Bounds for Fourier transforms of regular orbital integrals on -adic Lie algebras
Author:
Rebecca A. Herb
Journal:
Represent. Theory 5 (2001), 504-523
MSC (2000):
Primary 22E30, 22E45
DOI:
https://doi.org/10.1090/S1088-4165-01-00125-X
Published electronically:
November 16, 2001
MathSciNet review:
1870601
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Abstract | References | Similar Articles | Additional Information
Abstract: Let be a connected reductive
-adic group and let
be its Lie algebra. Let
be a
-orbit in
. Then the orbital integral
corresponding to
is an invariant distribution on
, and Harish-Chandra proved that its Fourier transform
is a locally constant function on the set
of regular semisimple elements of
. Furthermore, he showed that a normalized version of the Fourier transform is locally bounded on
. Suppose that
is a regular semisimple orbit. Let
be any semisimple element of
, and let
be the centralizer of
. We give a formula for
(in terms of Fourier transforms of orbital integrals on
), for regular semisimple elements
in a small neighborhood of
in
and
sufficiently large. We use this result to prove that Harish-Chandra's normalized Fourier transform is globally bounded on
in the case that
is a regular semisimple orbit.
- 1.
Harish-Chandra, Admissible invariant distributions on reductive
-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
-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
-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
Email:
rah@math.umd.edu
DOI:
https://doi.org/10.1090/S1088-4165-01-00125-X
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