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

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Abstract: This article examines in detail the matrix-valued gamma function \[ {\Gamma ^{{\lambda ^0}}} (\alpha ) = \int _P {{e^{ - {\text {tr}}(r)}}{\lambda ^0}(r, \bar r){{(\det r)}^{\alpha - 2}} } dr\] 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.

- L. C. Biedenharn and H. van Dam (eds.),
*Quantum theory of angular momentum. A collection of reprints and original papers*, Academic Press, New York-London, 1965. MR**0198829**
D. M. Brink and G. R. Satchler, - 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**
R. Godement, - 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** - 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** - 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**, DOI https://doi.org/10.1016/0022-1236%2877%2990030-1 - 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**435290**, DOI https://doi.org/10.1090/S0002-9904-1977-14302-4
K. I. Gross and W. J. Holman III, - L. K. Hua,
*Harmonic analysis of functions of several complex variables in the classical domains*, American Mathematical Society, Providence, R.I., 1963. Translated from the Russian by Leo Ebner and Adam Korányi. MR**0171936** - 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**, DOI https://doi.org/10.1016/0022-1236%2877%2990005-2 - A. P. Jucis and A. A. Bandzaĭtis,
*Teoriya momenta kolichestva dvizheniya v kvantovoĭ mekhanike*, Izdat. “Mintis”, Vilnius, 1965 (Russian). Academy of Sciences of the Lithuanian SSR. Institute of Physics and Mathematics; Publication No. 6. MR**0200023** - A. W. Knapp and K. Okamoto,
*Limits of holomorphic discrete series*, J. Functional Analysis**9**(1972), 375–409. MR**0299726**, DOI https://doi.org/10.1016/0022-1236%2872%2990017-1 - 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**, DOI https://doi.org/10.1016/0022-1236%2873%2990056-6 - 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**480883**, DOI https://doi.org/10.1007/BF02392042 - Lucy Joan Slater,
*Generalized hypergeometric functions*, Cambridge University Press, Cambridge, 1966. MR**0201688** - N. Ja. Vilenkin,
*Special functions and the theory of group representations*, Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, R. I., 1968. Translated from the Russian by V. N. Singh. MR**0229863**
N. Wallach, - 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**
W. J. Holman III,

*Angular momentum*, Clarendon Press, Oxford, 1962.

*Fonctions automorphes*, Séminaire Cartan, University of Paris, 1957-1958.

*Matrix-valued special functions and representation theory of the conformal group*. II:

*The generalized Bessel functions*(in preparation). M. Hammermesh,

*Group theory and its application to physical problems*, Addison-Wesley, Reading, Mass., 1964. 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.

*Analytic continuation for the holomorphic discrete series*. I, II, Trans. Amer. Math. Soc.

**251**(1979), 1-17; 19-37.

*Summation theorems for hypergeometric series in U*(

*n*), SLAM J. Math. Anal. (to appear). ---,

*Generalized Bessel functions and the representation theory of U*(2) $U(2) \text {\textcircled {$𝜎$}} \textbf {C}^{2 \times 2}$, J. Math. Phys. (to appear).

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Keywords:
Conformal group,
matrix-valued gamma function,
representation theory,
holomorphic discrete series,
special functions

Article copyright:
© Copyright 1980
American Mathematical Society