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

Abstract:

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