  We construct the NHEG phase space, the classical phase space of Near-Horizon
Extremal Geometries with fixed angular momenta and entropy, and with the
largest symmetry algebra. We focus on vacuum solutions to $d$ dimensional
Einstein gravity. Each element in the phase space is a geometry with
$SL(2,\mathbb R)\times U(1)^{d-3}$ isometries which has vanishing $SL(2,\mathbb
R)$ and constant $U(1)$ charges. We construct an on-shell vanishing symplectic
structure, which leads to an infinite set of symplectic symmetries. In four
spacetime dimensions, the phase space is unique and the symmetry algebra
consists of the familiar Virasoro algebra, while in $d>4$ dimensions the
symmetry algebra, the NHEG algebra, contains infinitely many Virasoro
subalgebras. The nontrivial central term of the algebra is proportional to the
black hole entropy. This phase space and in particular its symmetries might
serve as a basis for a semiclassical description of extremal rotating black
hole microstates.
