Symbolic calculi and the duality of homogeneous spaces. (English) Zbl 0536.58031

Microlocal analysis, Proc. Conf., Boulder/Colo. 1983, Contemp. Math. 27, 237-252 (1984).
[For the entire collection see Zbl 0527.00007.]
This is a report on some recent developments on the calculus of operators, the origin of which is traced back to the well-known calculi of Weyl and Wick. With the common aim of introducing new correspondences between symbols and operators, two different suggestions towards generalizations of these calculi to the case of non-Euclidean phase space have been made by F. A. Berezin in the seventies, and by the author in 1980: a comparison is made between the two methods in the case when the phase space is the Poincaré half-plane. Finally, we indicate how links between this latter theory and the classical Weyl calculus enable one to recover (in the special case of the group \(SL(2,{\mathbb{R}}))\) results related to the non-Euclidean Radon transformation.


58J40 Pseudodifferential and Fourier integral operators on manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
43A85 Harmonic analysis on homogeneous spaces


Zbl 0527.00007