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On the computability of some positive-depth supercuspidal characters near the identity

Authors: Raf Cluckers, Clifton Cunningham, Julia Gordon and Loren Spice
Journal: Represent. Theory 15 (2011), 531-567
MSC (2010): Primary 22E50, 03C98
Published electronically: July 7, 2011
MathSciNet review: 2833466
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Abstract: This paper is concerned with the values of Harish-Chandra characters of a class of positive-depth, toral, very supercuspidal representations of $ p$-adic symplectic and special orthogonal groups, near the identity element. We declare two representations equivalent if their characters coincide on a specific neighbourhood of the identity (which is larger than the neighbourhood on which the Harish-Chandra local character expansion holds). We construct a parameter space $ B$ (that depends on the group and a real number $ r>0$) for the set of equivalence classes of the representations of minimal depth $ r$ satisfying some additional assumptions. This parameter space is essentially a geometric object defined over $ \mathbb{Q}$. Given a non-Archimedean local field $ \mathbb{K}$ with sufficiently large residual characteristic, the part of the character table near the identity element for $ G(\mathbb{K})$ that comes from our class of representations is parameterized by the residue-field points of $ B$. The character values themselves can be recovered by specialization from a constructible motivic exponential function, in the terminology of Cluckers and Loeser in a recent paper. The values of such functions are algorithmically computable. It is in this sense that we show that a large part of the character table of the group $ G(\mathbb{K})$ is computable.

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Raf Cluckers
Affiliation: Université Lille 1, Laboratoire Painlevé, CNRS - UMR 8524, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex, France, and, Katholieke Universiteit Leuven, Department of Mathematics, Celestijnenlaan 200B, B-3001 Leuven, Belgium

Clifton Cunningham
Affiliation: Department of Mathematics, University of Calgary

Julia Gordon
Affiliation: Department of Mathematics, University of British Columbia

Loren Spice
Affiliation: Department of Mathematics, Texas Christian University

Keywords: Character, orbital integral, motivic integration, supercuspidal representation
Received by editor(s): April 19, 2009
Received by editor(s) in revised form: January 29, 2010, and February 4, 2011
Published electronically: July 7, 2011
Article copyright: © Copyright 2011 by the authors