Branching theorems for compact symmetric spaces

Author:
A. W. Knapp

Journal:
Represent. Theory **5** (2001), 404-436

MSC (2000):
Primary 20G20, 22E45; Secondary 05E15

Published electronically:
October 26, 2001

MathSciNet review:
1870596

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[Ba]**M. Welleda Baldoni Silva,*Branching theorems for semisimple Lie groups of real rank one*, Rend. Sem. Mat. Univ. Padova**61**(1979), 229–250 (1980). MR**569662****[BGG]**I. N. Bernstein, I. M. Gelfand, and S. I. Gelfand,*Structure of representations generated by highest weight vectors*, Funct. Anal. and Its Appl.**5**(1971), 1-8.**[Bo]**Hermann Boerner,*Representations of groups. With special consideration for the needs of modern physics*, Translated from the German by P. G. Murphy in cooperation with J. Mayer-Kalkschmidt and P. Carr. Second English edition, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1970. MR**0272911****[C]**P. Cartier,*On H. Weyl’s character formula*, Bull. Amer. Math. Soc.**67**(1961), 228–230. MR**0146306**, 10.1090/S0002-9904-1961-10583-1**[Da]**A. van Daele,*General formula to derive branching rules*, J. Mathematical Phys.**11**(1970), 3275–3282. MR**0270641****[DQ]**J. Deenen and C. Quesne,*Canonical solution of the state labelling problem for 𝑆𝑈(𝑛)⊃𝑆𝑂(𝑛) and Littlewood’s branching rule. I. General formulation*, J. Phys. A**16**(1983), no. 10, 2095–2104. MR**713171****[Ge]**Stephen S. Gelbart,*A theory of Stiefel harmonics*, Trans. Amer. Math. Soc.**192**(1974), 29–50. MR**0425519**, 10.1090/S0002-9947-1974-0425519-8**[GeC]**I. M. Gel′fand and M. L. Cetlin,*Finite-dimensional representations of the group of unimodular matrices*, Doklady Akad. Nauk SSSR (N.S.)**71**(1950), 825–828 (Russian). MR**0035774****[GoW]**Roe Goodman and Nolan R. Wallach,*Representations and invariants of the classical groups*, Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge University Press, Cambridge, 1998. MR**1606831****[Gr]**S. Greenleaf,*Decompositions of Group Actions on Symmetric Tensors*, Ph.D. Dissertation, State University of New York at Stony Brook, August 2000.**[Heg]**G. C. Hegerfeldt,*Branching theorem for the symplectic groups*, J. Mathematical Phys.**8**(1967), 1195–1196. MR**0224740****[Hel]**Sigurđur Helgason,*A duality for symmetric spaces with applications to group representations*, Advances in Math.**5**(1970), 1–154 (1970). MR**0263988****[J]**Kenneth D. Johnson,*A note on branching theorems*, Proc. Amer. Math. Soc.**129**(2001), no. 2, 351–353. MR**1709755**, 10.1090/S0002-9939-00-05646-X**[Kn]**Anthony W. Knapp,*Lie groups beyond an introduction*, Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 1996. MR**1399083****[Ko]**Bertram Kostant,*A formula for the multiplicity of a weight*, Trans. Amer. Math. Soc.**93**(1959), 53–73. MR**0109192**, 10.1090/S0002-9947-1959-0109192-6**[Lee]**C. Y. Lee,*On the branching theorem of the symplectic groups*, Canad. Math. Bull.**17**(1974), no. 4, 535–545. MR**0447485****[Lep]**J. Lepowsky,*Multiplicity formulas for certain semisimple Lie groups*, Bull. Amer. Math. Soc.**77**(1971), 601–605. MR**0301142**, 10.1090/S0002-9904-1971-12767-2**[Lim]**P. Littlemann,*Characters of representations and paths in*, Representation Theory and Automorphic Forms, Proc. Symp. Pure Math., vol. 61, 1997, pp. 29-49.**[Liw]**Dudley E. Littlewood,*The Theory of Group Characters and Matrix Representations of Groups*, Oxford University Press, New York, 1940. MR**0002127****[LiR]**D. E. Littlewood and A. R. Richardson,*Group characters and algebra*, Phil. Trans. Royal Soc. A**233**(1934), 99-141.**[Ma]**Mihalis Maliakas,*The universal form of the branching rule for the symplectic groups*, J. Algebra**168**(1994), no. 1, 221–248. MR**1289098**, 10.1006/jabr.1994.1227**[Mu]**F. D. Murnaghan,*The Theory of Group Representations*, Johns Hopkins Press, Baltimore, 1938.**[N]**M. J. Newell,*Modification rules for the orthogonal and symplectic groups*, Proc. Roy. Irish Acad. Sect. A.**54**(1951), 153–163. MR**0043093****[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), no. 13, 2329–2340. MR**1003734****[Pr]**Robert A. Proctor,*Young tableaux, Gel′fand patterns, and branching rules for classical groups*, J. Algebra**164**(1994), no. 2, 299–360. MR**1271242**, 10.1006/jabr.1994.1064**[Q]**C. Quesne,*Canonical solution of the state labelling problem for 𝑆𝑈(𝑛)⊃𝑆𝑂(𝑛) and Littlewood’s branching rule. II. Use of modification rules*, J. Phys. A**17**(1984), no. 4, 777–789. MR**738886****[Sc]**Wilfried Schmid,*Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen*, Invent. Math.**9**(1969/1970), 61–80 (German). MR**0259164****[St]**Robert Steinberg,*A general Clebsch-Gordan theorem*, Bull. Amer. Math. Soc.**67**(1961), 406–407. MR**0126508**, 10.1090/S0002-9904-1961-10644-7**[V]**David A. Vogan Jr.,*Lie algebra cohomology and a multiplicity formula of Kostant*, J. Algebra**51**(1978), no. 1, 69–75. MR**487465**, 10.1016/0021-8693(78)90135-7**[W]**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 finite-dimensional representations*, Russian Math. Surveys**17**(1962), 1-94.

Retrieve articles in *Representation Theory of the American Mathematical Society*
with MSC (2000):
20G20,
22E45,
05E15

Retrieve articles in all journals with MSC (2000): 20G20, 22E45, 05E15

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/S1088-4165-01-00139-X

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