Degenerate Principal Series Representations of GLn(C) and GLn(R)

Let F be the field C or R. For 1⩽k⩽[n/2], let Pk be the maximal parabolic subgroup of GLn(F) with its Levi factor isomorphic to GLn−k(F)×GLk(F). In this paper, we explicitly calculate the action of the enveloping algebra of gln(F) on the degenerate principal series representations induced from characters of Pk. We address composition series, K-types and unitarity.