On a multiplicity formula of weights of representations of and reciprocity theorems for symplectic groups

Authors:
Eric Y. Leung and Tuong Ton-That

Journal:
Proc. Amer. Math. Soc. **123** (1995), 1281-1288

MSC:
Primary 22E46; Secondary 22E45

MathSciNet review:
1227520

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A formula relating the multiplicity of a weight of a holomorphic discrete series of the group to the frequency of occurrence of an irreducible holomorphically induced representation of the group in an *n*-fold tensor product of irreducible symmetric representations of is given. Reciprocity theorems relating holomorphic discrete series of (resp. ) to holomorphically induced representations of (resp. ) are also derived.

**[DG]**R. M. Delaney and B. Gruber,*Inner and restriction multiplicity for classical groups*, J. Mathematical Phys.**10**(1969), 252–265. MR**0243780****[DL]**Dudley E. Littlewood,*The Theory of Group Characters and Matrix Representations of Groups*, Oxford University Press, New York, 1940. MR**0002127****[HS]**Henryk Hecht and Wilfried Schmid,*A proof of Blattner’s conjecture*, Invent. Math.**31**(1975), no. 2, 129–154. MR**0396855****[Ho1]**Roger Howe,*Reciprocity laws in the theory of dual pairs*, Representation theory of reductive groups (Park City, Utah, 1982) Progr. Math., vol. 40, Birkhäuser Boston, Boston, MA, 1983, pp. 159–175. MR**733812****[Ho2]**Roger Howe,*(𝐺𝐿_{𝑛},𝐺𝐿_{𝑚})-duality and symmetric plethysm*, Proc. Indian Acad. Sci. Math. Sci.**97**(1987), no. 1-3, 85–109 (1988). MR**983608**, 10.1007/BF02837817**[Hu]**James E. Humphreys,*Introduction to Lie algebras and representation theory*, Springer-Verlag, New York-Berlin, 1972. Graduate Texts in Mathematics, Vol. 9. MR**0323842****[KT]**W. H. Klink and Tuong Ton-That,*𝑛-fold tensor products of 𝐺𝐿(𝑁,𝐶) and decomposition of Fock spaces*, J. Funct. Anal.**84**(1989), no. 1, 1–18. MR**999487**, 10.1016/0022-1236(89)90109-2**[Ko]**Kazuhiko Koike,*On new multiplicity formulas of weights of representations for the classical groups*, J. Algebra**107**(1987), no. 2, 512–533. MR**885808**, 10.1016/0021-8693(87)90100-1**[Kn]**Anthony W. