Behavior of the Borel-Weil-Bott construction under pullbacks

Let $G\hookrightarrow \tilde{G}$ be an embedding of semisimple complex Lie groups, let $B\hookrightarrow\tilde{B}$ be a pair of nested Borel subgroups and let $\varphi:G/B \hookrightarrow \tilde{G}/\tilde{B}$ be the associated embedding of flag manifolds. Let $\tilde{\mathcal L}$ be a $\tilde{G}$-equivariant line bundle on $\tilde{G}/\tilde{B}$ and let ${\mathcal L}$ be its restriction to $G/B$. Consider the $G$-equivariant pullback on cohomology 

$$\pi : H^\cdot(\tilde{G}/\tilde{B},\tilde{\mathcal L}) \longrightarrow H^\cdot(G/B,{\mathcal L}) \;.$$

The Borel-Weil-Bott theorem implies that the two cohomology spaces above are irreducible modules over $\tilde{G}$ and $G$ respectively. By Schur's lemma, $\pi$ is either surjective or zero. Nonvanishing implies the existence of certain $G$-irreducible component in an irreducible $\tilde{G}$-module, so the above setting gives rise to some ``geometric branching laws''.

The purpose of the talk is to present a necessary and sufficient condition for nonvanishing of $\pi$. In some special cases this condition takes particularly nice forms and allows easy computations; such are diagonal embeddings and regular embeddings. The methods rely on Kostant's theory of Lie algebra cohomology. 

This work is done under the supervision of Dr. Ivan Dimitrov.