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An equivariant Guillemin trace formula

Peter Hochs, Hemanth Saratchandran

TL;DR

This work extends Guillemin's trace formula to an equivariant setting for proper, cocompact actions by unimodular groups, introducing a flat $g$-trace that handles distributional kernels. Under a $g$-nondegeneracy condition, the authors prove a global trace formula that expresses the equivariant trace as a delocalised orbital sum over $l$ and conjugacy data, weighted by determinants of linearised Poincaré-type maps and primitive periods. The construction connects naturally to the classical flat trace (when the group is trivial) and to universal-cover analyses via sums over conjugacy classes, ensuring consistency with known non-equivariant results. These results underpin the authors' broader program to develop an equivariant Ruelle dynamical zeta-function and explore links to equivariant analytic torsion and conjectures of Fried, highlighting potential applications in index theory and dynamical systems with symmetry.

Abstract

Guillemin's trace formula is an expression for the distributional trace of an operator defined by pulling back functions along a flow on a compact manifold. We obtain an equivariant generalisation of this formula, for proper, cocompact group actions. This is motivated by the construction of an equivariant version of the Ruelle dynamical $ζ$-function in another paper by the same authors, which is based on the equivariant Guillemin trace formula. To obtain this formula, we first develop an equivariant version of the distributional trace that appears in Guillemin's formula and other places.

An equivariant Guillemin trace formula

TL;DR

This work extends Guillemin's trace formula to an equivariant setting for proper, cocompact actions by unimodular groups, introducing a flat -trace that handles distributional kernels. Under a -nondegeneracy condition, the authors prove a global trace formula that expresses the equivariant trace as a delocalised orbital sum over and conjugacy data, weighted by determinants of linearised Poincaré-type maps and primitive periods. The construction connects naturally to the classical flat trace (when the group is trivial) and to universal-cover analyses via sums over conjugacy classes, ensuring consistency with known non-equivariant results. These results underpin the authors' broader program to develop an equivariant Ruelle dynamical zeta-function and explore links to equivariant analytic torsion and conjectures of Fried, highlighting potential applications in index theory and dynamical systems with symmetry.

Abstract

Guillemin's trace formula is an expression for the distributional trace of an operator defined by pulling back functions along a flow on a compact manifold. We obtain an equivariant generalisation of this formula, for proper, cocompact group actions. This is motivated by the construction of an equivariant version of the Ruelle dynamical -function in another paper by the same authors, which is based on the equivariant Guillemin trace formula. To obtain this formula, we first develop an equivariant version of the distributional trace that appears in Guillemin's formula and other places.

Paper Structure

This paper contains 14 sections, 24 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and results
  3. The flat $g$-trace
  4. Equivariant flows
  5. Relation with the classical flat trace
  6. The equivariant Guillemin trace formula
  7. Proofs of properties of the flat $g$-trace
  8. Independence of the cutoff function
  9. The flat $g$-trace for equivariant flows
  10. A relation with the classical flat trace
  11. Proof of the equivariant trace formula
  12. A local expression near individual fixed points
  13. Globalising Lemma \ref{['lem fixed pt loc']}
  14. Proof of Theorem \ref{['prop fixed pt gen']}

Key Result

Proposition 2.9

The integral eq def flat g trace is independent of the function $\chi \in C^{\infty}_c(M)$ satisfying eq cutoff fn.

Theorems & Definitions (67)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Proposition 2.9
  • Lemma 2.10
  • ...and 57 more