Angular momentum in curved spacetime It is known that the angular momentum components are also a representation of the $SU(2)$ generators. Given a non-trivial spacetime, say a black hole of some kind or AdS space, how can one define the action of $SU(2)$? Does one compute generalized angular momentum components for that space or is there another procedure? 
 A: First, let's go back to flat space.
The mentioned SU(2) algebra is just a part of the 10-dimensional Poincare algebra - the algebra of isometries of Minkowski space-time. The generators of isometries are called Killing vector fields, you can easily show that these (they come with the Lie bracket, as usual) obey the Poincare algebra.
Because isometries (by definition) do not change the metric, the flat-space action is left invariant. This gives you ten Noether currents, specifically the four currents forming the stress-energy tensor $T_{\nu}^{\mu}$ and six forming the angular-momentum tensor $M_{\alpha \beta}^{\mu}$.
Now to the general case.
Because each diffeomorphism is now a symmetry of the action, there are infinitely many conserved currents, but none of them can actually be called 'angular momentum'. There still is a notion of stress-energy tensor, because infinitesimal deffeomorphisms are kind of like 'local (gauge) translations'. But this tensor is only covariantly conserved (the so-called problem of energy in GR).
In order to find the conserved currents analogous to the usual angular momentum, you have to recover the algebra of Killing vector fields of your space.
