Positive definite spherical functions on Olshanskii domains
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- by Joachim Hilgert and Karl-Hermann Neeb PDF
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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$’$shanskiĭ 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.References
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Additional Information
- Joachim Hilgert
- Affiliation: Mathematisches Institut, Technische Universität Clausthal, Erzstr. $1$, 38678 Claus- thal-Zellerfeld, Germany
- Email: hilgert@math.tu-clausthal.de
- 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
- MR Author ID: 288679
- Email: neeb@mi.uni.erlangen.de
- Published electronically: May 21, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 1345-1380
- MSC (1991): Primary 22E46; Secondary 22A25
- DOI: https://doi.org/10.1090/S0002-9947-99-02184-4
- MathSciNet review: 1473443