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Representation Theory

ISSN 1088-4165



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.

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Additional Information

Rebecca A. Herb
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland, 20742
MR Author ID: 84600

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