On the coefficients of an asymptotic expansion of spherical functions on symmetric spaces
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- by John J. H. Miller and D. J. Simms PDF
- Proc. Amer. Math. Soc. 37 (1973), 448-452 Request permission
Abstract:
The asymptotic expansion of a spherical function on a symmetric space of noncompact type, obtained by Harish-Chandra, is a finite linear combination of expansions of the form ${\Phi _\theta } = {\sum _\mu }’{\Gamma _\mu }(\theta ){e^{\theta - \mu }}$. In this paper it is proved that ${\lim _{t \to \infty }}’{\Gamma _\mu }(te - \bar \rho )$ is finite and rational for any e, where $\bar \rho$ is the restriction of half the sum of the positive roots.References
-
N. Bourbaki, Éléments de mathématiques. XXVI. Groupes et algèbres de Lie, Actualités Sci. Indust., no. 1285, Hermann, Paris, 1960. MR 24 #A2641.
- R. Gangolli, On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Lie groups, Ann. of Math. (2) 93 (1971), 150–165. MR 289724, DOI 10.2307/1970758
- Harish-Chandra, Spherical functions on a semisimple Lie group. I, Amer. J. Math. 80 (1958), 241–310. MR 94407, DOI 10.2307/2372786
- Sigurđur Helgason, An analogue of the Paley-Wiener theorem for the Fourier transform on certain symmetric spaces, Math. Ann. 165 (1966), 297–308. MR 223497, DOI 10.1007/BF01344014
- Sigurđur Helgason, A duality for symmetric spaces with applications to group representations, Advances in Math. 5 (1970), 1–154 (1970). MR 263988, DOI 10.1016/0001-8708(70)90037-X
- Ottmar Loos, Symmetric spaces. I: General theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0239005
- John J. H. Miller and D. J. Simms, Radial limits of the rational functions $^{\prime } \Gamma _{\mu }$ of Harish-Chandra, Proc. Roy. Irish Acad. Sect. A 68 (1969), 41–47 (1969). MR 254181
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 37 (1973), 448-452
- MSC: Primary 43A90; Secondary 22E30
- DOI: https://doi.org/10.1090/S0002-9939-1973-0312159-9
- MathSciNet review: 0312159