  This paper concerns the investigation of the stability properties of relative
equilibria which are rigidly rotating vortex configurations sometimes called
vortex crystals, in the N-vortex problem. Such a configurations can be
characterized as critical point of the Hamiltonian function restricted on the
constant angular impulse hypersurface in the phase space (topologically a
pseudo-sphere whose coefficients are the circulation strengths of the
vortices). Relative equilibria are generated by the circle action on the
so-called shape pseudo-sphere (which generalize the standard shape sphere
appearing in the study of the N-body problem). Inspired by the planar
gravitational $N$-body problem, and after a geometrical and dynamical
discussion, we investigate the relation intertwining the stability of relative
equilibria and the inertia indices of the central configurations generating
such equilibria. In the last section we apply our main results to some
symmetric three and four vortices relative equilibria.
