## Derivatives and asymptotics of Whittaker functions

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- by Nadir Matringe
- Represent. Theory
**15**(2011), 646-669 - DOI: https://doi.org/10.1090/S1088-4165-2011-00397-6
- Published electronically: September 26, 2011
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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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## Bibliographic 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
- Email: Nadir.Matringe@math.univ-poitiers.fr
- 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.
- © Copyright 2011 American Mathematical Society
- Journal: Represent. Theory
**15**(2011), 646-669 - MSC (2010): Primary 22E50, 22E35
- DOI: https://doi.org/10.1090/S1088-4165-2011-00397-6
- MathSciNet review: 2833471