Fock model and Segal–Bargmann transform for minimal representations of Hermitian Lie groups

Hilgert J, Kobayashi T, Möllers J, Ørsted B (2012)

Publication Type: Journal article

Publication year: 2012

Journal

Journal of Functional Analysis Elsevier

Publisher: Elsevier

Book Volume: 263

Pages Range: 3492-3563

Journal Issue: 11

DOI: 10.1016/j.jfa.2012.08.026

Abstract

For any Hermitian Lie group G of tube type we construct a Fock model of its minimal representation. The Fock space is defined on the minimal nilpotent K_C-orbit X in p_C and the L²-inner product involves a K-Bessel function as density. Here K⊆G is a maximal compact subgroup and g_C=k_C+p_C is a complexified Cartan decomposition. In this realization the space of k-finite vectors consists of holomorphic polynomials on X. The reproducing kernel of the Fock space is calculated explicitly in terms of an I-Bessel function. We further find an explicit formula of a generalized Segal–Bargmann transform which intertwines the Schrödinger and Fock model. Its kernel involves the same I-Bessel function. Using the Segal–Bargmann transform we also determine the integral kernel of the unitary inversion operator in the Schrödinger model which is given by a J-Bessel function.

Authors with CRIS profile

Jan Frahm Juniorprofessur für Mathematik (Darstellungstheorie und Operatoralgebren)

Involved external institutions

Aarhus University

Denmark (DK) University of Tokyo

Japan (JP)

How to cite

APA:

Hilgert, J., Kobayashi, T., Möllers, J., & Ørsted, B. (2012). Fock model and Segal–Bargmann transform for minimal representations of Hermitian Lie groups. Journal of Functional Analysis, 263(11), 3492-3563. https://doi.org/10.1016/j.jfa.2012.08.026

MLA:

Hilgert, Joachim, et al. "Fock model and Segal–Bargmann transform for minimal representations of Hermitian Lie groups." Journal of Functional Analysis 263.11 (2012): 3492-3563.

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