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

TL;DR

This paper derives explicit formulas for Bessel models of the Steinberg representation of GSp(4), providing criteria for their existence, and constructs integral representations for local and global L-functions, including special value results.

Contribution

It introduces explicit formulas and criteria for Bessel models of Steinberg representations of GSp(4), and develops integral representations for associated L-functions, linking local and global aspects.

Findings

Explicit formulas for Bessel models of Steinberg representations.

Criteria for existence and uniqueness of Bessel models.

Integral representations for local and global 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 \pi 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 \pi to be a test vector. We apply the formulas for the test vector to obtain an integral representation of the local L-function of \pi twisted by any irreducible, admissible representation of GL(2,F). Together with results in \cite{Fu} and \cite{PS2}, 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 GL(2,A). A special value result for this L-function in the spirit of Deligne's conjecture is obtained.