On the spectral side of Arthur's trace formula-combinatorial setup

Abstract

In Arthur's trace formula, a ubiquitous role is played by certain limiting expressions arising from piecewise smooth functions with respect to projec-tions of the Coxeter fan ((G, M)-families). These include terms resulting from intertwining operators on the spectral side and volumes of polytopes on the geometric side. We introduce the combinatorial concept of a com-patible family with respect to an arbitrary polyhedral fan and obtain new formulas for the corresponding limiting expressions in this general frame-work. Our formulas can be regarded as algebraic generalizations of certain volume formulas for convex polytopes. In a companion paper, the results are used to study the spectral side of the trace formula.