Coherent states and Berezin quantization for non-scalar holomorphic representations

Abstract

Let \(G\) be a quasi-Hermitian Lie group and let \(K\) be a maximal compactly embedded subgroup of \(G\). Let \(\pi \) be a unitary representation of \(G\) which is holomorphically induced from a unitary representation \(\rho \) of \(K\). Then \(\pi \) is usually realized in a Hilbert space of vector-valued holomorphic functions. Here we introduce a realization of \(\pi \) in a reproducing kernel Hilbert space of complex-valued functions and we compute the (scalar-valued) coherent states. We study the corresponding Berezin quantization map and, in particular, we show that this map is equivalent to that introduced by Cahen (Rend Istit Mat Univ Trieste 46:157–180, 2014).