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Representation Theory

ISSN 1088-4165



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

Author: Bertram Kostant
Journal: Represent. Theory 4 (2000), 181-224
MSC (2000): Primary 22D10, 22E70, 33Cxx, 33C10, 33C45, 42C05, 43-xx, 43A65
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 $Sl(2,\mathbb R)$ 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 $\mathfrak g=\text{\it Lie}\,G$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 $D_r\vert{\mathcal H}_r^{HC}$ 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 $0$.

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  • [BL] L. Biedenharn and J. Louck, The Racah-Wigner algebra in quantum theory, Addison-Wesley, Reading MA, 1981. MR 83d:81002
  • [Ca] P. Cartier, Vecteurs différentiables dans les représentations unitaires des groupes de Lie, Séminaire Bourbaki, 454 (1974-1975). MR 57:534
  • [DG] H. Ding, K. J. Gross, Operator-valued Bessel functions on Jordan algebras, J. Reine Angew. Math. 435 (1993), 157-196. MR 93m:33010
  • [Ha] G. Hardy, Summation of a series of polynomials of Laguerre, Journ. London Math. Soc., 7 (1932), 138-139, 192.
  • [He] C. Herz, Bessel functions of matrix argument, Ann. Math. 61 (1955), 474-523. MR 16:1107e
  • [Ja] D. Jackson, Fourier Series and Orthogonal Polynomials, Carus Math. Monographs, 6, MAA, 1941. MR 3:230f
  • [Ko] B. Kostant, On Whittaker Vectors and Representation Theory, Inventiones math., 48, (1978), 101-184. MR 80b:22020
  • [Ne] E. Nelson, Analytic vectors, Ann. of Math. 70 (1959), 572-615. MR 21:5901
  • [Pu] L. Pukanszky, The Plancherel Formula for the Universal Covering group of $SL(\mathbb R,2)$, Math. Annalen 156 (1964), 96-143. MR 30:1215
  • [R-V] H. Rossi, M. Vergne, Analytic continuation of the holomorphic discrete series of a semi-simple Lie group, Acta Math. 136 (1976), 1-59. MR 58:1032
  • [Sc] L. Schwartz, Théorie des Distributions, II, Hermann, 1951. MR 12:833d
  • [Se] I. Segal, A class of operator algebras which are determined by groups, Duke Math. J. 18 (1951), 221-265. MR 13:534b
  • [Sz] G. Szego, Orthogonal Polynomials, AMS Colloquium Publications, 23, 1939. MR 1:14b
  • [Ta] M. Taylor, Noncommutative Harmonic Analysis, Math. Surveys and Monographs, 22, Amer. Math. Soc., 1986. MR 88a:22021
  • [Wa] G. Warner, Harmonic Analysis on Semi-Simple Lie Groups I, Grundlehren, 188, Springer-Verlag, 1972. MR 58:16979
  • [W$\ell$-1] N. Wallach, The analytic continuation of the discrete series, I. Trans. Amer. Math. Soc., 251 (1979), 1-17. MR 81a:22009
  • [W$\ell$-2] N. Wallach, The analytic continuation of the discrete series, II. Trans. Amer. Math. Soc., 251 (1979), 18-37. MR 81a:22009
  • [Wt] G. Watson, Theory of Bessel Functions, Cambridge Univ. Press, 1966.
  • [Yo] K. Yosida, Functional Analysis, Grundlehren, 123, Springer-Verlag, 1971.

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Additional Information

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

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

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