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

TL;DR

This paper characterizes modules over basic classical Lie superalgebras that are simultaneously Verma and dual Verma modules with respect to different Borel subalgebras, showing they are essentially Verma modules for distinguished or anti-distinguished Borel choices.

Contribution

It proves that such modules are isomorphic to Verma modules for distinguished or anti-distinguished Borel subalgebras, using combinatorial analysis of Young lattice contractions.

Findings

Modules are isomorphic to Verma modules for specific Borel choices.

Method involves analyzing edge contractions in Young lattice.

Strategy applicable to all basic classical Lie superalgebras.

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.