$\mathfrak b_1$-Verma $\mathfrak b_2$-dual Verma supermodules

Abstract

We show that if a module M over a basic classical Lie superalgebra of type type I is simultaneously a Verma module with respect to some Borel \(\mathfrak b_1\) and a dual Verma module with respect to Borel \(\mathfrak b_2\), then M is isomorphic to a Verma module with respect to either distinguished or an anti-distinguished Borel. Our method proceeds by analyzing edge contractions of the finite Young lattice that controls the combinatorics of odd reflections. In principle, the same strategy, for the most part, applies to all basic classical Lie superalgebras.