## 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
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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

- 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**, DOI 10.1090/ulect/016 - Rebecca A. Herb,
*Orbital integrals on $p$-adic Lie algebras*, Canad. J. Math.**52**(2000), no. 6, 1192–1220. MR**1794302**, DOI 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 10.1515/crll.1995.465.41

## 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