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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

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Bounds for Fourier transforms of regular orbital integrals on $p$-adic Lie algebras
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by Rebecca A. Herb
Represent. Theory 5 (2001), 504-523
DOI: https://doi.org/10.1090/S1088-4165-01-00125-X
Published electronically: November 16, 2001

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
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Bibliographic 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
  • © Copyright 2001 American Mathematical Society
  • Journal: Represent. Theory 5 (2001), 504-523
  • MSC (2000): Primary 22E30, 22E45
  • DOI: https://doi.org/10.1090/S1088-4165-01-00125-X
  • MathSciNet review: 1870601