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

Joachim Hilgert, Toshiyuki Kobayashi, Jan Möllers, Bent Ørsted

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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 L^2-inner product involves a K-Bessel function as density. Here K is a maximal compact subgroup of G, 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 Schroedinger 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 Schroedinger model which is given by a J-Bessel function.
Original languageEnglish
JournalJournal of Functional Analysis
Pages (from-to)3492–3563
Number of pages72
Publication statusPublished - 25 Mar 2012


  • Minimal representation
  • Schrödinger model
  • Fock model
  • Jordan algebra
  • Segal–Bargmann transform
  • Branching law
  • Bessel function
  • Spherical harmonics


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