On the spectrum of exterior algebra, and generalized exponents of small representations

Abstract

AbstractWe present some results about the irreducible representations appearing in the exterior algebra $$\Lambda \mathfrak {g}$$ Λ g , where $$\mathfrak {g}$$ g is a simple Lie algebra over $${\mathbb {C}}$$ C . For Lie algebras of type B, C or D we prove that certain irreducible representations, associated to weights characterized in a combinatorial way, appear as irreducible components of $$\Lambda \mathfrak {g}$$ Λ g . Moreover, we propose an analogue of a conjecture of Kostant, about irreducibles appearing in the exterior algebra of the little adjoint representation. Finally, we give some closed expressions, in type B, C and D, for generalized exponents of small representations that are fundamental representations and we propose a generalization of some results of De Concini, Möseneder Frajria, Procesi and Papi about the module of special covariants of adjoint and little adjoint type.