On the symmetric square. Unstable Twisted Characters

TL;DR

This paper computes the local twisted character of a specific PGL(3) representation, providing a new proof of the fundamental lemma related to symmetric square lifting without relying on prior automorphic theory.

Contribution

It offers a purely local, representation-theoretic computation of twisted characters, independently establishing the fundamental lemma for symmetric square lifting.

Findings

Confirmed matching orbital integrals for PGL(2) and PGL(3) spherical functions.

Provided a local proof of the fundamental lemma in this context.

Demonstrated independence from automorphic lifting theory.

Abstract

We provide a purely local computation of the (elliptic) twisted (by "transpose-inverse") character of the representation \pi=I(\1) of PGL(3) over a p-adic field induced from the trivial representation of the maximal parabolic subgroup. This computation is independent of the theory of the symmetric square lifting of [IV] of automorphic and admissible representations of SL(2) to PGL(3). It leads to a proof of the (unstable) fundamental lemma in the theory of the symmetric square lifting, namely that corresponding spherical functions (on PGL(2) and PGL(3)) are matching: they have matching orbital integrals.