Heisenberg parabolically induced representations of Hermitian Lie groups, Part I: Unitarity and subrepresentations
| Autor: | Frahm, Jan, Weiske, Clemens, Zhang, Genkai |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Adv. Math. 422 (2023), 109001 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.aim.2023.109001 |
| Popis: | 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 $I(\nu)$ of functions on the opposite unipotent radical $\bar{N}$, we apply the Heisenberg group Fourier transform mapping functions on $\bar N$ to operators on Fock spaces. The main result is an explicit expression for the Knapp-Stein intertwining operators $I(\nu)\to I(-\nu)$ 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 $\bar{N}$ and the decomposition of the metaplectic representation under the non-compact group $M$. Comment: 44 pages, v2: final published version |
| Databáze: | arXiv |
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