Integrals of equivariant forms and a Gauss–Bonnet theorem for constructible sheaves

The Berline–Vergne integral localization formula for equivariantly closed forms ([N. Berline, M. Vergne, Classes caractéristiques équivariantes. Formules de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris 295 (1982) 539–541], Theorem 7.11 in [N. Berline, E. Getzler, M. Vergne, Heat Kernels and Dirac Operators, Springer-Verlag, 1992]) is well-known and requires the acting Lie group to be compact. In this article, we extend this result to real reductive Lie groups $G_R$.

As an application of this generalization, we prove an analogue of the Gauss–Bonnet theorem for constructible sheaves. If $F$ is a $G_R$-equivariant sheaf on a complex projective manifold $M$, then the Euler characteristic of $M$ with respect to $F$$\chi (M,F)=\frac{1}{{\left(2\pi \right)}^{\underset{C}{dim}M}}{\int }_{\mathrm{Ch}\left(F\right)}\widetilde{{\chi }_{g_C}}$ as distributions on $g_R$, where $\mathrm{Ch}\left(F\right)$ is the characteristic cycle of $F$ and $\widetilde{{\chi }_{g_C}}$ is the Euler form of $M$ extended to the cotangent space $T^{\ast }M$ (independently of $F$). We also consider an analogue of Duistermaat–Heckman measures for real reductive Lie groups acting on symplectic manifolds.

In [M. Libine, Riemann–Roch–Hirzebruch integral formula for characters of reductive Lie groups, Represent. Theory 9 (2005) 507–524. Also math.RT/0312454] I apply the results of this article to obtain a Riemann–Roch–Hirzebruch type integral formula for characters of representations of reductive groups.