Extremal twist and tensor product of highest weight modules

We give a criterion for complete reducibility of tensor product of two highest weight modules over a quantum group. It is found to be controlled by an extremal twist operator related to the Shapovalov inverse of either of the modules. As an application, we construct homogeneous vector bundles over quantum projective spaces $\mathbb{P}^n$ on $\mathbb{C}$-homs between certain parabolic Verma modules. Using an alternative realization of $\mathbb{C}_q[\mathbb{P}^n]$ as a subalgebra in $\mathbb{C}_q[GL(n+1)]$, we reformulate quantum vector bundles in terms of symmetric pairs. In this way, we prove complete reducibility of modules over the coideal stabilizer subalgebras, via the quantum Frobenius reciprocity.