A Global Trace Formula for Reductive Lie Algebras and the Harish-Chandra Transform on the Space of Characteristic Polynomials

TL;DR

This paper develops a trace formula analogue for reductive Lie algebras, establishing an integral transform on characteristic polynomial spaces with a Poisson-type summation formula derived from the Arthur-Selberg trace formula.

Contribution

It introduces a new global trace formula for reductive Lie algebras and constructs an integral transform on characteristic polynomial spaces, extending harmonic analysis tools.

Findings

Derived a Poisson-type summation formula for the transform

Established an analogue of the Arthur-Selberg trace formula for Lie algebras

Constructed an integral transform between nonstandard test function spaces

Abstract

In this paper an integral transform between spaces of nonstandard test functions on the affine space of dimension n is constructed. The integral transform satisfies a summation formula of Poisson type, which is derived from an analogue of the Arthur-Selberg trace formula for the Lie algebra of n by n matrices.