Derivatives and asymptotics of Whittaker functions

Nadir Matringe

Derivatives and asymptotics of Whittaker functions

Representation Theory 2016-09-27 v2

Authors: Nadir Matringe

Abstract

Let $F$ be a $p$ -adic field, and $G_n$ one of the groups $GL(n,F)$ , $GSO(2n-1,F)$ , $GSp(2n,F)$ , 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 $G_n$ .

Keywords

Whittaker functions p-adic groups mirabolic subgroup derivative functors generic representations discrete series

Cite

@article{arxiv.1004.1315,
  title  = {Derivatives and asymptotics of Whittaker functions},
  author = {Nadir Matringe},
  journal= {arXiv preprint arXiv:1004.1315},
  year   = {2016}
}

Comments

We correct the statement of Theorem 2.1. We had forgotten the contribution of the zero derivatives to the asymptotic expansion of Whittaker functions on the torus. In fact they don't contribute, or more precisely they do via Schwartz functions which vanish at zero. The proof remains unchanged except for some details. Submission arXiv:1005.5629 was a duplication of this one

Derivatives and asymptotics of Whittaker functions

Abstract

Keywords

Cite

Comments

Related papers