  We discuss the geometry of the Marsden-Ratiu reduction theorem for a
bihamiltonian manifold. We consider the case of the manifolds associated with
the Gel'fand-Dickey theory, i.e., loop algebras over sl(n+1). We provide an
explicit identification, tailored on the MR reduction, of the
Adler-Gel'fand-Dickey brackets with the Poisson brackets on the MR-reduced
bihamiltonian manifold N. Such an identification relies on a suitable immersion
of the space of sections of the cotangent bundle of N into the algebra of
pseudo differential operators connected to geometrical features of the theory
of (classical) W_n algebras.
