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.

Joseph Ideals and Harmonic Analysis for

“…In particular it would also be of interest to construct general theories as in [1,2,18,26] for Lie superalgebras. For instance there seems to exist an interesting link with (co)adjoint orbits, see [29]. As will become apparent in the current paper, the representations corresponding to the Joseph ideal for Lie superalgebras can not always be expected to be unitarizable.…”

Super Vust theorem and Schur-Sergeev duality for principal finite $W$-superalgebras

“…Now set Y = G.e and Z = G.e. Thanks to [24,Theorem 6.8], dim Z = dim Y = dim G.e = dim G ev .e. By the above arguments, we have that…”

“…(1) Note that the superscheme G.e is of dimension (dim G ev .e, dim g1 − dim(g1) e ) (see [24,Theorem 6.8]). And the superscheme G.e has the same dimension as the one of G.e .…”