Zeta integrals for \(\operatorname{GSp}(4)\) via Bessel models
type: Item
instance of: scholarly article
title: Zeta integrals for \(\operatorname{GSp}(4)\) via Bessel models
zbMATH Open document ID: 1412.11076
author: Ralf Schmidt
author: Long Tran
published in: Pacific Journal of Mathematics
publication date: 2018-07-24T00:00:00Z
review text: In this paper, the authors give ``a revised treatment of Piatetski-Shapiro's theory of zeta integrals and \(L\)-factors for irreducible, admissible representations of \(\operatorname{GSp}(4,F)\) via Bessel models.'' (from the abstract). Let us explain in more detail the main results and the tools used. Using ideas of Tate, Jacquet, Langlands, among others, Piatetski-Shapiro, Shalika and others developed the theory of local and global \(L\)-functions for \(\mathrm{GL}(n),\) primarily using Whittaker models. It is well known that every infinite dimensional, irreducible and admissible representation of \(\mathrm{GL}(2)\) is generic (i.e., has a non-zero Whittaker model). For other algebraic groups, even those of small rank, this is not true. Already \(\operatorname{GSp}(4)\) has many irreducible admissible representations which are not generic. Novodvorsky and Piatetski-Shapiro introduced Bessel models in 1970's. The important thing is that every irreducible, admissible infinite-dimensional representation of \(\operatorname{GSp}(4,F)\) (where we fix, form now on, \(F\) to be a non-Archimedean field of characteristic zero) has a Bessel model of some kind. We can divide Bessel models of an irreducible admissible representation of \(\operatorname{GSp}(4,F)\) to split and non-split ones. Similarly to Whittaker model, the existing non-zero Bessel model on an irreducible admissible representation \(\pi\) of \(\operatorname{GSp}(4,F)\) gives rise to the space of functions on \(\operatorname{GSp}(4,F),\) the so-called Bessel functions. Similarly to the case of \(\mathrm{GL}(n,F)\) and the Whittaker functions, one can define (and the authors use Piatetski-Shapiro definition from 1997) certain zeta integral using these Bessel functions, where one integrates over certain subgroup of \(\operatorname{GSp}(4,F).\) Following Piatetski-Shapiro, the authors consider two types of this zeta integral-the simplified zeta integral and (full) zeta integral. In an expected way, those families…
reviewed by: Marcela Hanzer
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 6909767
zbMATH Keywords: \(L\)-factors
zbMATH Keywords: \(\operatorname{GSp}(4)\)
zbMATH Keywords: Bessel models
zbMATH Keywords: Jacquet-Waldspurger modules
MaRDI profile type: Publication
full work available at URL: pjm.2018.296.437
OpenAlex ID: W2883394594
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DOI: 10.2140/PJM.2018.296.437
description: scientific article; zbMATH DE number 6909767