  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, M\"oseneder Frajria, Procesi and Papi about the module of special
covariants of adjoint and little adjoint type.
