On special covariants in the exterior algebra of a simple Lie algebra

Corrado De Concini, Pierluigi Möseneder Frajria, Paolo Papi, Claudio Procesi

Published 2014-04-16Version 1

For a simple complex Lie algebra $\mathfrak g$ we study the space $Hom_\mathfrak g(L,\bigwedge \mathfrak g)$ when $L$ is either the little adjoint representation or, in type $A_{n-1}$, the $n$-th symmetric power of the defining representation. As main result we prove that $Hom_\mathfrak g(L,\bigwedge \mathfrak g)$ is a free module, of rank twice the dimension of the $0$-weight space of $L$, over the exterior algebra generated by all primitive invariants in $(\bigwedge \mathfrak g^*)^{\mathfrak g}$, with the exception of the one of highest degree.