Nonzero n cohomology of Totally Degenerate Limit of Discrete Series representations

We show that a totally degenerate limit of discrete series representation admits a choice of n cohomology group that is nonvanishing at a canonically defined degree. We then show that the combinatorial complexes used by Soergel to compute these cohomology groups satisfies Serre duality. We conclude that this produces two n cohomology groups, each for a totally degenerate limit of discrete series of U(n+1) and U(n), which are nonvanishing at the same degree. This suggests Gan Gross Prasad type branching laws for the TDLDS of unitary groups of any rank.