Branching theorems for compact symmetric spaces
Author:
A. W. Knapp
Journal:
Represent. Theory 5 (2001), 404436
MSC (2000):
Primary 20G20, 22E45; Secondary 05E15
Published electronically:
October 26, 2001
MathSciNet review:
1870596
Fulltext PDF Free Access
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Abstract: A compact symmetric space, for purposes of this article, is a quotient , where is a compact connected Lie group and is the identity component of the subgroup of fixed points of an involution. A branching theorem describes how an irreducible representation decomposes upon restriction to a subgroup. The article deals with branching theorems for the passage from to , where is any of , , or , with . For each of these compact symmetric spaces, one associates another compact symmetric space with the following property: To each irreducible representation of whose space of fixed vectors is nonzero, there corresponds a canonical irreducible representation of such that the representations and are equivalent. For the situations under study, is equal respectively to , , and , independently of . Hints of the kind of ``duality'' that is suggested by this result date back to a 1974 paper by S. Gelbart.
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 S. Greenleaf, Decompositions of Group Actions on Symmetric Tensors, Ph.D. Dissertation, State University of New York at Stony Brook, August 2000.
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 K. D. Johnson, A note on branching theorems, Proc. Amer. Math. Soc. 129 (2001), 351353. MR 2001e:17018
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 C. Y. Lee, On the branching theorem of the symplectic groups, Canad. Math. Bull. 17 (1974), 535545. MR 56:5796
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 D. E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, Oxford University Press, New York, 1940, second edition, 1950. MR 2:3a
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 D. E. Littlewood and A. R. Richardson, Group characters and algebra, Phil. Trans. Royal Soc. A 233 (1934), 99141.
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 M. Maliakas, The universal form of the branching rule for the symplectic groups, J. Algebra 168 (1994), 221248. MR 95j:20040
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 M. J. Newell, Modification rules for the orthogonal and symplectic groups, Proc. Roy. Irish Acad. Sect. A 54 (1951), 153163. MR 13:204e
 [PaS]
 J. Patera and R. T. Sharp, Branching rules for representations of simple Lie algebras through Weyl group orbit reduction, J. Phys. A 22 (1989), 23292340. MR 90h:17009
 [Pr]
 R. A. Proctor, Young tableaux, Gelfand patterns, and branching rules for classical groups, J. Algebra 164 (1994), 299360. MR 96e:05180
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 W. Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math. 9 (1969), 6180. MR 41:3806
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 D. A. Vogan, Lie algebra cohomology and a multiplicity formula of Kostant, J. Algebra 51 (1978), 6975. MR 81e:22027
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 H. Weyl, The Theory of Groups and Quantum Mechanics, Methuen and Co., Ltd., London, 1931, reprinted Dover Publications, Inc., New York, 1950.
 [Z]
 D. P. Zhelobenko, The classical groups. Spectral analysis of their finitedimensional representations, Russian Math. Surveys 17 (1962), 194.
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Additional Information
A. W. Knapp
Affiliation:
School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540, and Department of Mathematics, State University of New York, Stony Brook, New York 11794
Address at time of publication:
81 Upper Sheep Pasture Road, East Setauket, New York 11733–1729
Email:
aknapp@math.sunysb.edu
DOI:
http://dx.doi.org/10.1090/S108841650100139X
PII:
S 10884165(01)00139X
Keywords:
Branching rule,
branching theorem,
representation
Received by editor(s):
March 20, 2001
Received by editor(s) in revised form:
September 10, 2001
Published electronically:
October 26, 2001
Article copyright:
© Copyright 2001
Anthony W. Knapp
