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 GRGR.As an application of this generalization, we prove an analogue of the Gauss–Bonnet theorem for constructible sheaves. If FF is a GRGR-equivariant sheaf on a complex projective manifold MM, then the Euler characteristic of MM with respect to FFχ(M,F)=1(2π)dimCM∫Ch(F)χgC˜ as distributions on gRgR, where Ch(F) is the characteristic cycle of FF and χgC˜ is the Euler form of MM extended to the cotangent space T∗MT∗M (independently of FF). 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.