 
Summary: Weyl Calculus for Complex and Real Symmetric
Domains
Jonathan Arazy Harald Upmeier
Abstract
We define the Weyl functional calculus for real and complex symmetric do
mains, and compute the associated Weyl transform in the rank 1 case.
0 Introduction
In the theory of pseudodifferential operators the Weyl calculus (a quantization method
for the cotangent bundle T # (R n )) is of basic importance since it allows the full sym
plectic group Sp(2n; R) as covariance group and the relationship between operators
and symbols has optimal continuity properties. Unterberger [10, 11] has introduced
an analogous Weyl calculus for (curved) hermitian symmetric spaces of noncompact
type and computed the Weyl transform in the simplest case of the unit disk. The
higher dimensional case is more difficult. In this paper we define the Weyl calculus
for real symmetric domains and then determine the Weyl transform for all symmetric
spaces of rank 1. The new feature is the appearance of a hypergeometric function
in the spectral decomposition, indicating that the harmonic analysis underlying the
Weyl calculus involves (multivariable) special functions in a significant way.
1 Real symmetric domains and quantization Hilbert spaces
Real bounded symmetric domains, as defined in [7], are those Riemannian symmetric
