  Group representations play a central role in theoretical physics. In
particular, in quantum mechanics unitary --- or, in general, projective unitary
--- representations implement the action of an abstract symmetry group on
physical states and observables. More specifically, a major role is played by
the so-called square integrable representations. Indeed, the properties of
these representations are fundamental in the definition of certain families of
generalized coherent states, in the phase-space formulation of quantum
mechanics and the associated star product formalism, in the definition of an
interesting notion of function of quantum positive type, and in some recent
applications to the theory of open quantum systems and to quantum information.
