Algèbres de Jordan et équations de Hua. (Jordan algebras and Hua's equations) (Q1084236)

Suppose \(X=G/K\) is a hermitian symmetric domain of tube type and \(S=K/L\) is its Shilov boundary. It is possible to characterize those functions f on X which are Poisson transforms of hyperfunctions on S; if \(\tilde f\) is the right - K - invariant extension of \(f\) to \(G\), \textit{K. D. Johnson} and \textit{A. Koranyi} [Ann. Math. II. Ser. 111, 589-608 (1980; Zbl 0468.32007)] showed that these f's are characterized by the condition that \(\tilde f\) be in the kernel of a certain differential operator. This generalizes some results of L. K. Hua in particular cases. In Invent. Math. 77, 129-161 (1984; Zbl 0582.32042)], the author sharpened the results by using a different differential operator, namely the contraction of \(\partial\) and \({\bar \partial}\) by the Jordan product associated to \(X\). The present article is devoted to translating the latter result, which involves a differential operator on \(G\), to a condition expressed in terms of \(X\), under the additional assumption that \(f\) is bounded. This is done for each of three canonical realizations of \(X\): (i) the unbounded realization as a tube over a domain of positivity, (ii) the realization as a bounded symmetric domain à la Harish-Chandra, and (iii) the author's realization using polar coordinates [Bull. Soc. Math. Fr. 111, 181-192 (1983; Zbl 0576.32042)]. In each case, the condition can be expressed in terms of Jordan algebras, and the characterizations associated to realizations (i) and (iii) exhibit a striking similarity in terms of the Jordan triple product.