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

TY - JOUR

T1 - Heisenberg parabolically induced representations of Hermitian Lie groups, Part I:

T2 - Intertwining operators and Weyl transform

AU - Frahm, Jan

AU - Weiske, Clemens

AU - Zhang, Genkai

PY - 2023/4

Y1 - 2023/4

N2 - For a Hermitian Lie group $G$, we study the family of representations inducedfrom a character of the maximal parabolic subgroup $P=MAN$ whose unipotentradical $N$ is a Heisenberg group. Realizing these representations in thenon-compact picture on a space $I(\nu)$ of functions on the opposite unipotentradical $\bar{N}$, we apply the Heisenberg group Fourier transform mappingfunctions on $\bar N$ to operators on Fock spaces. The main result is anexplicit expression for the Knapp-Stein intertwining operators $I(\nu)\toI(-\nu)$ on the Fourier transformed side. This gives a new construction of thecomplementary series and of certain unitarizable subrepresentations at pointsof reducibility. Further auxiliary results are a Bernstein-Sato identity forthe Knapp-Stein kernel on $\bar{N}$ and the decomposition of the metaplecticrepresentation under the non-compact group $M$.

AB - For a Hermitian Lie group $G$, we study the family of representations inducedfrom a character of the maximal parabolic subgroup $P=MAN$ whose unipotentradical $N$ is a Heisenberg group. Realizing these representations in thenon-compact picture on a space $I(\nu)$ of functions on the opposite unipotentradical $\bar{N}$, we apply the Heisenberg group Fourier transform mappingfunctions on $\bar N$ to operators on Fock spaces. The main result is anexplicit expression for the Knapp-Stein intertwining operators $I(\nu)\toI(-\nu)$ on the Fourier transformed side. This gives a new construction of thecomplementary series and of certain unitarizable subrepresentations at pointsof reducibility. Further auxiliary results are a Bernstein-Sato identity forthe Knapp-Stein kernel on $\bar{N}$ and the decomposition of the metaplecticrepresentation under the non-compact group $M$.

KW - Complementary series

KW - Heisenberg parabolic subgroups

KW - Hermitian Lie groups

KW - Induced representations

KW - Unitarizable subrepresentations

UR - http://www.scopus.com/inward/record.url?scp=85152121875&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2209.04273

DO - 10.48550/arXiv.2209.04273

M3 - Tidsskriftartikel

SN - 0001-8708

VL - 422

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 109001

ER -