Berezin–Toeplitz quantization on Lie groups

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 “Schrödinger” Hilbert space L2(K) by a unitary map, the generalized Segal–Bargmann transform. This paper considers certain natural operators on L2(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.