Se p 20 09 Steinberg representation of GSp 4 : Bessel models and integral representation of L - functions

Abstract. We obtain explicit formulas for the test vector in the Bessel model and derive the criteria for existence and uniqueness for Bessel models for the unramified, quadratic twists of the Steinberg representation π of GSp 4 (F ), where F is a non-archimedean 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 GL2(F ). Together with results in [4] and [11], we derive 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 GL2(A). A special value result for this L-function in the spirit of Deligne’s conjecture is obtained.