Index of transversally elliptic operators

In 1996, Berline and Vergne gave a cohomological formula for the index of a transversally elliptic operator. In this paper we propose a new point of view where the cohomological formulae make use of equivariant Chern characters with generalized coefficients and with compact suppport. This kind of Chern characters was studied by the authors in a previous paper (see arXiv:0801.2822).

💡 Research Summary

This paper revisits the index theory of transversally elliptic operators, providing a fresh cohomological framework that unifies and extends the earlier formulas of Berline and Vergne. A transversally elliptic operator (P) on a compact manifold (M) equipped with a compact Lie group (K) is elliptic only in directions transverse to the (K)-orbits. Consequently, its principal symbol (\sigma) is invertible on the complement of the conormal bundle (T^{}_{K}M), but the support of (\sigma) may be non‑compact in the full cotangent bundle. The classical Berline‑Vergne formula expresses the equivariant index as an integral over (T^{}M) of the product of the equivariant Chern character (\operatorname{Ch}{c}(\sigma)(X)) and the square of the equivariant (\hat A)-genus (\hat A(M)^{2}(X)). However, the non‑compact support of (\operatorname{Ch}{c}(\sigma)) makes the integral ill‑defined in many transversally elliptic situations.

The authors introduce two key constructions to overcome this difficulty. First, they define a generalized inverse of the equivariant differential (D\omega) associated with the Liouville one‑form (\omega) on (T^{*}M). The inverse is realized as the distributional integral \