  Let $M$ be a compact Riemannian manifold endowed with an isometric action of
a compact Lie group. The method of the Witten deformation is used to compute
the virtual representation-valued equivariant index of a transversally
elliptic, first order differential operator on $M$. The multiplicities of
irreducible representations in the index are expressed in terms of local
quantities associated to the isolated singular points of an equivariant bundle
map that is locally Clifford multiplication by a Killing vector field near
these points.
