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Bergman and Szegö kernels for strictly pseudo-convex domains which generalize the unit ball. (Les noyaux de Bergman et Szegö pour des domaines strictement pseudo- convexes qui généralisent la boule.) (French) Zbl 0765.32014
Let $$G$$ be a complex semi-simple Lie group and $$K$$ a maximal compact subgroup of $$G$$. Let $$\Lambda$$ be a dominant weight of $$G$$ and $$(\pi_ \Lambda,E_ \Lambda)$$ be the irreducible representation of $$G$$ associated to the dominant weight $$\Lambda$$. We fix a $$K$$-invariant hermitian scalar product $$(\;|\;)$$ on $$E_ \Lambda$$. Let $$\Omega$$ be the intersection of the unit ball in $$E_ \Lambda$$ with the orbit $$G.v_ \Lambda$$, where $$v_ \Lambda$$ is the dominant vector. In the case $$G=SL(n,\mathbb{C})$$, $$E_ \Lambda=\mathbb{C}^ n$$ with usual scalar product $$(\;|\;)$$ on $$\mathbb{C}^ n$$, $$\Omega$$ becomes the usual ball of $$\mathbb{C}^ n$$. Generalizing the above special case, the author proves that the Bergman (resp. Szegö) kernel of $$\Omega$$ is a rational fraction of $$(\pi_ \Lambda(g1)v_ \Lambda\mid\pi_ \Lambda(g2)v_ \Lambda)$$ $$(g1,g2\in G)$$ and can be expressed in terms of some invariants of $$G$$ and $$\pi_ \Lambda$$.

##### MSC:
 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.) 32T99 Pseudoconvex domains 22E46 Semisimple Lie groups and their representations
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