Berezin-Toeplitz quantization on Lie groups

TL;DR

This paper explores Berezin-Toeplitz quantization on complexified Lie groups, showing how certain operators on L^2(K) correspond to Toeplitz operators on the Segal-Bargmann space, with symbols linked to subelliptic heat kernels.

Contribution

It demonstrates the representation of natural operators as Toeplitz operators on the Segal-Bargmann space for Lie groups, connecting heat kernel analysis with quantization.

Findings

Operators conjugated by the Segal-Bargmann transform are Toeplitz operators.

Symbols are expressed via subelliptic heat kernels on Kc.

Infinite-dimensional analysis perspectives are discussed.

Abstract

Let K be a connected compact semisimple Lie group and Kc its complexification. The generalized Segal-Bargmann space for Kc, is a space of square-integrable holomorphic functions on Kc, with respect to a K-invariant heat kernel measure. This space is connected to the "Schrodinger" Hilbert space L^2(K) by a unitary map, the generalized Segal-Bargmann transform. This paper considers certain natural operators on L^2(K), namely multiplication operators and differential operators, conjugated by the generalized Segal-Bargmann transform. The main results show that the resulting operators on the generalized Segal-Bargmann space can be represented as Toeplitz operators. The symbols of these Toeplitz operators are expressed in terms of a certain subelliptic heat kernel on Kc. I also examine some of the results from an infinite-dimensional point of view based on the work of L. Gross and P. Malliavin.