  Near Horizon Extremal Geometries (NHEG), are geometries which may appear in
the near horizon region of the extremal black holes. These geometries have
$SL(2,\mathbb{R})\!\times\!U(1)^n$ isometry, and constitute a family of
solutions to the theory under consideration. In the first part of this report,
their thermodynamic properties are reviewed, and their three universal laws are
derived. In addition, at the end of the first part, the role of these laws in
black hole thermodynamics is presented.
  In the second part of this thesis, we review building their classical phase
space in the Einstein-Hilbert theory. The elements in the NHEG phase space
manifold are built by appropriately chosen coordinate transformations of the
original metric. These coordinate transformations are generated by some vector
fields, dubbed "symplectic symmetry generators." To fully specify the phase
space, we also need to identify the symplectic structure. In order to fix the
symplectic structure, we use the formulation of Covariant Phase Space method.
The symplectic structure has two parts, the Lee-Wald term and a boundary
contribution. The latter is fixed requiring on-shell vanishing of the
symplectic current, which guarantees the conservation and integrability of the
symplectic structure, and leads to the new concept of "symplectic symmetry."
Given the symplectic structure, we construct the corresponding conserved
charges, the "symplectic symmetry generators." We also specify the explicit
expression of the charges as a functional over the phase space. These symmetry
generators constitute the "NHEG algebra," which is an infinite dimensional
algebra (may be viewed as a generalized Virasoro), and admits a central
extension which is equal to the black hole entropy.

General Relativity and Quantum Cosmology;High Energy Physics - Theory