Contraction of discrete series via Berezin quantization. (English) Zbl 1185.22007

Cahen, Benjamin

J. Lie Theory 19, No. 2, 291-310 (2009).

Let \(G\) be a noncompact semisimple connected real Lie group with finite center and let \(K\) be a maximal compact subgroup of \(G\) such that the center of its Lie algebra is nontrivial. Denote by \(\pi_\chi\) a holomorphic discrete series representation of \(G\), induced from a unitary character \(\chi\) of \(K\), on a reproducing kernel Hilbert space of holomorphic functions on a bounded symmetric domain which is diffeomorphic to the Hermitian symmetric space of the noncompact type \(G/K\) [M. P. de Oliveira, Geom. Dedicata 86, No. 1–3, 227–247 (2001; Zbl 0996.32011); A. W. Knapp, Representation theory of semisimple Lie groups. An overview based on examples. Princeton Mathematical Series, 36. Princeton, New Jersey: Princeton University Press (1986; Zbl 0604.22001)]. Let \(\rho\) be a nondegenerate irreducible unitary representation of the Heisenberg group of dimension \(\dim G- \dim K+1\).
In this paper, the author extends to a wider class of groups the results on contractions of the discrete series representations of \(\text{SU}(1,n)\) to the irreducible unitary representations of the \((2n + 1)\)-dimensional Heisenberg group obtained in a previous work [B. Cahen, J. Anal. Math. 97, 83–101 (2006; Zbl 1131.22005)]. Again, this is done by using the quantization method introduced by F. A. Berezin [Math. USSR, Izv. 8, 1109–1165 (1974); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 38, 1116–1175 (1974; Zbl 0312.53049)], but exploiting a new explicit formula for the Berezin symbol of \(\pi_\chi(g)\) \((g\in G)\). It is shown that the matrix coefficients of \(\rho\) are limits of those of the sequence \((\pi_{\chi^n})_{n\in\mathbb N \backslash\{0\}}\) and that \(\rho\) is a contraction of the sequence \((\pi_{\chi^n})\) in the sense of J. Mickelsson and J. Niederle [Commun. Math. Phys. 27, 167–180 (1972; Zbl 0236.22021)].

Reviewer: Uberto Cattaneo (Maggia)

MSC:

22E46	Semisimple Lie groups and their representations
22E27	Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
46E22	Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
81R30	Coherent states

Keywords:

contraction of representations; holomorphic discrete series; reproducing kernel Hilbert space; Heisenberg group; Berezin quantization; Berezin symbol

Citations:

Zbl 0996.32011; Zbl 0604.22001; Zbl 1131.22005; Zbl 0312.53049; Zbl 0236.22021