Analytic uniformly bounded representations of
Author:
Ronald J. Stanke
Journal:
Trans. Amer. Math. Soc. 290 (1985), 281302
MSC:
Primary 22E46; Secondary 22E30
MathSciNet review:
787966
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: By analytically continuing suitably normalized spherical principal series, a family of uniformly bounded representations of , all of which act on the same Hilbert space , is constructed which is parametrized by complex numbers lying in the strip . The proper normalization of the principal series representations involves the intertwining operators of equivalent principal series representations. These intertwining operators are first analyzed using Fourier analysis on the Heisenberg group.
 [1]
Michael
Cowling, Unitary and uniformly bounded representations of some
simple Lie groups, Harmonic analysis and group representations,
Liguori, Naples, 1982, pp. 49–128. MR 777340
(86h:22012)
 [2]
, Harmonic analysis on some nilpotent groups (preprint).
 [3]
Jacek
Cygan, Subadditivity of homogeneous norms on
certain nilpotent Lie groups, Proc. Amer. Math.
Soc. 83 (1981), no. 1, 69–70. MR 619983
(82k:22009), http://dx.doi.org/10.1090/S00029939198106199838
 [4]
Daryl
Geller, Fourier analysis on the Heisenberg group. I. Schwartz
space, J. Funct. Anal. 36 (1980), no. 2,
205–254. MR
569254 (81g:43008), http://dx.doi.org/10.1016/00221236(80)901007
 [5]
Einar
Hille and Ralph
S. Phillips, Functional analysis and semigroups, American
Mathematical Society Colloquium Publications, vol. 31, American
Mathematical Society, Providence, R. I., 1957. rev. ed. MR 0089373
(19,664d)
 [6]
A.
W. Knapp and E.
M. Stein, Intertwining operators for semisimple groups, Ann.
of Math. (2) 93 (1971), 489–578. MR 0460543
(57 #536)
 [7]
R.
A. Kunze and E.
M. Stein, Uniformly bounded representations and harmonic analysis
of the 2×2 real unimodular group, Amer. J. Math.
82 (1960), 1–62. MR 0163988
(29 #1287)
 [8]
R.
A. Kunze and E.
M. Stein, Uniformly bounded representations. II. Analytic
continuation of the principal series of representations of the
𝑛×𝑛 complex unimodular group, Amer. J. Math.
83 (1961), 723–786. MR 0163989
(29 #1288)
 [9]
R.
A. Kunze and E.
M. Stein, Uniformly bounded representations. III. Intertwining
operators for the principal series on semisimple groups, Amer. J.
Math. 89 (1967), 385–442. MR 0231943
(38 #269)
 [10]
N.
N. Lebedev, Special functions and their applications, Dover
Publications, Inc., New York, 1972. Revised edition, translated from the
Russian and edited by Richard A. Silverman; Unabridged and corrected
republication. MR 0350075
(50 #2568)
 [11]
Ronald
L. Lipsman, Uniformly bounded representations of the Lorentz
groups, Amer. J. Math. 91 (1969), 938–962. MR 0267044
(42 #1946)
 [12]
Yudell
L. Luke, Integrals of Bessel functions, McGrawHill Book Co.,
Inc., New YorkTorontoLondon, 1962. MR 0141801
(25 #5198)
 [13]
Walter
Rudin, Function theory in the unit ball of 𝐶ⁿ,
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of
Mathematical Science], vol. 241, SpringerVerlag, New YorkBerlin,
1980. MR
601594 (82i:32002)
 [14]
Paul
J. Sally Jr., Analytic continuation of the irreducible unitary
representations of the universal covering group of
𝑆𝐿(2,𝑅), Memoirs of the American Mathematical
Society, No. 69, American Mathematical Society, Providence, R. I., 1967. MR 0235068
(38 #3380)
 [15]
E.
M. Stein and Guido
Weiss, Fractional integrals on 𝑛dimensional Euclidean
space, J. Math. Mech. 7 (1958), 503–514. MR 0098285
(20 #4746)
 [16]
Edward
N. Wilson, Uniformly bounded representations for
the Lorentz groups, Trans. Amer. Math. Soc.
166 (1972),
431–438. MR 0293011
(45 #2091), http://dx.doi.org/10.1090/S00029947197202930118
 [1]
 M. Cowling, Unitary and uniformly bounded representations of some simple Lie groups (preprint). MR 777340 (86h:22012)
 [2]
 , Harmonic analysis on some nilpotent groups (preprint).
 [3]
 J. Cygan, Subadditivity of homogeneous norms on certain nilpotent Lie groups, Proc. Amer. Math. Soc. 83 (1981). MR 619983 (82k:22009)
 [4]
 D. Geller, Fourier analysis on the Heisenberg group, I. Schwartz space, J. Funct. Anal. 36 (1980), 205254. MR 569254 (81g:43008)
 [5]
 E. Hille and R. S. Phillips, Functional analysis and semigroups, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R.I., 1957. MR 0089373 (19:664d)
 [6]
 A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups, Ann. of Math. (2) 93 (1971), 489578. MR 0460543 (57:536)
 [7]
 R. A. Kunze and E. M. Stein, Uniformly bounded representations and harmonic analysis of the real unimodular group, Amer. J. Math. 82 (1960), 162. MR 0163988 (29:1287)
 [8]
 , Uniformly bounded representations, II. Analytic continuation of the principal series of representations of the complex unimodular group, Amer. J. Math. 83 (1961), 723786. MR 0163989 (29:1288)
 [9]
 , Uniformly bounded representations, III. Intertwining operators for the principal series on semisimple groups, Amer. J. Math. 83 (1967), 385442. MR 0231943 (38:269)
 [10]
 N. N. Lebedev, Special functions and their applications (R. A. Silverman, transl. and ed.), Dover, New York, 1972. MR 0350075 (50:2568)
 [11]
 R. L. Lipsman, Uniformly bounded representations of the Lorentz groups, Amer. J. Math. 91 (1969), 938962. MR 0267044 (42:1946)
 [12]
 Y. L. Luke, Integrals of Bessel functions, McGrawHill, New York, 1962. MR 0141801 (25:5198)
 [13]
 W. Rudin, Function theory in the unit ball of , SpringerVerlag, New York, 1980. MR 601594 (82i:32002)
 [14]
 P. J. Sally, Jr., Analytic continuation of the irreducible unitary representations of the universal covering group of , Mem. Amer. Math. Soc. No. 69 (1967). MR 0235068 (38:3380)
 [15]
 E. M. Stein and Guido Weiss, Fractional integrals on dimensional Euclidean space, J. Math. Mech. 7 (1958), 503514. MR 0098285 (20:4746)
 [16]
 E. N. Wilson, Uniformly bounded representations for the Lorentz groups, Trans. Amer. Math. Soc. 166 (1972), 431438. MR 0293011 (45:2091)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
22E46,
22E30
Retrieve articles in all journals
with MSC:
22E46,
22E30
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198507879661
PII:
S 00029947(1985)07879661
Keywords:
Fourier analysis on the Heisenberg group,
homogeneous norms,
spherical principal series,
intertwining operators,
Laguerre polynomials,
gamma function,
analytic continuation of operators,
uniformly bounded representations
Article copyright:
© Copyright 1985
American Mathematical Society
