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

TY - JOUR

T1 - Fock model and Segal-Bargmann transform for minimal representations of Hermitian Lie groups

AU - Hilgert, Joachim

AU - Kobayashi, Toshiyuki

AU - Möllers, Jan

AU - Ørsted, Bent

PY - 2012/3/25

Y1 - 2012/3/25

N2 - 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.

AB - 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.

KW - Minimal representation

KW - Schrödinger model

KW - Fock model

KW - Jordan algebra

KW - Segal–Bargmann transform

KW - Branching law

KW - Bessel function

KW - Spherical harmonics

U2 - 10.1016/j.jfa.2012.08.026

DO - 10.1016/j.jfa.2012.08.026

M3 - Journal article

SN - 0022-1236

VL - 263

SP - 3492

EP - 3563

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 11

ER -