On combinatorics of the Arthur trace formula, convex polytopes, and toric varieties

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