I am trying to understand the Berline - Vergne Localization formula for the equivariant Index of the Dirac operator on a spin manifold M which states that the G - equivariant index of Dirac operator $Ind_{D}(X)$ where $X \in \mathfrak{g}$ is given by the following integral :
$Ind_{D}(X)= \int_{M} Ch(L,X)\hat{A}(M,X)$
Where Ch(L,X) is the G - equivariant Chern character of the twisting bundle L and $\hat{A}(M,X)$ is the G-equivariant A - genus of M. These terms are defined in e.g Berline - Getzler - Vergne (Heat Kernels and Dirac operators). For a compact Riemannian manifold $(M,g)$ acted by a compact Lie group preserving the metric,
$\hat{A}(M,X) = det^{1/2} ( \frac{(T + R)/2}{sinh (T + R)/2} )$
Where T is the endomorphis of the tangent bundle induced by the G - action and R is the curvature of the Levi - Civita connection on M.
I'm having trouble in understanding these quantities especially the $\hat{A}(M,X)$ (the square root of the determinant of sinh is bothering me). How does one think of these classes. I would greatly appreciate if someone can explain these equivariant Clases, may be in a simple example say action of rotations about z- axis on sphere. Unfortunately in the book by Berline - Getzler - Vergne there are very few examples. Please provide any simple examples which can be helpful.
