Matrixvalued 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), 319350
MSC:
Primary 22E70; Secondary 33A75, 43A75, 81C40
MathSciNet review:
558177
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: This article examines in detail the matrixvalued gamma function associated to the conformal group . Here, is a continuous complex parameter, runs through a family of ``weights'' of , P is the cone of positivedefinite 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 vectorvalued holomorphic functions is described. Analogs are given for some of the wellknown formulas for the classical gamma function. As an epilogue, applications of the matrixvalued 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]
Quantum theory of angular momentum. A collection of reprints and
original papers, Edited by L. C. Biedenharn and H. van Dam, Academic
Press, New YorkLondon, 1965. MR 0198829
(33 #6983)
 [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
(49 #9119)
 [4]
R. Godement, Fonctions automorphes, Séminaire Cartan, University of Paris, 19571958.
 [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
(58 #6075)
 [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
(49 #9120)
 [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
(56 #12181)
 [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
(55 #8250), http://dx.doi.org/10.1090/S000299041977143024
 [9]
K. I. Gross and W. J. Holman III, Matrixvalued 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, AddisonWesley, Reading, Mass., 1964.
 [11]
HarishChandra, Representations of semisimple Lie groups. IV, V, VI, Amer. J. Math. 77 (1955), 743777; 78 (1956), 141; 78 (1956), 564628.
 [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 (30 #2162)
 [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
(55 #12876)
 [14]
A.
P. Jucis and A.
A. Bandzaĭtis, Teoriya momenta kolichestva dvizheniya v
kvantovoi mekhanike, Academy of Sciences of the Lithuanian SSR.
Institute of Physics and Mathematics. Publication No. 6, Izdat.
“Mintis”, Vilnius, 1965 (Russian). MR 0200023
(33 #8163)
 [15]
A.
W. Knapp and K.
Okamoto, Limits of holomorphic discrete series, J. Functional
Analysis 9 (1972), 375–409. MR 0299726
(45 #8774)
 [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
(53 #10989)
 [17]
M.
Vergne and H.
Rossi, Analytic continuation of the holomorphic discrete series of
a semisimple Lie group, Acta Math. 136 (1976),
no. 12, 1–59. MR 0480883
(58 #1032)
 [18]
Lucy
Joan Slater, Generalized hypergeometric functions, Cambridge
University Press, Cambridge, 1966. MR 0201688
(34 #1570)
 [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
(37 #5429)
 [20]
N. Wallach, Analytic continuation for the holomorphic discrete series. I, II, Trans. Amer. Math. Soc. 251 (1979), 117; 1937.
 [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
(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).
 [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. 343345. MR 0344380 (49:9119)
 [4]
 R. Godement, Fonctions automorphes, Séminaire Cartan, University of Paris, 19571958.
 [5]
 K. I. Gross and R. A. Kunze, FourierBessel transforms and holomorphic discrete series, Conference on Harmonic Analysis, Lecture Notes in Math., vol. 266, SpringerVerlag, Berlin and New York, 1972, pp. 79122. 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. 347350. MR 0344381 (49:9120)
 [7]
 , Bessel functions and representation theory, II: Holomorphic discrete series and metaplectic representations, J. Functional Anal. 25 (1977), 149. 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), 412415. MR 0435290 (55:8250)
 [9]
 K. I. Gross and W. J. Holman III, Matrixvalued 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, AddisonWesley, Reading, Mass., 1964.
 [11]
 HarishChandra, Representations of semisimple Lie groups. IV, V, VI, Amer. J. Math. 77 (1955), 743777; 78 (1956), 141; 78 (1956), 564628.
 [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), 52106. 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), 375409. 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 semisimple Lie group, J. Functional Anal. 13 (1973), 324389. MR 0407206 (53:10989)
 [17]
 , Analytic continuation of the holomorphic discrete series of a semisimple Lie group, Acta Math. 136 (1976), 159. 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), 117; 1937.
 [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).
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:
http://dx.doi.org/10.1090/S00029947198005581775
PII:
S 00029947(1980)05581775
Keywords:
Conformal group,
matrixvalued gamma function,
representation theory,
holomorphic discrete series,
special functions
Article copyright:
© Copyright 1980
American Mathematical Society
