On certain modules of covariants in exterior algebras

Abstract

We study the structure of the space of covariants $B:=\left(\bigwedge (\mathfrak g/\mathfrak k)^*\otimes \mathfrak g\right)^{\mathfrak k},$ for a certain class of infinitesimal symmetric spaces $(\mathfrak g,\mathfrak k)$ such that the space of invariants $A:=\left(\bigwedge (\mathfrak g/\mathfrak k)^*\right)^{\mathfrak k}$ is an exterior algebra $\wedge (x_1,...,x_r),$ with $r=rk(\mathfrak g)-rk(\mathfrak k)$. We prove that they are free modules over the subalgebra $A_{r-1}=\wedge (x_1,...,x_{r-1})$ of rank $4r$. In addition we will give an explicit basis of $B$. As particular cases we will recover same classical results. In fact we will describe the structure of $\left(\bigwedge (M_n^{\pm})^*\otimes M_n\right)^G$, the space of the $G-$equivariant matrix valued alternating multilinear maps on the space of (skew-symmetric or symmetric with respect to a specific involution) matrices, where $G$ is the symplectic group or the odd orthogonal group. Furthermore we prove new polynomial trace identities.