Derivatives and asymptotics of Whittaker functions

Matringe, Nadir

doi:10.1090/S1088-4165-2011-00397-6

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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