Stratonovich-Weyl correspondence for discrete series representations

Benjamin Cahen

Address: Université de Metz, UFR-MIM, Département de mathématiques, LMMAS, ISGMP-Bât. A Ile du Saulcy 57045, Metz cedex 01, France

E-mail: cahen@univ-metz.fr

Abstract: Let $M=G/K$ be a Hermitian symmetric space of the noncompact type and let $\pi $ be a discrete series representation of $G$ holomorphically induced from a unitary character of $K$. Following an idea of Figueroa, Gracia-Bondìa and Vàrilly, we construct a Stratonovich-Weyl correspondence for the triple $(G, \pi , M)$ by a suitable modification of the Berezin calculus on $M$. We extend the corresponding Berezin transform to a class of functions on $M$ which contains the Berezin symbol of $d\pi (X)$ for $X$ in the Lie algebra $\mathfrak{g}$ of $G$. This allows us to define and to study the Stratonovich-Weyl symbol of $d\pi (X)$ for $X\in \mathfrak{g}$.

AMSclassification: primary 22E46; secondary 81S10, 46E22, 32M15.

Keywords: Stratonovich-Weyl correspondence, Berezin quantization, Berezin transform, semisimple Lie group, coadjoint orbits, unitary representation, Hermitian symmetric space of the noncompact type, discrete series representation, reproducing kernel Hilbert space, coherent states.