Charge orbits of symmetric special geometries and attractors

We study the critical points of the black hole scalar potential V _BH in N = 2, d = 4 super-gravity coupled to n_V vector multiplets, in an asymptotically flat extremal black hole background described by a 2(n_V + 1)-dimensional dyonic charge vector and (complex) scalar fields which are coordinates of a special Kähler manifold. For the case of homogeneous symmetric spaces, we find three general classes of regular attractor solutions with nonvanishing Bekenstein-Hawking entropy. They correspond to three (inequivalent) classes of orbits of the charge vector, which is in a 2(n_V + 1)-dimensional representation R_V of the U-duality group. Such orbits are nondegenerate, namely they have nonvanishing quartic invariant (for rank-3 spaces). Other than the 1/2-BPS one, there are two other distinct non-BPS classes of charge orbits, one of which has vanishing central charge. The three species of solutions to the N = 2 extremal black hole attractor equations give rise to different mass spectra of the scalar fluctuations, whose pattern can be inferred by using invariance properties of the critical points of V_BH and some group theoretical considerations on homogeneous symmetric special Kähler geometry.