On the Spectrum of Exterior Algebra, and Generalized Exponents of Small Representations

Abstract

We present some results about the irreducible representations appearing in the exterior algebra $\Lambda \mathfrak{g}$, where $\mathfrak{g}$ is a simple Lie algebra over $\mathbb{C}$. 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 $\Lambda \mathfrak{g}$. 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.

28 pages. Comments are welcome. To appear in Bollettino dell'Unione Matematica Italiana