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

*P*is the cone of positive-definite Hermitian matrices, and the integral is well known to converge absolutely for . However, until now very little has been known about the analytic continuation for the general weight . The results of this paper include the following: The complete analytic continuation of is determined for all weights . In analogy to the case of the classical gamma function it is proved that for any weight the mapping 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 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.

**[1]**L. C. Biedenharn and H. Van Dam, Editors,*Quantum theory of angular momentum*, Academic Press, New York, 1965. MR**0198829 (33:6983)****[2]**D. M. Brink and G. R. Satchler,*Angular momentum*, Clarendon Press, Oxford, 1962.**[3]**S. Gelbart,*Bessel functions, representation theory, and automorphic functions*, Proc. Sympos. Pure Math., vol. 26, Amer. Math. Soc., Providence, R. I., 1973, pp. 343-345. MR**0344380 (49:9119)****[4]**R. Godement,*Fonctions automorphes*, Séminaire Cartan, University of Paris, 1957-1958.**[5]**K. I. Gross and R. A. Kunze,*Fourier-Bessel transforms and holomorphic discrete series*, Conference on Harmonic Analysis, Lecture Notes in Math., vol. 266, Springer-Verlag, Berlin and New York, 1972, pp. 79-122. MR**0486318 (58:6075)****[6]**-,*Generalized Bessel transforms and unitary representations*, Proc. Sympos. Pure Math., vol. 26, Amer. Math. Soc., Providence, R. I., 1973, pp. 347-350. MR**0344381 (49:9120)****[7]**-,*Bessel functions and representation theory*, II:*Holomorphic discrete series and metaplectic representations*, J. Functional Anal.**25**(1977), 1-49. MR**0453928 (56:12181)****[8]**K. I. Gross, W. J. Holman III and R. A. Kunze,*The generalized gamma function, new Hardy spaces, and representations of holomorphic type for the conformal group*, Bull. Amer. Math. Soc.**83**(1977), 412-415. MR**0435290 (55:8250)****[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*, Transl. Math. Mono., vol. 6, Amer. Math. Soc., Providence, R. I., 1963. MR**0171936 (30:2162)****[13]**H. Jakobsen and M. Vergne,*Wave and Dirac operators and representations of the conformal group*, J. Functional Anal.**24**(1977), 52-106. MR**0439995 (55:12876)****[14]**A. P. Jucys and A. A. Bandzaitis,*Theoriya momenta kolichestva dvizheniya v kvantovoi mechanike*, ``Mintis", Vilnius, 1965. MR**0200023 (33:8163)****[15]**A. Knapp and K. Okamoto,*Limits of holomorphic discrete series*, J. Functional Anal.**9**(1972), 375-409. MR**0299726 (45:8774)****[16]**H. Rossi and M. Vergne,*Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions and the application to the holomorphic discrete series of a semi-simple Lie group*, J. Functional Anal.**13**(1973), 324-389. MR**0407206 (53:10989)****[17]**-,*Analytic continuation of the holomorphic discrete series of a semi-simple Lie group*, Acta Math.**136**(1976), 1-59. MR**0480883 (58:1032)****[18]**L. J. Slater,*Generalized hypergeometric functions*, Cambridge Univ. Press, New York, 1966. MR**0201688 (34:1570)****[19]**N. J. Vilenkin,*Special functions and the theory of group representations*, Amer. Math. Soc. Transl., Amer. Math. Soc., Providence, R. I., 1968. MR**0229863 (37:5429)****[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*, 4th ed., Cambridge Univ. Press, New York, 1927. MR**1424469 (97k:01072)****[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) , J. Math. Phys. (to appear).

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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