Degenerate principal series representations of GLn(C) and GLn(R)

Let F be the field C or R. For 1 less than or equal to k less than or equal to [n/2], let P-k be the maximal parabolic subgroup of GL(n)(F) with its Levi factor isomorphic to GL(n-k)(F) x GL(k)(F). In this paper, we explicitly calculate the action of the enveloping algebra of gl,,(F) on the degenerate principal series representations induced from characters of P-k. We address composition series, K-types and unitarity. (C) 1999 Academic Press.