ON LAGUERRE POLYNOMIALS, BESSEL FUNCTIONS, HANKEL TRANSFORM AND A SERIES IN THE UNITARY DUAL OF THE SIMPLY-CONNECTED COVERING GROUP OF Sl(2,R)

  • BERTRAM KOSTANT
  • Published 2000

Abstract

Analogous to the holomorphic discrete series of Sl(2,R) there is a continuous family {πr}, −1 < r <∞, 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,∞)). First of all one exhibits a representation, Dr , of g = LieG by second order differential operators on C∞((0,∞)). For x ∈ (0,∞), −1 < r < ∞ and n ∈ Z+ let φ n (x) = e−xx r 2L (r) n (2x) where L (r) n (x) is the Laguerre polynomial with parameters {n, r}. Let HHC r be the span of φ (r) n for n ∈ Z+. Next one shows, using a famous result of E. Nelson, that Dr |HHC r exponentiates to the unitary representation πr of G. The power of Nelson’s theorem is exhibited here by the fact that if 0 < r < 1, one has Dr = D−r, whereas πr is inequivalent to π−r . For r = 1 2 , the elements in the pair {π 1 2 , π− 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 πr(a) where a ∈ 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 Jr is the classical Bessel function, then for any y ∈ (0,∞), the function Jr,y(x) = Jr(2 √ xy) is a Whittaker vector. Other weight vectors are given and the highest weight vector is given by a limiting behavior at 0. 0. Introduction 0.1. Throughout this paper r is a real number where −1 < r <∞. The classical Laguerre polynomials, {L n (x)}, are defined for those values of r and non-negative integers n. See e.g. [Ja], p. 184 or [Sz], p. 96. We will take the normalization as defined in [Ja]. Let H be the Hilbert space L2((0,∞)) with respect to Lebesgue measure dx. Let φ n ∈ C∞((0,∞)) be defined by putting φ n (x) = e−xx r 2L (r) n (2x). We refer to the {φ n } as Laguerre functions. Then for each value of r the subset {φ n }, n = 0, 1, ..., of Laguerre functions is an orthogonal basis of H. In particular, H r is a dense subspace of H where H r is defined to be the linear span of this subset. Received by the editors December 2, 1999 and, in revised form, January 21, 2000. 2000 Mathematics Subject Classification. Primary 22D10, 22E70, 33Cxx, 33C10, 33C45, 42C05, 43-xx, 43A65. Research supported in part by NSF grant DMS-9625941 and in part by the KG&G Foundation. c ©2000 American Mathematical Society

Cite this paper

@inproceedings{KOSTANT2000ONLP, title={ON LAGUERRE POLYNOMIALS, BESSEL FUNCTIONS, HANKEL TRANSFORM AND A SERIES IN THE UNITARY DUAL OF THE SIMPLY-CONNECTED COVERING GROUP OF Sl(2,R)}, author={BERTRAM KOSTANT}, year={2000} }