Riemann-Roch-Hirzebruch integral formula for characters of reductive Lie groups

Let $G_{\mathbb R}$ be a real reductive Lie group acting on a manifold $M$. M. Kashiwara and W. Schmid constructed representations of $G_{\mathbb R}$ using sheaves and quasi- $G_{\mathbb R}$-equivariant ${\mathcal D}$-modules on $M$. In this article we prove an integral character formula for these representations (Theorem 1). Our main tools will be the integral localization formula recently proved by the author and the integral character formula proved by W. Schmid and K. Vilonen (originally established by W. Rossmann) in the important special case when the manifold $M$ is the flag variety of $\mathbb C \otimes_{\mathbb R} \mathfrak g_{\mathbb R}$--the complexified Lie algebra of $G_{\mathbb R}$. In the special case when $G_{\mathbb R}$ is commutative and the ${\mathcal D}$-module is the sheaf of sections of a $G_{\mathbb R}$-equivariant line bundle over $M$ this integral character formula will reduce to the classical Riemann-Roch-Hirzebruch formula. As an illustration we give a concrete example on the enhanced flag variety.