Eigencones and the PRV conjecture

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 ${\text{Li}}_i$ be a $G$-linearized line bundle on $G\mathrm{/}B$ such that ${\mathrm{H}}^{q_i}(G\mathrm{/}B,{\text{Li}}_i)$ is isomorphic to $V_i$. Assume that the cup product $${\mathrm{H}}^{q_1}(G\mathrm{/}B,{\text{Li}}_1)\otimes {\mathrm{H}}^{q_2}(G\mathrm{/}B,{\text{Li}}_2)\text{longto}{\mathrm{H}}^{q_1+q_2}(G\mathrm{/}B,{\text{Li}}_1\otimes {\text{Li}}_2)$$ is non zero. Then, ${\rm H}^{q_1+q_2}(G/B,\Li_1\otimes\Li_2)$isanirreduciblecomponentof$V_1\otimes V_2$; 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 ${\mathrm{H}}^{\ast }(G\mathrm{/}B,\text{ZZ})$ in the Schubert basis are zero or one.