On Some Degenerate Principal Series Representations of U(n,n)

We study the infinitesimal action of u(n, n) on the degenerate principal series representations of U(n, n) associated with a parabolic subgroup which stabilizes a maximal isotropic subspace of C2n relative to the indefinite Hermitian form with signature (n, n). This allows us in particular to determine the reducibility, socle series, and unitarity of these representations.