Kostant's theorem for Lie super-algebra gl ( m, n)

Let p be a parabolic subalgebra of Lie super-algebra gl(m, n), such that its Levi subalgebra l of purely even degree. Denote by n by the nilpotent radical of p. We give a combinatorial description of the irreducible representations of l appearing in the H•(n, k). In particular we show that each irreducible representation appears with multiplicity at most one. This extends the classical result of Kostant regarding the cohomology of nilpotent radical of a parabolic subalgebra in a semi-simple Lie algebra.;On the other hand, for a finite dimensional graded vector space V•, one can associate Buchsbaum-Eisenbud variety of complexes Com(V•). Groups GL(V•) act on this variety by composition of differentials with automorphisms of vector spaces Vi. It has been shown that the each irreducible representation of producti GL(Vi) appears in the ring of functions k[Com(V•)] with multiplicity at most one. The result of this paper can be also interpreted as an extension of this fact to an appropriately defined derived variety of complexes.