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Local Character Expansions for Supercuspidal Representations of U(3)

Published online by Cambridge University Press:  20 November 2018

Fiona Murnaghan
Fiona Murnaghan*
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, M5S1A1
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Abstract

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The topic of this paper is the relationship between characters of irreducible supercuspidal representations of the p-adic unramified 3 x 3 unitary group and Fourier transforms of invariant measures on elliptic adjoint orbits in the Lie algebra. We prove that most supercuspidal representations have the property that, on some neighbourhood of zero, the character composed with the exponential map coincides with the formal degree of the representation times the Fourier transform of a measure on one elliptic orbit. For the remainder, a linear combination of the Fourier transforms of measures on two elliptic orbits must be taken. As a consequence of these relations between characters and Fourier transforms, the coefficients in the local character expansions are expressed in terms of values of Shalika germs. By calculating which of the values of the Shalika germs associated to regular nilpotent orbits are nonzero, we determine which irreducible supercuspidal representations have Whittaker models. Finally, the coefficients in the local character expansions of three families of supercuspidal representations are computed.

Keywords

22E5022E35
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 3 , 01 June 1995 , pp. 606 - 640
Copyright
Copyright © Canadian Mathematical Society 1995

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