Let G be a complex semi-simple group with a compact maximal group K and an irreducible holomorphic representation ρ on a finite dimensional space V. There exists on V a K-invariant Hermitian scalar product. Let Ω be the intersection of the unit ball of V with the G-orbit of a dominant vector. Ω is a generalization of the unit ball (case obtained for G = SL(n,C) and ρ the natural representation on Cn).
We prove that for such manifolds, the Bergman and Szegö kernels as for the ball are rational fractions of the scalar products and these fractions can be computed explicitely, using invariants of ρ. To compute this kernels, one uses a good orthonormal basis related to ρ, and then proves that one has a rational fraction, using Schur's orthogonality relations and Weyl's dimensional formula for V.
@article{urn:eudml:doc:41167,
title = {Les noyaux de Bergman et Szeg\"o pour des domaines strictment pseudo-convexes qui g\'en\'eralisent la boule.},
journal = {Publicacions Matem\`atiques},
volume = {36},
year = {1992},
pages = {65-72},
zbl = {0765.32014},
language = {fr},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41167}
}
Loeb, Jean-Jacques. Les noyaux de Bergman et Szegö pour des domaines strictment pseudo-convexes qui généralisent la boule.. Publicacions Matemàtiques, Tome 36 (1992) pp. 65-72. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41167/