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

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 L 2 -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.

Varna Lecture on L 2-Analysis of Minimal Representations

“…In the case G = O(p+1, q+1) with p + q even > 2, it was proved in [28] that (i) holds if and only if either min(p, q) = 1 (equivalently, (ii) holds) or (p, q) = (3, 3) (equivalently, g = o(3, 3) ≃ sl(4, R) is of type A 3 ). The implication (ii) ⇒ (i) was proved in [12] for tube type, see (8.5). The implication (i) ⇒ (ii) is an open problem except for the above mentioned case g = o(p + 1, q + 1).…”

“…• new orthogonal polynomials for fourth order differential operators [9,10,29], • a generalization of the Fock model and Bargmann transforms [12].…”

“…The aim of this article is to provide a brief guidance to the rapid progress on our program, [3,9,10,11,12,23,24,28,29,34,38]. We should mention that in order to avoid an overlap with a recent publication [24], we do not include here some other constructions such as a conformal model of minimal representations (e.g.…”

“…Any coadjoint orbit of a Lie group is naturally a symplectic manifold endowed with the Kirillov-Kostant-Souriau symplectic form. For a reductive Lie group G, 'geometric quantization' of semisimple coadjoint orbits has been considerably well-understood -this corresponds to the ordinary or cohomological parabolic induction in representation theory, whereas 'geometric quantization' of nilpotent coadjoint orbits is more mysterious (see [4,12,23]).…”

“…• Dunkl operators (with Ben Saïd and Ørsted [3]), • Conformal group of Euclidean Jordan algebras (with Hilgert and Möllers [12]).…”