Berezin transform for non-scalar holomorphic discrete series. (Q2897410)

In the article, the author studies a Berezin type transform defined using the so called Berezin map introduced by the author in [\textit{B. Cahen}, ``Berezin quantization for holomorphic discrete series representations: the non scalar case'', online first articles in: Beitr. Algebra Geom., (2011; \url{doi:10.1007/s13366-011-0066-2})]. He investigates a more general transform than the classical one and also more general than the one studied, e.g., in [\textit{A. Unterberger} and \textit{A. Upmeier}, Commun. Math. Phys. 164, No. 3, 563--597 (1994; Zbl 0843.32019)].NEWLINENEWLINENEWLINEThe generalization concerns the definition domain of the transformed functions as well as their values. The functions need not be defined on a unit disc in \(\mathbb {C}\) as in the classical case but on hermitian symmetric spaces \(G/K\) of non-compact type. The center \(K\) is supposed to be of positive dimension. The functions may have their values in any holomorphically induced discrete representations of \(G\) generalizing the results in the above mentioned paper [A. Unterberger, loc. cit.].NEWLINENEWLINEThe author proves several properties of the considered transform, e.g., that the transform is an integral one. This new transform turns out to be bounded on the space of square integrable functions on \(G/K \times o,\) where \(o\) denotes the co-adjoint orbit of \(K\) associated with \(\rho ,\) a regularity condition. At the end, an appropriate generalization of the Stratonovich-Weyl correspondence is presented.