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Restriction of square integrable representations: Discrete spectrum
Jorge Vargas and Bent Ørsted
Source: Duke Math. J. Volume 123, Number 3
(2004), 609-633.
Abstract
In this paper, we study the problem of restricting a square integrable representation of a connected semisimple Lie group to a reductive subgroup. Using a geometric method of restricting sections of a vector bundle to a submanifold, we obtain information about both the discrete and the continuous spectrum. We also show the (L2,L2)-continuity of the associated Berezin transform and that, under suitable general conditions, the Berezin transform is (L2,L2)-continuous for 1≤p≤∞
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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1086957717
Digital Object Identifier: doi:10.1215/S0012-7094-04-12336-X
Zentralblatt MATH identifier: 02103767
Mathematical Reviews number (MathSciNet): MR2068970
References
M. Atiyah and W. Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1--62.
Mathematical Reviews (MathSciNet): MR0463358
Digital Object Identifier: doi:10.1007/BF01389783
E. P. van den Ban, Invariant differential operators on a semisimple symmetric space and finite multiplicities in a Plancherel formula, Ark. Mat. 25 (1985), 175--187.
Mathematical Reviews (MathSciNet): MR0923405
Digital Object Identifier: doi:10.1007/BF02384442
Zentralblatt MATH: 0645.43009
L. Bers and M. Schechter, ``Elliptic equations'' in Partial Differential Equations (Boulder, Colo., 1957), Lectures in Appl. Math. 3, Interscience, New York, 1964, 131--299.
Mathematical Reviews (MathSciNet): MR0165224
M. Cowling, The Kunze-Stein phenomenon, Ann. of Math. (2) 107 (1978), 209--234.
Mathematical Reviews (MathSciNet): MR0507240
Digital Object Identifier: doi:10.2307/1971142
JSTOR: links.jstor.org
B. Gross and N. Wallach, ``Restriction of small discrete series representations to symmetric subgroups'' in The Mathematical Legacy of Harish-Chandra (Baltimore, 1998), ed. R. S. Doran and V. S. Varadarajan, Proc. Sympos. Pure Math. 68, Amer. Math. Soc., Providence, 2000, 255--272.
Mathematical Reviews (MathSciNet): MR1767899
Harish-Chandra, ``Some results on differential equations'' in Collected Papers, Vol. III ( 1959--1968.), ed. V. S. Varadarajan, Springer, New York, 1984, 4--48.
Mathematical Reviews (MathSciNet): MR0726024
H. Hecht and W. Schmid, A proof of Blattner's conjecture, Invent. Math. 31 (1975), 129--154.
Mathematical Reviews (MathSciNet): MR0396855
Digital Object Identifier: doi:10.1007/BF01404112
Zentralblatt MATH: 0319.22012
R. Hotta and R. Parthasarathy, Multiplicity formulae for discrete series, Invent. Math. 26 (1974), 133--178.
Mathematical Reviews (MathSciNet): MR0348041
Digital Object Identifier: doi:10.1007/BF01435692
Zentralblatt MATH: 0298.22013
R. Howe, ``Reciprocity laws in the theory of dual pairs'' in Representation Theory of Reductive Groups (Park City, Utah, 1982), ed. P. C. Trombi, Progr. Math. 40, Birkhäuser, Boston, 1983, 156--175.
Mathematical Reviews (MathSciNet): MR0733812
H. P. Jacobsen and M. Vergne, Restrictions and expansions of holomorphic representations, J. Funct. Anal. 34 (1979), 29--53.
Mathematical Reviews (MathSciNet): MR0551108
Digital Object Identifier: doi:10.1016/0022-1236(79)90023-5
Zentralblatt MATH: 0433.22011
A. Knapp, Representation Theory of Semisimple Lie Groups: An Overview Based on Examples, Princeton Math. Ser. 36, Princeton Univ. Press, Princeton, 1986.
Mathematical Reviews (MathSciNet): MR0855239
--------, Lie Groups beyond an Introduction, Progr. Math. 140, Birkhäuser, Boston, 1996.
Mathematical Reviews (MathSciNet): MR1399083
T. Kobayashi, The restricton of $A_\mathfrak q(\lambda)$ to reductive subgroups, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 262--267.
Mathematical Reviews (MathSciNet): MR1249224
Digital Object Identifier: doi:10.3792/pjaa.69.262
Project Euclid: euclid.pja/1195511349
--. --. --. --., Discrete decomposability of the restriction of $A_\mathfrak q(\lambda)$ with respect to reductive subgroups and its applications, Invent. Math. 117 (1994), 181--205.
Mathematical Reviews (MathSciNet): MR1273263
Digital Object Identifier: doi:10.1007/BF01232239
--. --. --. --., Discrete decomposability of the restriction of $A_\mathfrak q(\lambda)$ with respect to reductive subgroups, II: Micro-local analysis and asymptotic $K$-support, Ann. of Math. (2) 147 (1998), 709--729.
Mathematical Reviews (MathSciNet): MR1637667
Digital Object Identifier: doi:10.2307/120963
JSTOR: links.jstor.org
Zentralblatt MATH: 0910.22016
--. --. --. --., Discrete decomposability of the restriction of $A_\mathfrak q(\lambda) $ with respect to reductive subgroups, III: Restriction of Harish-Chandra modules and associated varieties, Invent. Math. 131 (1998), 229--256.
