Are the quasinormal modes scalar quantities? I am studying the so-called quasinormal modes (QNMs) in the context of the AdS/CFT correspondence and I got stuck.
For instance, if I choose a weird patch of coordinates for the, say, AdS5-Schwarzschild black hole, will I get the same frequencies for the QNMs? Or it will transform as a component of a 4-vector?
I think that the frequencies should be scalar quantities because one can relate the poles of the Green's functions to transport coefficients of the field theory. And, of course, Green's functions are Lorentz scalars...
E.g.: http://arxiv.org/abs/hep-th/0506184 
I know that the equations of motion for the fluctuations are diffeo invariant, but the background invariance puzzles me.
 A: The quasinormal mode frequencies are complex-valued numbers. For a given type of field (for example, scalar, vector, gravitational, fermionic, etc) there are a discrete infinity of these frequencies which are of course independent of the metric used to describe the background. If the background possess some symmetry, for example spherical symmetry, then the quasinormal modes are often decomposed into different sectors corresponding to different spherical harmonics.
More concretely, in the case of a massive scalar field the equation of motion is
$$ (\square - M^2) \phi = 0 \, , $$
and if the background spacetime is a static, spherically symmetric black hole in 4 spacetime dimensions, then a specific solution to the equations of motion corresponding can be written as
$$ \phi_{n \ell m}(x) = \psi_{n \ell m}(r) e^{-i \omega_{n \ell m} t} Y_{\ell,m}(\Omega)\, .$$
The virture of introducing the spherical harmonics and complex exponential is that the PDE equation of motion for $\phi$ simplies to an ODE for the radial profile $\psi_{n \ell m}(r)$. There are three discrete indices, $n$ is the overtone number, and $(\ell,m)$ are the usual quantum numbers of a spherical harmonic. The frequencies $\omega_{n \ell m}$ are the quasinormal mode frequencies, assuming proper boundary conditions are imposed (a very important point which I will not discuss here).
This was all worked out in a specific coordinate system. A general diffeomorphism will mix up the coordinates, so that the above form will no longer be separable, which is perhaps what you are concerned about. But This doesn't have any physical consequence, it just means that now you're solving the problem in a less-than-ideal way. 
