Character Formulae For Representations Research Articles

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 χ ( M , F ) = 1 ( 2 π ) dim C M ∫ Ch ( F ) χ g C ˜ as distributions on g R , where Ch ( F ) is the characteristic cycle of F and χ g C ˜ is the Euler form of M extended to the cotangent space T ∗ 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.