I am trying to understand cohomology of $G := GL(N)$. For this I need to understand representations of $G(\mathbb{R})$ with nontrivial $(\mathfrak{g},K_\infty)$-cohomology, where $\mathfrak{g}$ is the Lie algebra of $G(\mathbb{R})$ and $K_\infty$ is $O(N,\mathbb{R})$. Is there a classification of such representations?
Thank you.