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Analogous to the holomorphic discrete series of Sl(2;R )t here is a continuous familyfrg, 1 <r <1, of irreducible unitary representations of G, the simply-connected covering group of Sl(2;R). A construction of this series is given in this paper using classical function theory. For all r the Hilbert space is L2((0;1)). First of all one exhibits a representation, Dr, of g = LieG by second order dierential operators on C1((0;1)). For x2 (0;1), 1 <r< 1 and n2 Z+ let ' (r) n (x )= e xx r 2L (r) n (2x )w hereL (r) n (x) (r) n for n2 Z+. Next one shows, using a famous result of E. Nelson, that DrjHHC r exponentiates to the unitary representation r of G.T he power of Nelson's theorem is exhibited here by the fact that if 0 <r< 1, one has Dr = D r ,w hereasr is inequivalent to r .F orr = 1 2 , the elements in the pairf 1 2 ; 1 2g are the two components of the metaplectic representation. Using a result of G.H. Hardy one shows that the Hankel transform is given by r(a )w herea2 G induces the non-trivial element of a Weyl group. As a consequence, continuity properties and enlarged domains of denition, of the Hankel transform follow from standard facts in representation theory. Also, if Jr is the classical Bessel function, then for any y 2 (0;1), the function Jr;y (x )= Jr(2 p xy) is a Whittaker vector. Other weight vectors are given and the highest weight vector is given by a limiting behavior at 0.

On Laguerre polynomials, Bessel functions, Hankel transform and a series in the unitary dual of the simply-connected covering group of 𝑆𝑙(2,ℝ)