Equivariant de Rham torsions

The purpose of this paper is to give an explicit local formula for the difference of two natural versions of equivariant analytic torsion in de Rham theory. This difference is the sum of the integral of a Chern-Simons current and of a new invariant, the V -invariant of an odd dimensional manifold equipped with an action of a compact Lie group. The V -invariant localizes on the critical manifolds of invariant Morse-Bott functions.