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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Positive definite spherical functions
on Ol$'$shanskii domains

Authors: Joachim Hilgert and Karl-Hermann Neeb
Journal: Trans. Amer. Math. Soc. 352 (2000), 1345-1380
MSC (1991): Primary 22E46; Secondary 22A25
Published electronically: May 21, 1999
MathSciNet review: 1473443
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Abstract: Let $G$ be a simply connected complex Lie group with Lie algebra $\mathfrak{g}$, $\mathfrak{h}$ a real form of $\mathfrak{g}$, and $H$ the analytic subgroup of $G$ corresponding to $\mathfrak{h}$. The symmetric space ${\mathcal{M}}=H\backslash G$ together with a $G$-invariant partial order $\le $ is referred to as an Ol$'$shanskii space. In a previous paper we constructed a family of integral spherical functions $\phi _{\mu }$ on the positive domain ${\mathcal{M}}^{+} := \{Hx\colon Hx\ge H\}$ of ${\mathcal{M}}$. In this paper we determine all of those spherical functions on ${\mathcal{M}}^{+}$ which are positive definite in a certain sense.

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Additional Information

Joachim Hilgert
Affiliation: Mathematisches Institut, Technische Universität Clausthal, Erzstr. $1$, 38678 Claus- thal-Zellerfeld, Germany

Karl-Hermann Neeb
Affiliation: Mathematisches Institut, Universität Erlangen, Bismarckstr. $1\frac{1}2$, 91054 Erlangen, Germany
Address at time of publication: Fachbereich Mathematik, Technische Universität Darmstadt, Schloßgartenstr. 7, 64289 Darmstadt, Germany

Keywords: Positive definite function, ordered symmetric space, holomorphic representation, spherical function, involutive semigroup
Published electronically: May 21, 1999
Article copyright: © Copyright 1999 American Mathematical Society