2005/07/11 by M. Duflo, Duflo, M., V. Serganova +1 · 2 citations
Mathematics · #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT
paper · pdf · doi:10.48550/arxiv.math/0507198
21 pages
arxiv created 2005/07/11 · arxiv updated 2009/12/01
We define the associated variety XM of a module M over a finite-dimensional superalgebra \mathfrak g , and show how to extract information about M from these geometric data. XM is a subvariety of the cone X of self-commuting odd elements. For finite-dimensional M , XM is invariant under the action of the underlying Lie group G0 . For simple superalgebra with invariant symmetric form, X has finitely many G0 -orbits; we associate a number (rank) to each such orbit. One can also associate a number (degree of atypicality) to an irreducible finite-dimensional representation. We prove that if M is an irreducible \mathfrak g -module of degree of atypicality k , then XM lies in the closure of all orbits on X of rank k . If \mathfrak g=\mathfrak g\mathfrak l(m|n) we prove that XM coincides with this closure.