Some results on Bessel functionals for GSp(4)

TL;DR

This paper proves the existence and uniqueness of Bessel functionals for irreducible admissible representations of GSp(4) over non-archimedean fields, providing explicit classifications in various cases.

Contribution

It establishes the existence and uniqueness of Bessel functionals for a broad class of GSp(4) representations and explicitly classifies these functionals.

Findings

Every irreducible admissible GSp(4) representation admits a Bessel functional.

Explicit classification of split Bessel functionals for these representations.

Uniqueness of Bessel functionals in the considered cases.

Abstract

We prove that every irreducible, admissible representation of GSp(4,F), where F is a non-archimedean local field of characteristic zero, admits a Bessel functional, provided the representation is not one-dimensional. Given such a representation, we explicitly determine the set of all split Bessel functionals admitted by the representation, and prove that these functionals are unique. If the representation is not supercuspidal, or in an L-packet with a non-supercuspidal representation, we explicitly determine the set of all Bessel functionals admitted by the representation, and prove that these functionals are unique.