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

Författare

Jan Frahm

Aarhus Universitet

Clemens Weiske

Chalmers, Matematiska vetenskaper, Analys och sannolikhetsteori

Genkai Zhang

Chalmers, Matematiska vetenskaper, Analys och sannolikhetsteori

Advances in Mathematics

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

Vol. 422 109001

Representationer av Liegrupper. Harmonisk och komplex analys på symmetriska och lokalt symmetriska rum

Vetenskapsrådet (VR), 2019-01-01 -- 2022-12-31.

Vetenskapsrådet (VR) (2022-02861), 2023-01-01 -- 2026-12-31.

Ämneskategorier

Algebra och logik

Geometri

Matematisk analys

DOI

10.1016/j.aim.2023.109001

Mer information

Senast uppdaterat

2023-05-03