Heisenberg parabolically induced representations of Hermitian Lie groups, Part I: Intertwining operators and Weyl transform

Abstract

For a Hermitian Lie group G, we study the family of representations induced from a character of the maximal parabolic subgroup P=MAN whose unipotent radical N is a Heisenberg group. Realizing these representations in the non-compact picture on a space I(ν) of functions on the opposite unipotent radical N¯, we apply the Heisenberg group Fourier transform mapping functions on N¯ to operators on Fock spaces. The main result is an explicit expression for the Knapp–Stein intertwining operators I(ν)→I(−ν) on the Fourier transformed side. This gives a new construction of the complementary series and of certain unitarizable subrepresentations at points of reducibility. Further auxiliary results are a Bernstein–Sato identity for the Knapp–Stein kernel on N‾ and the decomposition of the metaplectic representation under the non-compact group M.