Mathematical Reviews (MathSciNet): MR1608642
Digital Object Identifier: doi:10.1007/s002220050203
Zentralblatt MATH: 0907.22016
--. --. --. --., Discrete series representations for the orbit spaces arising from two involutions of real reductive Lie groups, J. Funct. Anal. 152 (1998), 100--135.
Mathematical Reviews (MathSciNet): MR1600074
Digital Object Identifier: doi:10.1006/jfan.1997.3128
Zentralblatt MATH: 0937.22008
--. --. --. --., ``Harmonic analysis on homogeneous manifolds of reductive type and unitary representation theory'' in Selected Papers on Harmonic Analysis: Groups and Invariants, ed. K. Nomizu, Amer. Math. Soc. Transl. Ser. 2 183, Amer. Math. Soc., Providence, 1998, 1--31.
Mathematical Reviews (MathSciNet): MR1615135
--. --. --. --., ``Discretely decomposable restrictions of unitary representations of reductive Lie groups --.-examples and conjectures'' in Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, ed. T. Kobayashi, Adv. Stud. Pure Math. 26, Math. Soc. Japan, Tokyo, 2000, 99--127.
Mathematical Reviews (MathSciNet): MR1770719
H. Y. Loke, Restrictions of quaternionic representations, J. Funct. Anal. 172 (2000), 377--403.
Mathematical Reviews (MathSciNet): MR1753179
Digital Object Identifier: doi:10.1006/jfan.1999.3450
Zentralblatt MATH: 0953.22018
G. W. Mackey, The Theory of Unitary Group Representations, Chicago Lectures in Math., Univ. of Chicago Press, Chicago, 1967.
Mathematical Reviews (MathSciNet): MR0396826
S. Martens, The characters of the holomorphic discrete series, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 3275--3276.
Mathematical Reviews (MathSciNet): MR0419687
Digital Object Identifier: doi:10.1073/pnas.72.9.3275
Zentralblatt MATH: 0308.22013
G. Ólafsson, Symmetric spaces of Hermitian type, Differential Geom. Appl. 1 (1991), 195--233.
Mathematical Reviews (MathSciNet): MR1244444
Digital Object Identifier: doi:10.1016/0926-2245(91)90001-P
Zentralblatt MATH: 0785.22021
G. Ólafsson and B. \Orsted, The holomorphic discrete series for affine symmetric spaces, I, J. Funct. Anal. 81 (1988), 126--159.
Mathematical Reviews (MathSciNet): MR0967894
Digital Object Identifier: doi:10.1016/0022-1236(88)90115-2
Zentralblatt MATH: 0678.22008
R. Takahashi, Sur les représentations unitaires des groupes de Lorentz généralisés, Bull. Soc. Math. France 91 (1963), 289--433.
Mathematical Reviews (MathSciNet): MR0179296
F. Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, 1967.
Mathematical Reviews (MathSciNet): MR0225131
P. C. Trombi and V. S. Varadarajan, Asymptotic behaviour of eigen functions on a semisimple Lie group: The discrete spectrum, Acta Math. 129 (1972), 237--280.
Mathematical Reviews (MathSciNet): MR0393349
Digital Object Identifier: doi:10.1007/BF02392217
Zentralblatt MATH: 0244.43006
J. A. Vargas, Restriction of some discrete series representations, Algebras Groups Geom. 18 (2001), 85--100.
Mathematical Reviews (MathSciNet): MR1834195
--. --. --. --., Restriction of holomorphic discrete series to real forms, Rend. Sem. Mat. Univ. Politec. Torino 60 (2002), 45--53.
Mathematical Reviews (MathSciNet): MR1980327
N. R. Wallach, On the Enright-Varadarajan modules: A construction of the discrete series, Ann. Sci. École Norm. Sup. (4) 9 (1996), 81--101.
Mathematical Reviews (MathSciNet): MR0422518
N. R. Wallach and J. A. Wolf, ``Completeness of Poincare series for automorphic forms associated to the integrable discrete series'' in Representation Theory of Reductive Groups (Park City, Utah, 1982), ed. P. C. Trombi, Progr. Math. 40, Birkhäuser, Boston, 1983, 265--281.
Mathematical Reviews (MathSciNet): MR0733818
G. Warner, Harmonic Analysis on Semi-Simple Lie Groups, I, II, Grundlehren Math. Wiss. 188, 189, Springer, New York, 1972.
J. A. Wolf, ``Representations that remain irreducible on parabolic subgroups'' in Differential Geometrical Methods in Mathematical Physics (Aix-en-Provence and Salamanca, 1979), Lecture Notes in Math. 836, Springer, Berlin, 1980, 129--144.
Mathematical Reviews (MathSciNet): MR0607689
G. Zhang, Berezin transform on real bounded symmetric domains, Trans. Amer. Math. Soc. 353 (2001), 3769--3787.
Mathematical Reviews (MathSciNet): MR1837258
Digital Object Identifier: doi:10.1090/S0002-9947-01-02832-X
JSTOR: links.jstor.org
Zentralblatt MATH: 0965.22015
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