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Derivatives and asymptotics of Whittaker functions

Author: Nadir Matringe
Journal: Represent. Theory 15 (2011), 646-669
MSC (2010): Primary 22E50, 22E35
Published electronically: September 26, 2011
MathSciNet review: 2833471
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Abstract: Let $ F$ be a $ p$-adic field and $ G_n$ be one of the groups $ GL(n,F)$, $ GSO(2n-1,F)$, $ GSp(2(n-1),F)$, or $ GSO(2(n-1),F)$. Using the mirabolic subgroup or analogues of it, and related ``derivative'' functors, we give an asymptotic expansion of functions in the Whittaker model of generic representations of $ G_n$, with respect to a minimal set of characters of subgroups of the maximal torus. Denoting by $ Z_n$ the center of $ G_n$ and by $ N_n$ the unipotent radical of its standard Borel subgroup, we characterize generic representations occurring in $ L^2(Z_nN_n\backslash G_n)$ in terms of these characters.

This is related to a conjecture of Lapid and Mao for general split groups, asserting that the generic representations occurring in $ L^2(Z_nN_n\backslash G_n)$ are the generic discrete series; we prove it for the group $ G_n$.

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Additional Information

Nadir Matringe
Affiliation: School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, United Kingdom
Address at time of publication: Laboratoire de Mathématiques et Applications, Université de Poitiers, Téléport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962, Futuroscope Chasseneuil Cedex, France

Received by editor(s): April 7, 2010
Received by editor(s) in revised form: September 12, 2010, and October 6, 2010
Published electronically: September 26, 2011
Additional Notes: This work was supported by the EPSRC grant EP/G001480/1.
Article copyright: © Copyright 2011 American Mathematical Society

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