Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Matrix-valued special functions and representation theory of the conformal group. I. The generalized gamma function


Authors: Kenneth I. Gross and Wayne J. Holman
Journal: Trans. Amer. Math. Soc. 258 (1980), 319-350
MSC: Primary 22E70; Secondary 33A75, 43A75, 81C40
DOI: https://doi.org/10.1090/S0002-9947-1980-0558177-5
MathSciNet review: 558177
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This article examines in detail the matrix-valued gamma function

$\displaystyle {\Gamma ^{{\lambda ^0}}}\,(\alpha )\, = \,\int_P {{e^{ - {\text{tr}}(r)}}{\lambda ^0}(r,\,\bar r){{(\det \,r)}^{\alpha - 2}}\,} dr$

associated to the conformal group $ G\, = \,U(2,\,2)$. Here, $ \alpha $ is a continuous complex parameter, $ {\lambda ^0}$ runs through a family of ``weights'' of $ K\, = \,U(2)\, \times \,U(2)$, P is the cone of $ 2\, \times \,2$ positive-definite Hermitian matrices, and the integral is well known to converge absolutely for $ {\text{Re}}(\alpha )\, > \,1$. However, until now very little has been known about the analytic continuation for the general weight $ {\lambda ^0}$. The results of this paper include the following: The complete analytic continuation of $ {\Gamma ^{{\lambda ^0}}}$ is determined for all weights $ {\lambda ^0}$. In analogy to the case of the classical gamma function it is proved that for any weight $ {\lambda ^0}$ the mapping $ \alpha \, \to \,{\Gamma ^{{\lambda ^0}}}\,{(\alpha )^{ - 1}}$ is entire. A new integral formula is given for the inverse of the gamma function. An explicit calculation is given for the normalized variant of the gamma matrix that arises in the reproducing kernel for the spaces in which the holomorphic discrete series of G is realized, and one observes that the behavior of the analytic continuation for weights ``in general position'' is markedly different from the special cases in which the gamma function ``is scalar". The full analytic continuation of the holomorphic discrete series for G is determined. The gamma function for the forward light cone (the boundary orbit) is found, and the associated Hardy space of vector-valued holomorphic functions is described. Analogs are given for some of the well-known formulas for the classical gamma function. As an epilogue, applications of the matrix-valued gamma function, such as generalizations to $ 2\, \times \,2$ matrix space of the classical binomial theorem, are announced. These applications require a detailed understanding of the (generalized) Bessel functions associated to the conformal group that will be treated in the sequel to this paper.

References [Enhancements On Off] (What's this?)

  • [1] Quantum theory of angular momentum. A collection of reprints and original papers, Edited by L. C. Biedenharn and H. van Dam, Academic Press, New York-London, 1965. MR 0198829
  • [2] D. M. Brink and G. R. Satchler, Angular momentum, Clarendon Press, Oxford, 1962.
  • [3] Stephen Gelbart, Bessel functions, representation theory, and automorphic functions, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 343–345. MR 0344380
  • [4] R. Godement, Fonctions automorphes, Séminaire Cartan, University of Paris, 1957-1958.
  • [5] Kenneth I. Gross and Ray A. Kunze, Fourier Bessel transforms and holomorphic discrete series, Conference on Harmonic Analysis (Univ. Maryland, College Park, Md., 1971), Springer, Berlin, 1972, pp. 79–122. Lecture Notes in Math., Vol. 266. MR 0486318
  • [6] Kenneth I. Gross and Ray A. Kunze, Generalized Bessel transforms and unitary representations, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 347–350. MR 0344381
  • [7] Kenneth I. Gross and Ray A. Kunze, Bessel functions and representation theory. II. Holomorphic discrete series and metaplectic representations, J. Functional Analysis 25 (1977), no. 1, 1–49. MR 0453928
  • [8] Kenneth I. Gross, Wayne J. Holman III, and Ray A. Kunze, The generalized gamma function, new Hardy spaces, and representations of holomorphic type for the conformal group, Bull. Amer. Math. Soc. 83 (1977), no. 3, 412–415. MR 0435290, https://doi.org/10.1090/S0002-9904-1977-14302-4
  • [9] K. I. Gross and W. J. Holman III, Matrix-valued special functions and representation theory of the conformal group. II: The generalized Bessel functions (in preparation).
  • [10] M. Hammermesh, Group theory and its application to physical problems, Addison-Wesley, Reading, Mass., 1964.
  • [11] Harish-Chandra, Representations of semi-simple Lie groups. IV, V, VI, Amer. J. Math. 77 (1955), 743-777; 78 (1956), 1-41; 78 (1956), 564-628.
  • [12] L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, Translated from the Russian by Leo Ebner and Adam Korányi, American Mathematical Society, Providence, R.I., 1963. MR 0171936
  • [13] Hans Plesner Jakobsen and Michele Vergne, Wave and Dirac operators, and representations of the conformal group, J. Functional Analysis 24 (1977), no. 1, 52–106. MR 0439995
  • [14] A. P. Jucis and A. A. Bandzaĭtis, \cyr Teoriya momenta kolichestva dvizheniya v kvantovoĭ mekhanike., Academy of Sciences of the Lithuanian SSR. Institute of Physics and Mathematics. Publication No. 6, Izdat. “Mintis”, Vilnius, 1965 (Russian). MR 0200023
  • [15] A. W. Knapp and K. Okamoto, Limits of holomorphic discrete series, J. Functional Analysis 9 (1972), 375–409. MR 0299726
  • [16] Hugo Rossi and Michèle Vergne, Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions and the application to the holomorphic discrete series of a semisimple Lie group, J. Functional Analysis 13 (1973), 324–389. MR 0407206
  • [17] M. Vergne and H. Rossi, Analytic continuation of the holomorphic discrete series of a semi-simple Lie group, Acta Math. 136 (1976), no. 1-2, 1–59. MR 0480883, https://doi.org/10.1007/BF02392042
  • [18] Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. MR 0201688
  • [19] N. Ja. Vilenkin, Special functions and the theory of group representations, Translated from the Russian by V. N. Singh. Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, R. I., 1968. MR 0229863
  • [20] N. Wallach, Analytic continuation for the holomorphic discrete series. I, II, Trans. Amer. Math. Soc. 251 (1979), 1-17; 19-37.
  • [21] E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469
  • [22] W. J. Holman III, Summation theorems for hypergeometric series in U(n), SLAM J. Math. Anal. (to appear).
  • [23] -, Generalized Bessel functions and the representation theory of U(2) $ U(2)\circlebin{\sigma}{\textbf{C}^{2 \times 2}}$, J. Math. Phys. (to appear).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E70, 33A75, 43A75, 81C40

Retrieve articles in all journals with MSC: 22E70, 33A75, 43A75, 81C40


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0558177-5
Keywords: Conformal group, matrix-valued gamma function, representation theory, holomorphic discrete series, special functions
Article copyright: © Copyright 1980 American Mathematical Society