of irreducible G-representations? Conversely, given generic irreducible representations V' and V" of G, which irreducible components of V' x V" may appear in the right hand side of the map above? We also give bounds on the multiplicities appearing in a tensor product, and relate these considerations to the boundary of the Littlewood-Richardson cone.