On certain modules of covariants in exterior algebras

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 $A:=\left(\bigwedge (\mathfrak g/\mathfrak k)^\right)^{\mathfrak k}$isanexterioralgebra$\wedge (x_1,…,x_r),$with$r=rk(\mathfrak g)-rk(\mathfrak k)$. We prove that they are free modules over the subalgebra $A_{r-1}=\wedge (x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_{r-1})$ 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 $\left(\bigwedge (M_n^{\pm})^*\otimes M_n\right)^G$,thespaceofthe$G-$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.