Eigencones and the PRV conjecture

TL;DR

This paper characterizes cohomological components of tensor products of representations of complex semisimple groups, proving they are exactly the PRV components with stable multiplicity one, and relates this to the geometry of the Eigencone.

Contribution

It proves that cohomological components coincide with PRV components of stable multiplicity one, solving a conjecture and linking to Eigencone geometry.

Findings

Cohomological components are exactly PRV components with stable multiplicity one.

Structure coefficients of the Belkale-Kumar product are zero or one.

Characterization of these components via Eigencone geometry.

Abstract

Let $G$ be a complex semisimple simply connected algebraic group. Given two irreducible representations $V_1$ and $V_2$ of $G$, we are interested in some components of $V_1\otimes V_2$. Consider two geometric realizations of $V_1$ and $V_2$ using the Borel-Weil-Bott theorem. Namely, for $i=1, 2$, let $\Li_i$ be a $G$-linearized line bundle on $G/B$ such that ${\rm H}^{q_i}(G/B,\Li_i)$ is isomorphic to $V_i$. Assume that the cup product $$ {\rm H}^{q_1}(G/B,\Li_1)\otimes {\rm H}^{q_2}(G/B,\Li_2)\longto {\rm H}^{q_1+q_2}(G/B,\Li_1\otimes\Li_2) $$ is non zero. Then, ${\rm H}^{q_1+q_2}(G/B,\Li_1\otimes\Li_2)$ is an irreducible component of $V_1\otimes V_2$; such a component is said to be {\it cohomological}. Solving a Dimitrov-Roth conjecture, we prove here that the cohomological components of $V_1\otimes V_2$ are exactly the PRV components of stable multiplicity one. Note that Dimitrov-Roth already obtained some particular cases. We also characterize these components in terms of the geometry of the Eigencone of $G$. Along the way, we prove that the structure coefficients of the Belkale-Kumar product on ${\rm H}^*(G/B,\ZZ)$ in the Schubert basis are zero or one.