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

Abstract

Let $X=G∕B$ and let $L_1$ and $L_2$ be two line bundles on $X$. Consider the cup-product map

$$H^{d_1}\left(X,L_1\right)\otimes H^{d_2}\left(X,L_2\right)\underset{}{\overset{\cup }{\to }}{H}^d\left(X,L\right),$$

where $L=L_1\otimes L_2$ and $d=d_1+d_2$. We answer two natural questions about the map above: When is it a nonzero homomorphism of representations of $G$? Conversely, given generic irreducible representations $V_1$ and $V_2$, which irreducible components of $V_1\otimes V_2$ may appear in the right hand side of the equation above? For the first question we find a combinatorial condition expressed in terms of inversion sets of Weyl group elements. The answer to the second question is especially elegant: the representations $V$ appearing in the right hand side of the equation above are exactly the generalized PRV components of $V_1\otimes V_2$ of stable multiplicity one. Furthermore, the highest weights $\left({\lambda }_1,{\lambda }_2,\lambda \right)$ corresponding to the representations $\left(V_1,V_2,V\right)$ fill up the generic faces of the Littlewood–Richardson cone of $G$ of codimension equal to the rank of $G$. In particular, we conclude that the corresponding Littlewood–Richardson coefficients equal one.