An equivariant Guillemin trace formula

TL;DR

This paper extends Guillemin's trace formula to an equivariant setting involving proper, cocompact group actions, enabling new applications in dynamical systems and representation theory.

Contribution

It develops an equivariant version of Guillemin's trace formula and the associated distributional trace, broadening its applicability to group actions.

Findings

Derived an equivariant Guillemin trace formula.

Constructed an equivariant distributional trace.

Enabled applications to equivariant dynamical zeta functions.

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 $\zeta$-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.