A geometric quantization of the Kostant–Sekiguchi correspondence for scalar type unitary highest weight representations

Jan Möllers

Documenta mathematica, Tome 18 (2013), pp. 785-855

Résumé

For any Hermitian Lie group G of tube type we give a geometric quantization procedure of certain KC​-orbits in pC∗​ 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)⊆pC∗​. The associated variety Ass(π) is the closure of a single nilpotent KC​-orbit OKC​⊆pC∗​ which corresponds by the Kostant–Sekiguchi correspondence to a nilpotent coadjoint G-orbit OG⊆g∗. The known Schrödinger model of π is a realization on L2(O), where O⊆OG 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 OKC​, 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.

Détail

DOI : 10.4171/dm/414

Classification : 17C30, 22E45, 30H20, 33C70, 44A15, 46E22
Mots-clés : unitary highest weight representation, orbit method, Schrödinger model, Fock model, Jordan algebra, Segal--Bargmann transform, Bessel function, Bessel operator, unitary inversion operator, Whittaker vectors, branching law