a normalization factor, which turns out to be dependent on only. These coherent states satisfy the temporal stability condition, and the action identity, While these generalized coherent states do form an overcomplete set in , the resolution of the identity is generally not given by an integral relation as above, but instead by an integral in Bohr's sense, like it is in use in the theory of almost periodic functions. Actually the construction of Gazeau–Klauder CS can be extended to vector CS and to Hamiltonians with degenerate spectra, as shown by Ali and Bagarello. Heat kernel coherent states Another type of coherent state arises when considering a particle whose configuration space is the group manifold of a compact Lie group K. Hall introduced coherent states in which the usual Gaussian on Euclidean space is replaced by the heat kernel on K. The parameter space for the coherent states is the "complexification" of ; e.g., if is then the complexification is . These coherent states have a resolution of the identity that leads to a Segal-Bargmann space over the complexification. Hall's results were extended to compact symmetric spaces, including spheres, by Stenzel. The heat kernel coherent states, in the case , have been applied in the theory of quantum gravity by Thiemann and his collaborators. Although there are two different Lie groups involved in the construction, the heat kernel coherent states are not of Perelomov type. The group-theoretical approach Gilmore and Perelomov, independently, realized that the construction of coherent states may sometimes be viewed as a group theoretical problem. In order to see this, let us go back for a while to the case of CCS. There, indeed, the displacement operator is nothing but the representative in Fock space of an element of the Heisenberg group (also called the Weyl–Heisenberg group), whose Lie algebra is generated by and . However, before going on with the CCS, take first the general case. Let be a locally compact group and suppose that it has a continuous, irreducible representation on a Hilbert space by unitary operators . This representation is called square integrable if there exists a non-zero vector in for which the integral converges. Here is the left invariant Haar measure on . A vector for which is said to be admissible, and it can be shown that the existence of one such vector guarantees the existence of an entire dense set of