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We discuss the degeneracies of 4D and 5D BPS black holes in toroidalcompactifications of M-theory or type II string theory, using U-duality as atool. We generalize the 4D/5D lift to include all charges in N=8 supergravity,and compute the exact indexed degeneracies of certain 4D 1/8-BPS black holes.Using the attractor formalism, we obtain the leading micro-canonical entropyfor arbitrary Legendre invariant prepotentials and non-vanishing D6-branecharge. In particular, we find that the N=8 prepotential is given to leadingorder by the cubic invariant of $E_6$. This suggests that the minimal unitaryrepresentation of $E_8$, based on the same cubic prepotential, underlies themicroscopic degeneracies of N=8 black holes. We propose that the exactdegeneracies are given by the Wigner function of the $E_8(Z)$ invariant vectorin this automorphic representation. A similar conjecture relates thedegeneracies of N=4 black holes to the minimal unipotent representation of$SO(8,24,Z)$.Comment: 36 pages, uses JHEP3.cls; v5: the "extra charge", which was,  mistakenly identified as angular momentum, is now correctly identified as NUT  charge (see historical note on p29), a few other cosmetic changes to

BPS black hole degeneracies and minimal automorphic representations