zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A weighted Plancherel formula. III: The case of the hyperbolic matrix ball. (English) Zbl 0836.43018

Summary: [Part I by J. Peetre, L. Peng and G. Zhang, A weighted Plancherel formula. The case of the unit disk. Application of Hankel operators. Technical report (Stockholm 1990); Part II by G. Zhang in Stud. Math. 102, 103-120 (1992; Zbl 0811.43003)].

The group SU(2,2) acts naturally on an L 2 -space on a hyperbolic matrix ball (type one bounded symmetric domain) with respect to the usual weighted measure. We will find the corresponding invariant Laplace operator and study its special resolution. The spherical functions (K- invariant eigenfunctions) can be expressed using hypergeometric functions. It turns out that, besides the weighted Bergman space, some discrete parts enter into the decomposition. The number of the discrete parts equals the number of the orbits of the Weyl group action on the zeros (in the “lower half plane”) of the generalized Harish-Chandra 𝐜-function. We calculate their reproducing kernels in a special case. As an application, we obtain decompositions of the tensor products of holomorphic discrete series representations. This improves an earlier result by J. Repka.

MSC:
43A85Analysis on homogeneous spaces
22E46Semisimple Lie groups and their representations
43A65Representations of groups, semigroups, etc. (abstract harmonic analysis)
43A90Spherical functions (abstract harmonic analysis)
32M15Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (analytic spaces)