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 GSp4(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 GL2(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 GSp4(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.
Keywords
Steinberg representation, Siegel modular forms,
L-functions, special values of
L-functions
