 
Summary: Covariant Symbolic Calculi on Real Symmetric Domains
Jonathan Arazy, Harald Upmeier
We introduce the concept of ''covariant symbolic calculus'' on real and complex sym
metric domains, prove a general product formula for the link transform (generalized
Berezin transform) between two such calculi, and describe a basic example (Toeplitz
calculus) in more detail.
1. Introduction
The complex hermitian spaces of noncompact type, realized as bounded sym
metric domains D ae C n , are a fundamental class of noncompact K¨ahler manifolds
whose quantization, e.g. by the wellknown BerezinToeplitz operators, has been
studied intensively [BLU], [UU]. Writing D = G=K for a semisimple Lie group
G of hermitian type and its maximal compact subgroup K, the quantization map
A : C 1 (D) ! L(H)
(1.1)
f 7! A f
realized by (possibly unbounded) operators on a complex Hilbert space H should
satisfy the covariance condition
A f ffig \Gamma1 = U(g) A f U(g \Gamma1 )
for all g 2 G, where U denotes an irreducible (projective) representation of G act
ing on H . In [AU1] a general theory concerning such ''covariant quantizations'' on
