Bounds for Fourier transforms of regular orbital integrals on -adic Lie algebras

Author:
Rebecca A. Herb

Journal:
Represent. Theory **5** (2001), 504-523

MSC (2000):
Primary 22E30, 22E45

Published electronically:
November 16, 2001

MathSciNet review:
1870601

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Abstract: Let be a connected reductive -adic group and let be its Lie algebra. Let be a -orbit in . Then the orbital integral corresponding to is an invariant distribution on , and Harish-Chandra proved that its Fourier transform is a locally constant function on the set of regular semisimple elements of . Furthermore, he showed that a normalized version of the Fourier transform is locally bounded on . Suppose that is a regular semisimple orbit. Let be any semisimple element of , and let be the centralizer of . We give a formula for (in terms of Fourier transforms of orbital integrals on ), for regular semisimple elements in a small neighborhood of in and sufficiently large. We use this result to prove that Harish-Chandra's normalized Fourier transform is globally bounded on in the case that is a regular semisimple orbit.

**1.**Harish-Chandra,*Admissible invariant distributions on reductive 𝑝-adic groups*, University Lecture Series, vol. 16, American Mathematical Society, Providence, RI, 1999. Preface and notes by Stephen DeBacker and Paul J. Sally, Jr. MR**1702257****2.**Rebecca A. Herb,*Orbital integrals on 𝑝-adic Lie algebras*, Canad. J. Math.**52**(2000), no. 6, 1192–1220. MR**1794302**, 10.4153/CJM-2000-050-x**3.**J.-L. Waldspurger,*Une formule des traces locale pour les algèbres de Lie 𝑝-adiques*, J. Reine Angew. Math.**465**(1995), 41–99 (French). MR**1344131**, 10.1515/crll.1995.465.41

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Additional Information

**Rebecca A. Herb**

Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland, 20742

Email:
rah@math.umd.edu

DOI:
https://doi.org/10.1090/S1088-4165-01-00125-X

Received by editor(s):
March 14, 2001

Published electronically:
November 16, 2001

Additional Notes:
Supported in part by NSF Grant DMS 0070649

Article copyright:
© Copyright 2001
American Mathematical Society