Lie Superalgebras, Clifford Algebras, Induced Modules and Nilpotent Orbits

2004/02/05 by Ian M. Musson, Musson, Ian M.
Mathematics · #17B35 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:17B35

paper · pdf · doi:10.48550/arxiv.math/0402089

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

arxiv created 2004/12/23 · arxiv updated 2009/12/01

Abstract

Let \FRAKg be a classical simple Lie superalgebra. To every nilpotent orbit \cal O in \FRAKg₀ 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(\FRAKg)-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⋅\cal O where G is a Lie supergroup with Lie superalgebra \mathfrak g.

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