Analogous to the holomorphic discrete series of S l ( 2 , R ) Sl(2,\mathbb R) there is a continuous family { π r } \{\pi _r\} , − 1 > r > ∞ -1>r>\infty , of irreducible unitary representations of G G , the simply-connected covering group of S l ( 2 , R ) Sl(2,\mathbb R) . A construction of this series is given in this paper using classical function theory. For all r r the Hilbert space is L 2 ( ( 0 , ∞ ) ) L_2((0,\infty )) . First of all one exhibits a representation, D r D_r , of g = L i e G \mathfrak g=Lie\,G by second order differential operators on C ∞ ( ( 0 , ∞ ) ) C^\infty ((0,\infty )) . For x ∈ ( 0 , ∞ ) x\in (0,\infty ) , − 1 > r > ∞ -1>r>\infty and n ∈ Z + n\in \mathbb Z_+ let φ n ( r ) ( x ) = e − x x r 2 L n ( r ) ( 2 x ) \varphi _n^{(r)}(x)= e^{-x}x^{\frac {r}{2}}L_n^{(r)}(2x) where L n ( r ) ( x ) L_n^{(r)}(x) is the Laguerre polynomial with parameters { n , r } \{n,r\} . Let H r H C \mathcal H_r^{HC} be the span of φ n ( r ) \varphi _n^{(r)} for n ∈ Z + n\in \mathbb Z_+ . Next one shows, using a famous result of E. Nelson, that D r | H r H C D_r|{\mathcal H}_r^{HC} exponentiates to the unitary representation π r \pi _r of G G . The power of Nelson’s theorem is exhibited here by the fact that if 0 > r > 1 0>r>1 , one has D r = D − r D_r=D_{-r} , whereas π r \pi _r is inequivalent to π − r \pi _{-r} . For r = 1 2 r=\frac 12 , the elements in the pair { π 1 2 , π − 1 2 } \{\pi _{\frac {1}{2}},\pi _{-\frac {1}{2}}\} are the two components of the metaplectic representation. Using a result of G.H. Hardy one shows that the Hankel transform is given by π r ( a ) \pi _r(a) where a ∈ G a\in G induces the non-trivial element of a Weyl group. As a consequence, continuity properties and enlarged domains of definition, of the Hankel transform follow from standard facts in representation theory. Also, if J r J_r is the classical Bessel function, then for any y ∈ ( 0 , ∞ ) y\in (0,\infty ) , the function J r , y ( x ) = J r ( 2 x y ) J_{r,y}(x)=J_r(2\sqrt {xy}) is a Whittaker vector. Other weight vectors are given and the highest weight vector is given by a limiting behavior at 0 0 .