The index of G-transversally elliptic families I

TL;DR

This paper develops an index theory for families of G-transversally elliptic operators, extending Atiyah-Singer results, and computes their Kasparov intersection product using functorial properties and advanced operator algebra techniques.

Contribution

It introduces a new index map for G-transversally elliptic families and establishes its fundamental properties, extending classical index theory to a broader setting.

Findings

Defined the index map for G-transversally elliptic families.

Proved axiomatic properties extending Atiyah-Singer index theorem.

Computed Kasparov intersection product for the index class.

Abstract

We define and study the index map for families of $G$-transversally elliptic operators and introduce the multiplicity for a given irreducible representation as a virtual bundle over the base of the fibration. We then prove the usual axiomatic properties for the index map extending the Atiyah-Singer results [1]. Finally, we compute the Kasparov intersection product of our index class against the K-homology class of an elliptic operator on the base. Our approach is based on the functorial properties of the intersection product, and relies on some constructions due to Connes-Skandalis and to Hilsum-Skandalis.