A unitary representation of SL(2,R)

A given unitary representation of the group SL(2,R), belonging to the discrete series, is shown to involve necessarily some special functions (in particular, Laguerre and Hardy–Pollaczek polynomials). Various realizations of this representation are investigated, including the coherent states one. More generally, it is shown that the representations of the discrete series of the universal covering of SL(2,R) involves generalized Laguerre and Pollaczek polynomials. The Riemann zeta function is shown to be concerned with these representations.