Möllers J (2013)
Publication Type: Journal article
Publication year: 2013
Publisher: Universität Bielefeld, Fakultät für Mathematik
Book Volume: 18
Pages Range: 785-855
For any Hermitian Lie group G of tube type we give a geometric quantization procedure of certain KC-orbits in p ∗ C to obtain all scalar type highest weight representations. Here KC is the complexification of a maximal compact subgroup K ⊆ G with corresponding Cartan decomposition g = k+p of the Lie algebra of G. We explicitly realize every such representation π on a Fock space consisting of square integrable holomorphic functions on its associated variety Ass(π) ⊆ p ∗ C. The associated variety Ass(π) is the closure of a single nilpotent KCorbit O KC ⊆ p C which corresponds by the Kostant–Sekiguchi correspondence to a nilpotent coadjoint G-orbit O G ⊆ g ∗. The known Schrödinger model of π is a realization on L 2 (O), where O ⊆ O G is a Lagrangian submanifold. We construct an intertwining operator from the Schrödinger model to the new Fock model, the generalized Segal–Bargmann transform, which gives a geometric quantization of the Kostant–Sekiguchi correspondence (a notion invented by Hilgert, Kobayashi, Ørsted and the author). The main tool in our construction are multivariable I- and K-Bessel functions on Jordan algebras which appear in the measure of O KC as reproducing kernel of the Fock space and as integral kernel of the Segal–Bargmann transform. As a corollary to our construction we also obtain the integral kernel of the unitary inversion operator in the Schrödinger model in terms of a multivariable J-Bessel function as well as explicit Whittaker vectors.
APA:
Möllers, J. (2013). A geometric quantization of the Kostant-Sekiguchi correpondence for scalar type unitary highest weight representations. Documenta Mathematica, 18, 785-855.
MLA:
Möllers, Jan. "A geometric quantization of the Kostant-Sekiguchi correpondence for scalar type unitary highest weight representations." Documenta Mathematica 18 (2013): 785-855.
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