Test vectors for non‐Archimedean Godement–Jacquet zeta integrals

Abstract

Given an induced representation of Langlands type (π, Vπ) of GLn(F) with F non-Archimedean, we show that there exist explicit choices of matrix coefficient β and Schwartz-Bruhat function Φ for which the Godement-Jacquet zeta integral Z(s, β, Φ) attains the L-function L(s, π).

The local period integrals and essential vectors

“…The strong test vector problem is pursued by Kurinczuk and Matringe [39] for a pair of discrete series representations π and σ, though the space $$\mathcal {W}(\sigma ,\psi ^{-1})$$ is enlarged to the Whittaker model for the standard module associated to σ. Recently, Humphries [21] completed the case of standard L ‐factors $$L(s,\pi )$$. There has been renowned work [4] involved in determining an explicit Whittaker function and a characteristic function for local Asai L ‐functions $$L(s,\pi ,As)$$, but again, the strong test vector is the Paskunas–Stevens's cut‐off Whittaker function relying on the type theory of Bushnell–Kutzko.…”

“…The key formulation is constructed independently by Matringe [44, Corollary 3.2] and Miyauchi [50, Theorem 4.1], which generalizes Shintani's method for spherical representations. In the spirit of [20], it would be interesting to find analogous weak test vectors attached to essential Whittaker functions for archimedean GL n ‐type local Zeta integrals [22] (slightly different problems concerning cohomological vectors have been suggested in [12, section 1.6‐2]).…”