  The Noether-charge realization and the Hamiltonian realization for the
$\diff({\cal M})$ algebra in diffeomorphism invariant gravitational theories
are studied in a covariant formalism. For the Killing vector fields, the
Nother-charge realization leads to the mass formula as an entire vanishing
Noether charge for the vacuum black hole spacetimes in general relativity and
the corresponding first law of the black hole mechanics. It is analyzed in
which sense the Hamiltonian functionals form the $\diff({\cal M})$ algebra
under the Poisson bracket and shown how the Noether charges with respect to the
diffeomorphism generated by vector fields and their variations in general
relativity form this algebra. The asymptotic behaviors of vector fields
generating diffeomorphism of the manifold with boundaries are discussed. In
order to get more precise estimation for the "central extension" of the
algebra, it is analyzed in the Newman-Penrose formalism and shown that the
"central extension" for a large class of vector fields is always zero on the
Killing horizon. It is also checked whether the Virasoro algebra may be picked
up by choosing the vector fields near the horizon. The conclusion is
unfortunately negative.
