Berezin-Toeplitz quantization on Lie groups

Let K be a connected compact semisimple Lie group and K-C its complexification. The generalized Segal-Bargmann space for K-C is a space of square-integrable holomorphic functions on K-C, 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 K-C. I also examine some of the results from an infinite-dimensional point of view based on the work of L. Gross and P. Malliavin. (C) 2008 Elsevier Inc. All rights reserved.