# Equivariant Algebraic Index Theorem

## TL;DR

This paper establishes a version of the algebraic index theorem that accounts for symmetries given by a discrete group of automorphisms, extending classical index theorems to equivariant settings.

## Contribution

It proves a -equivariant algebraic index theorem, generalizing previous index theorems to include group actions on deformation quantizations of symplectic manifolds.

## Key findings

- Proves a -equivariant algebraic index theorem.
- Connects to Connes-Moscovici's transversal index theorem.
- Extends the algebra of pseudodifferential operators with group actions.

## Abstract

We prove a {\Gamma}-equivariant version of the algebraic index theorem, where {\Gamma} is a discrete group of automorphisms of a formal deformation of a symplectic manifold. The particular cases of this result are the algebraic version of the transversal index theorem related to the theorem of A. Connes and H. Moscovici for hypoelliptic operators and the index theorem for the extension of the algebra of pseudodifferential operators by a group of diffeomorphisms of the underlying manifold due to A. Savin, B. Sternin, E. Schrohe and D. Perrot.