Let $\mathfrak{g}$ be the Lie algebra of $GL(n,\mathbb{R})$. Let $\theta(X) = - X^T$ be the Cartan involution on $\mathfrak{g}$; it induces decomposition as $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ of $+1$-eigenspace $\mathfrak{k}$ and $-1$ eigenspace $\mathfrak{p}$. Then $\mathfrak{k}$ is the Lie algebra of $K = O(n)$. I want to compute the dimensions of $K$-invariants $\textrm{Hom}_K(\wedge^q \mathfrak{p}, \mathbb{C})$, which, I suppose, is equal to $(W/ \mathfrak{k} W)^*$ where $W = \wedge^q \mathfrak{p}$, where $\mathfrak{p}$ is viewed as $\mathfrak{k}$-module by adjoint-representation. Could you someone point a way further?