Skip to Main Content

Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Bounds for Fourier transforms of regular orbital integrals on $p$-adic Lie algebras
HTML articles powered by AMS MathViewer

by Rebecca A. Herb PDF
Represent. Theory 5 (2001), 504-523 Request permission


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.
Similar Articles
  • Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 22E30, 22E45
  • Retrieve articles in all journals with MSC (2000): 22E30, 22E45
Additional Information
  • Rebecca A. Herb
  • Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland, 20742
  • MR Author ID: 84600
  • Email:
  • 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:
  • MathSciNet review: 1870601