Lie Superalgebras, Clifford Algebras, Induced Modules and Nilpotent Orbits

Details Lie Superalgebras, Clifford Algebras, Induced Modules and Nilpotent Orbits Lie Superalgebras, Clifford Algebras, Induced Modules and Nilpotent Orbits

2004-02-05

arxiv.org/abs/math/0402089v2

Mathematics

Representation Theory

Accepted for publication in Advances in Mathematics. Some minor changes have been made to the original version, mostly in the

Scientific paper

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 ${\mathfrak g.}$

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