Heisenberg parabolically induced representations of Hermitian Lie groups
Let G be a hermitian Lie group. Then G naturally contains a maximal parabolic subgroup MAN whose unipotent radical is a Heisenberg group. Consider the corresponding spherical principal series representations of G, realised on functions on the radical opposite to N, which is also a Heisenberg group. Under the Heisenberg group Fourier transform, this space transforms into operators on Fock spaces. We show that these Fock spaces decompose into a multiplicity free direct sum under the action of M, which is in general non-compact. We find an explicit expression for the Knapp-Stein intertwining operator on the Fourier transformed side, generalising classical results by Cowling for the case G=SU(1,n), where M is compact. This gives a new construction of the complementary series and of certain interesting unitarizable subrepresentatio. The presented results are joint work with Jan Frahm and Genkai Zhang.
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