  Let G_R be a real reductive Lie group acting on a manifold M. M.Kashiwara and
W.Schmid in [KaSchm] constructed representations of G_R using sheaves and
quasi-G_R-equivariant 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 in [L3] and the integral
character formula proved by W.Schmid and K.Vilonen in [SchV2] (originally
established by W. Rossmann in [Ro]) in the important special case when the
manifold M is the flag variety of the complexified Lie algebra of G_R. In the
special case when G_R is commutative and the D-module is the sheaf of sections
of a G_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.
