Index of transversally elliptic D-modules

We consider the action of a complex Lie group G on a complex manifold X, a G-quasi-equivariant D-X-module, M, and a R-constructible sheaf, F, on X, eq uivariant for the action of a real form, G(R), of G. Under transversal elli pticity hypothesis on the characteristic varieties of M and F, we associate to these data a hyperfunction on G(R), by a microlocal product of characte ristic classes. We show that if G(R) is compact this hyperfunction correspo nds to the generalized trace of the action of G(R) in the global solutions of M x F. This remains true if G(R) is a semi-simple Lie group acting on it s flag manifold, which gives a proof of a character formula of Kashiwara. ( C) 2001 Editions scientifiques et medicales Elsevier SAS.