Generalizations of the PRV conjecture, II

TL;DR

This paper combines recent generalizations of the PRV conjecture to broader classes of reductive groups, advancing the understanding of submodule structures in irreducible modules.

Contribution

It unifies previous generalizations of the PRV conjecture into a more comprehensive result for complex reductive groups.

Findings

Unified the PRV conjecture generalizations for spherical minimal rank cases and classical cases.

Provided new methods to identify sub-$G$-modules in irreducible $ ilde{G}$-modules.

Extended the applicability of the PRV conjecture to broader group embeddings.

Abstract

Let $G\subset\hat{G}$ be two complex connected reductive groups. We deals with the hard problem of finding sub-$G$-modules of a given irreducible $\hat{G}$-module. In the case where $G$ is diagonally embedded in $\hat{G}=G\times G$, S. Kumar and O. Mathieu found some of them, proving the PRV conjecture. Recently, the authors generalized the PRV conjecture on the one hand to the case where $\hat{G}/G$ is spherical of minimal rank, and on the other hand giving more sub-$G$-modules in the classical case $G\subset G\times G$. In this paper, these two recent generalizations are combined in a same more general result.