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

Author:
Bertram Kostant

Journal:
Represent. Theory **4** (2000), 181-224

MSC (2000):
Primary 22D10, 22E70, 33Cxx, 33C10, 33C45, 42C05, 43-xx, 43A65

DOI:
https://doi.org/10.1090/S1088-4165-00-00096-0

Published electronically:
April 26, 2000

MathSciNet review:
1755901

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Abstract | References | Similar Articles | Additional Information

Analogous to the holomorphic discrete series of there is a continuous family , , of irreducible unitary representations of , the simply-connected covering group of . A construction of this series is given in this paper using classical function theory. For all the Hilbert space is . First of all one exhibits a representation, , of by second order differential operators on . For , and let where is the Laguerre polynomial with parameters . Let be the span of for . Next one shows, using a famous result of E. Nelson, that exponentiates to the unitary representation of . The power of Nelson's theorem is exhibited here by the fact that if , one has , whereas is inequivalent to . For , the elements in the pair are the two components of the metaplectic representation. Using a result of G.H. Hardy one shows that the Hankel transform is given by where 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 is the classical Bessel function, then for any , the function is a Whittaker vector. Other weight vectors are given and the highest weight vector is given by a limiting behavior at .

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Additional Information

**Bertram Kostant**

Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Email:
kostant@math.mit.edu

DOI:
https://doi.org/10.1090/S1088-4165-00-00096-0

Received by editor(s):
December 2, 1999

Received by editor(s) in revised form:
January 21, 2000

Published electronically:
April 26, 2000

Additional Notes:
Research supported in part by NSF grant DMS-9625941 and in part by the KG&G Foundation

Article copyright:
© Copyright 2000
American Mathematical Society