Relation of Orbital Integrals on SO(5) and PGL(2)

TL;DR

This paper establishes a relationship between orbital integrals on SO(5) and PGL(2) groups via a lifting of representations, aiding the comparison of trace formulas and impacting the understanding of the Ramanujan conjecture.

Contribution

It introduces a local fundamental lemma connecting orbital integrals on SO(5) and PGL(2) through a representation lifting, facilitating trace formula comparisons.

Findings

Relation between orbital integrals on SO(5) and PGL(2)

Implications for representation lifting from PGL(2) to PGSp(4)

Counterexamples to the Ramanujan conjecture

Abstract

We relate the "Fourier" orbital integrals of corresponding spherical functions on the p-adic groups SO(5) and PGL(2). The correspondence is defined by a "lifting" of representations of these groups. This is a local "fundamental lemma" needed to compare the geometric sides of the global Fourier summation formulae (or relative trace formulae) on these two groups. This comparison leads to conclusions about a well known lifting of representations from PGL(2) to PGSp(4). This lifting produces counter examples to the Ramanujan conjecture.