  Let G be a finitely connected Lie group and let K be a maximal compact
subgroup. Let M be a cocompact G-proper manifold with boundary, endowed with a
G-invariant metric which is of product type near the boundary. Under additional
assumptions on G, for example that it satisfies the Rapid Decay condition and
is such that G/K has nonpositive sectional curvature, we define higher
Atiyah-Patodi-Singer C^*-indices associated to smooth group cocycles on G and
to a generalized G-equivariant Dirac operator D on M with L^2-invertible
boundary operator D_\partial. We then establish a higher index formula for
these C^*-indices and use it in order to introduce higher genera for M, thus
generalizing to manifolds with boundary the results that we have established in
Part 1. Our results apply in particular to a semisimple Lie group G. We use
crucially the pairing between suitable relative cyclic cohomology groups and
relative K-theory groups.
