Heisenberg parabolically induced representations of Hermitian Lie groups, Part I: Intertwining operators and Weyl transform
Journal article, 2023

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.

Hermitian Lie groups

Unitarizable subrepresentations

Induced representations

Complementary series

Heisenberg parabolic subgroups


Jan Frahm

Aarhus University

Clemens Weiske

Chalmers, Mathematical Sciences, Analysis and Probability Theory

Genkai Zhang

Chalmers, Mathematical Sciences, Analysis and Probability Theory

Advances in Mathematics

0001-8708 (ISSN) 1090-2082 (eISSN)

Vol. 422 109001

Representations of Lie groups. Harmonic and complex analysis on symmetric and locally symmetric spaces

Swedish Research Council (VR), 2019-01-01 -- 2022-12-31.

Swedish Research Council (VR) (2022-02861), 2023-01-01 -- 2026-12-31.

Subject Categories

Algebra and Logic


Mathematical Analysis



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5/3/2023 1