  We try to give hereafter an answer to some open questions about the
definition of conserved quantities in Chern-Simons theory, with particular
reference to Chern-Simons AdS_3 Gravity. Our attention is focused on the
problem of global covariance and gauge invariance of the variation of Noether
charges. A theory which satisfies the principle of covariance on each step of
its construction is developed, starting from a gauge invariant Chern-Simons
Lagrangian and using a recipe developed in gr-qc/0110104 and gr-qc/0107074 to
calculate the variation of conserved quantities. The problem to give a
mathematical well-defined expression for the infinitesimal generators of
symmetries is pointed out and it is shown that the generalized Kosmann lift of
spacetime vector fields leads to the expected numerical values for the
conserved quantities when the solution corresponds to the BTZ black hole. The
fist law of black holes mechanics for the BTZ solution is then proved and the
transition between the variation of conserved quantities in Chern-Simons AdS_3
Gravity theory and the variation of conserved quantities in General Relativity
is analysed in detail.
