On associated variety for Lie superalgebras

Abstract

We define the associated variety $ X_{M} $ of a module $ M $ over a finite-dimensional superalgebra $ {\mathfrak g} $, and show how to extract information about $ M $ from these geometric data. $ X_{M} $ is a subvariety of the cone $ X $ of self-commuting odd elements. For finite-dimensional $ M $, $ X_{M} $ is invariant under the action of the underlying Lie group $ G_{0} $. For simple superalgebra with invariant symmetric form, $ X $ has finitely many $ G_{0} $-orbits; we associate a number (rank) to each such orbit. One can also associate a number (degree of atypicality) to an irreducible finite-dimensional representation. We prove that if $ M $ is an irreducible $ {\mathfrak g} $-module of degree of atypicality $ k $, then $ X_{M} $ lies in the closure of all orbits on $ X $ of rank $ k $. If $ {\mathfrak g}={\mathfrak g}{\mathfrak l}(m|n) $ we prove that $ X_{M} $ coincides with this closure.