Lie superalgebras, Clifford algebras, induced modules and nilpotent orbits

Abstract

Let g be a classical simple Lie superalgebra. To every nilpotent orbit O in g 0 we associate a Clifford algebra over the field of rational functions on O . We find the rank, k ( O ) of the bilinear form defining this Clifford algebra, and deduce a lower bound on the multiplicity of a U ( g ) -module with O or an orbital subvariety of O as associated variety. In some cases we obtain modules where the lower bound on multiplicity is attained using parabolic induction. The invariant k ( O ) is in many cases, equal to the odd dimension of the orbit G ⋅ O , where G is a Lie supergroup with Lie superalgebra g .