Consider the classical action of ${ \rm GL}_n$ on a sum of $q$ copies of the defining representation and $p$ copies of its dual; by Howe duality, the polynomial functions on this space decompose under the joint action of ${ \rm GL}_n$ and $\mathfrak{ gl}_{ p+q}$. The modules for $\mathfrak{ gl}_{ p+q}$ are infinite-dimensional and their structure is complicated outside a certain stable range, although Enright and Willenbring (2005) constructed resolutions in terms of generalized Verma modules. We show that these resolutions can be read off from the coefficients in a formal series arising in an entirely different setting: discrete series representations of ${ \rm SU}(n,p+q)$ in the case of two noncompact simple roots.
Categories: math.RT