Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Mahdi Asgari, Kiumars Kaveh

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.

On Combinatorics of the Arthur Trace Formula, Convex Polytopes, and 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.

Paper Structure

This paper contains 20 sections, 27 theorems, 132 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cones and fans
  4. Polytopes
  5. Normal fan
  6. Nearest face partition
  7. Conical decomposition theorems
  8. Khovanskii-Pukhlikov virtual polytopes and convex chains
  9. Incidence algebra of a poset and Möbius inversion
  10. Convergence
  11. Polynomiality
  12. An extension of Langlands combinatorial lemma
  13. Proof of polynomiality
  14. Toric varieties
  15. Background on toric varieties
  16. ...and 5 more sections

Key Result

Theorem 1

Assume that the fan $\Sigma$ above is acute (cf. Definition acute). With the notation as above, suppose for any $\sigma_2 \preceq \sigma_1$, the function $K_{\sigma_1, \sigma_2}$ is rapidly decreasing on the shifted neighborhoods of $S^{\sigma_1}_{\sigma_2}$. (See Theorem th-growth-conv for the prec is absolutely convergent.

Figures (17)

  • Figure 1: Left: a complete simplicial fan in $V=\mathbb R^2$; we have labeled three cones in the fan. Right: a polygon normal to the fan and regions obtained by drawing the outward face cones; the function $k_\Delta$ in the shaded region is given by $K_0 -K_1-K_2+K_{12}$.
  • Figure 2:
  • Figure 3: Inward and outward tangent cones at a vertex (left inward, right outward)
  • Figure 4: Inward and outward tangent cones at an edge (left inward, right outward)
  • Figure 5: A polygon and its normal fan. Note that in our convention we use outward facet normals to define the cones in the normal fan.
  • ...and 12 more figures

Theorems & Definitions (60)

  • Theorem : Convergence
  • Theorem : Polynomiality
  • Theorem : Convergence, discrete version
  • Theorem : Polynomiality, discrete version
  • Theorem 2.4: Volume polynomial
  • Theorem 2.5: Ehrhart polynomial
  • Theorem 2.6
  • Remark 2.7
  • Proposition 2.9
  • Theorem 2.10: Brianchon-Gram
  • ...and 50 more