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.

21 pages