The Hua system on irreducible Hermitian symmetric spaces of nontube type

Buraczewski, Dariusz

Annales de l'Institut Fourier, Tome 54 (2004) no. 1, pp. 81-127.

Soit $G/K$ un espace hermitien symétrique non compact irréductible. On étudie un système invariant $H$ d’opérateurs differentiels sur $G/K$. Selon un théorème de K. Johnson et A. Korányi, une fonction sur un espace hermitien symétrique de type tube est annulée par le système $𝐇$ de Hua si et seulement si elle est l’intégrale de Poisson-Szegö d’une hyperfonction. N. Berline et M. Vergne ont posé la question de caractériser les fonctions $𝐇$ - harmoniques sur les espaces hermitiens de type II. Ici on montre que ce sont les fonctions pluriharmoniques.

Let $G/K$ be an irreducible Hermitian symmetric space of noncompact type. We study a $G$- invariant system of differential operators on $G/K$ called the Hua system. It was proved by K. Johnson and A. Korányi that if $G/K$ is a Hermitian symmetric space of tube type, then the space of Poisson-Szegö integrals is precisely the space of zeros of the Hua system. N. Berline and M. Vergne raised the question about the nature of the common solutions of the Hua system for Hermitian symmetric spaces of nontube type. In this paper we show that these are exactly the pluriharmonic functions.

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