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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
Published electronically: November 16, 2001
MathSciNet review: 1870601
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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.

References [Enhancements On Off] (What's this?)

  • 1. Harish-Chandra, Admissible invariant distributions on reductive $p$-adic groups, Preface and notes by S. DeBacker and P.J. Sally, Jr., University Lecture Series, Vol. 16, Amer. Math. Soc., Providence, R.I., 1999. MR 2001b:22015
  • 2. R. Herb, Orbital integrals on $p$-adic Lie algebras, Canadian J. Math. 52 (6) (2000), 1192-1220. MR 2001k:22021
  • 3. J.-L. Waldspurger, Une formule des traces locale pour les algebres de Lie $p$-adiques, J. Reine Angew. Math. 465 (1995), 41-99. MR 96i:22039

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Additional Information

Rebecca A. Herb
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland, 20742

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

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