Lie Superalgebras, Clifford Algebras, Induced Modules and Nilpotent Orbits
Abstract
Description
Let $\FRAK{g}$ be a classical simple Lie superalgebra. To every nilpotent orbit $\cal O$ in $\FRAK{g}_0$ we associate a Clifford algebra over the field of rational functions on $\cal O$. We find the rank, $k(\cal O)$ of the bilinear form defining this Clifford algebra, and deduce a lower bound on the multiplicity of a $U(\FRAK{g})$-module with $\cal O$ or an orbital subvariety of $\cal O$ as associated variety. In some cases we obtain modules where the lower bound on multiplicity is attained using parabolic induction. The invariant $k(\cal O)$ is in many cases, equal to the odd dimension of the orbit $G\cdot\cal O$ where $G$ is a Lie supergroup with Lie superalgebra ${\mathfrak g.}$
Accepted for publication in Advances in Mathematics. Some minor changes have been made to the original version, mostly in the last section
Accepted for publication in Advances in Mathematics. Some minor changes have been made to the original version, mostly in the last section