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

Author:
Rebecca A. Herb

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

MSC (2000):
Primary 22E30, 22E45

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

Published electronically:
November 16, 2001

MathSciNet review:
1870601

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Abstract | References | Similar Articles | Additional Information

Abstract: Let $G$ be a connected reductive $p$-adic group and let $\mathfrak g$ be its Lie algebra. Let $\mathcal O$ be a $G$-orbit in $\mathfrak g$. Then the orbital integral $\mu _{\mathcal O}$ corresponding to $\mathcal O$ is an invariant distribution on $\mathfrak g$, and Harish-Chandra proved that its Fourier transform $\hat \mu _{\mathcal O }$ is a locally constant function on the set $\mathfrak g’$ of regular semisimple elements of $\mathfrak g$. Furthermore, he showed that a normalized version of the Fourier transform is locally bounded on $\mathfrak g$. Suppose that $\mathcal O$ is a regular semisimple orbit. Let $\gamma$ be any semisimple element of $\mathfrak g$, and let $\mathfrak m$ be the centralizer of $\gamma$. We give a formula for $\hat \mu _{\mathcal O }(tH)$ (in terms of Fourier transforms of orbital integrals on $\mathfrak m$), for regular semisimple elements $H$ in a small neighborhood of $\gamma$ in $\mathfrak m$ and $t\in F^{\times }$ sufficiently large. We use this result to prove that Harish-Chandra’s normalized Fourier transform is globally bounded on $\mathfrak g$ in the case that $\mathcal O$ is a regular semisimple orbit.

- Harish-Chandra,
*Admissible invariant distributions on reductive $p$-adic groups*, University Lecture Series, vol. 16, American Mathematical Society, Providence, RI, 1999. With a preface and notes by Stephen DeBacker and Paul J. Sally, Jr. MR**1702257** - Rebecca A. Herb,
*Orbital integrals on $p$-adic Lie algebras*, Canad. J. Math.**52**(2000), no. 6, 1192–1220. MR**1794302**, DOI https://doi.org/10.4153/CJM-2000-050-x - 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**, DOI https://doi.org/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

MR Author ID:
84600

Email:
rah@math.umd.edu

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