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

We present some results about the irreducible representations appearing in the exterior algebra $\mathrm{\Lambda }\mathfrak{g}$, 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 $\mathrm{\Lambda }\mathfrak{g}$. 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.