The equivariant index of transversally elliptic operators (Q1911135)

This is a companion paper to [\textit{N. Berline} and \textit{M. Vergne}, Invent. Math. 124, 11-49 (1996; Zbl 0847.46037)], and contains the proofs of results previously announced by the authors in [Lect. Notes Math. 1565, 157-202 (1993; Zbl 0793.58032)]. It is shown that the cohomological index of a transversally elliptic operator (introduced in the first paper) coincides with its analytic index. The proof of the index theorem follows Atiyah's axiomatic approach: several functorial properties are established that characterize both indices uniquely, and both indices are explicitly calculated and compared in special cases.