Zeta integrals for \(\operatorname{GSp}(4)\) via Bessel models (Q723586)

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 of integrals form a fractional ideal in an appropriate field of rational functions \(\mathbb{C}(q^{-s}),\) where \(q\) is the cardinality of the residual field of \(F.\) In this way, one attaches an \(L\)-function to this representation (and this Bessel model). So, there are two types of \(L\)-functions introduced in this way and it turns out that the one attached to the simplified zeta integral is a sort of regularization of the second one (so they might be the same). The so-called exceptional poles appear when these two functions do not agree. To be able to do the analysis of the zeta integrals, the authors examine the asymptotics of the Bessel functions, restricted to a certain subgroup of \(\operatorname{GSp}(4,F).\) To be able to do that, the authors introduce the so-called Jacquet-Waldspurger module of an admissible representation of \(\operatorname{GSp}(4,F).\) This is, essentially, the usual Jacquet functor with respect to the Siegel parabolic subgroup followed by the Waldpurger functor on \(\mathrm{GL}(2,F).\) The authors main contributions lie in that they uniformly treat the asymptotics of the Bessel functions for all the irreducible representations, not just the unitarizable ones (like in the Piatetski-Shapiro's work). Their results are complete in the case of non-split Bessel models (there the Jacquet-Waldspurger functor behaves better). They also complete some missing arguments in the Piatetski-Shapiro's work.