Steinberg representation of GSp(4): Bessel models and integral representation ofL-functions

Abstract

We obtain explicit formulas for the test vector in the Bessel model, and derive the criteria for existence and uniqueness of Bessel models for the unramified quadratic twists of the Steinberg representation π of GSp 4 (F), where F is a nonarchimedean local field of characteristic zero. We also give precise criteria for the Iwahori spherical vector in π to be a test vector. We apply the formulas for the test vector to obtain an integral representation of the local L-function of π, twisted by any irreducible admissible representation of GL 2 (F). Using results of Furusawa and of Pitale and Schmidt, we derive from this an integral representation for the global L-function of the irreducible cuspidal automorphic representation of GSp 4 (A) obtained from a Siegel cuspidal Hecke newform, with respect to a Borel congruence subgroup of square-free level, twisted by any irreducible cuspidal automorphic representation of GL 2 (A). A special-value result for this L-function, in the spirit of Deligne's conjecture, is obtained.