On a generalized Forelli–Rudin construction

Abstract

We generalize the classical construction of Forelli, Rudin and Ligocka, expressing the Bergman kernel of a Hartogs domain in terms of weighted Bergman kernels on its base, to the situation when the “fiber” of the Hartogs domain is, instead of the disc or the ball, an arbitrary irreducible bounded symmetric domain.

Keywords:

• Cartan domain
• Bergman kernel
• Szegö kernel
• Hartogs domain
• Forelli–Rudin construction
• 2000 Mathematics Subject Classifications: Primary 32M15
• Secondary 32A07
• 32A25
• 32A36

Acknowledgements

This article was written while the first author was visiting the second; it is a pleasant duty for him to thank the Chalmers Tekniska Högskola and Göteborg University for their hospitality and support. The authors also thank the referee for pointing out several misprints and inaccuracies in the first version of this article.

The research of the first author was supported by GA AV ČR grant no. A1019304 and the research of the second author was supported by the Swedish Science Council (VR).

Notes

¹ The hypothesis of irreducibility is unessential and is made only for the sake of simplicity.

² The support of dσ is actually a fairly small subset of ∂Ω ^F , related to the Shilov boundary of F.

³ The polynomials Ψ _j ;ν coincide, up to a constant factor of jν ^j , with the Jack symmetric polynomials ; see 4, p. 378 and Example 1 on p. 383.