Natural Equivariant Dirac Operators

TL;DR

This paper introduces a new class of transversally elliptic differential operators on manifolds with group actions, showing they generate all equivariant index values and relate to elliptic operators.

Contribution

It defines a novel class of natural, explicitly constructed transversally elliptic operators and demonstrates their comprehensive role in generating equivariant indices.

Findings

Operators generate all equivariant index values

Representation components match those of related elliptic operators

Provides explicit construction of new transversally elliptic operators

Abstract

We introduce a new class of natural, explicitly defined, transversally elliptic differential operators over manifolds with compact group actions. Under certain assumptions, the symbols of these operators generate all the possible values of the equivariant index. We also show that the components of the representation-valued equivariant index coincide with those of an elliptic operator constructed from the original data.