Abstract:Given a Hamiltonian $G$-space $(M,\omega, G)$, one way to define a ``quantization'' of $M$ is in terms of a virtual representation $Q(M) = \ker(P) - \ker(P^*)$ of $G$, where $P$ is a $G$-invariant elliptic differential operator on $M$, whose trace is given in terms of the equivariant index of $P$.
If instead we are given the action of a compact Lie group $G$ on a contact manifold $(M,E)$ preserving the contact distribution $E\subset TM$, we must modify our approach: to obtain an analogous result of any interest, we must chose a differential operator that is $G$-transversally elliptic: that is, an operator whose principal symbol is invertible only in directions transverse to the $G$-orbits. In this case, the representation $Q(M)$ is no longer finite-dimensional, and its trace is defined only as a generalised function on $G$.
The equivariant index of such operators was defined by Atiyah, and a character formula was given by Berline and Vergne which involves the integration of non-compactly supported forms over $T^*M$. Using recent work by Paradan and Vergne (arxiv:0801:2822) we show how the formula of Berline and Vergne can be re-written in terms of compactly supported forms, provided one allows equivariant differential forms with generalised coefficients. It then becomes possible to integrate over the fibres of $T^*M$, and we obtain a surprisingly simple formula as a result.