Journal of Lie Theory, Vol. 10, No. 1, pp. 1-32 (2000)

Invariant differential operators and holomorphic function spaces

Zhimin Yan

28 Cranmore Road
Braintree, MA 02184, USA

Abstract: In this paper, we are interested in studying the algebra $D(\Omega)$ of invariant differential operators on a symmetric cone $\Omega$. We will give some sets of generators of $D(\Omega)$ and calculate the eigenvalues of spherical functions under those generators. The explicit construction of our invariant differential operators in $D(\Omega)$ leads to introducing some differential operators on an irreducible bounded symmetric domain $D$ in a complex vector space $Z$. Some interesting results are obtained about these differential operators and their applications to the study of spaces of holomorphic functions on $D$ are given.

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