On Combinatorics of the Arthur Trace Formula, Convex Polytopes, and Toric Varieties

TL;DR

This paper explores the combinatorial and geometric aspects of Arthur's trace formula, connecting convex polytopes, toric varieties, and Ehrhart polynomials to clarify convergence and polynomiality properties.

Contribution

It introduces a combinatorial truncation method based on fans and polytopes, extending Arthur's results and linking them to Ehrhart polynomials and toric geometry.

Findings

Established convergence and polynomiality of truncated integrals

Extended Ehrhart polynomial concepts to Arthur's trace formula

Provided geometric interpretations on toric varieties

Abstract

We explicate the combinatorial/geometric ingredients of Arthur's proof of the convergence and polynomiality, in a truncation parameter, of his non-invariant trace formula. Starting with a fan in a real, finite dimensional, vector space and a collection of functions, one for each cone in the fan, we introduce a combinatorial truncated function with respect to a polytope normal to the fan and prove the analogues of Arthur's results on the convergence and polynomiality of the integral of this truncated function over the vector space. The convergence statements clarify the important role of certain combinatorial subsets that appear in Arthur's work and provide a crucial partition that amounts to a so-called nearest face partition. The polynomiality statements can be thought of as far reaching extensions of the Ehrhart polynomial. Our proof of polynomiality relies on the Lawrence-Varchenko conical decomposition and readily implies an extension of the well-known combinatorial lemma of Langlands. The Khovanskii-Pukhlikov virtual polytopes are an important ingredient here. Finally, we give some geometric interpretations of our combinatorial truncation on toric varieties as a measure and a Lefschetz number.