Torus actions on compact quotients

TL;DR

This paper establishes a formula connecting the orbits of a noncompact torus acting on a compact quotient to its action on tangential cohomology, extending Lefschetz and index theorems to noncompact group actions.

Contribution

It introduces a novel formula relating compact orbits of a noncompact torus to tangential cohomology, providing a new perspective on equivariant index theory for noncompact actions.

Findings

Derived a Lefschetz-type formula for noncompact torus actions

Linked orbit structure to tangential cohomology via harmonic analysis

Extended index theorem concepts to noncompact group actions

Abstract

We consider the action of a noncompact torus H on the compact quotient G/L, where G is a Lie group containing H and L is a uniform lattice in G. Using harmonic analysis on G we prove a formula relating the compact orbits of H to the action of H on the (infinite dimensional) tangential cohomology. The formula may be viewed as a Lefschetz formula or an equivariant index theorem for noncompact group actions.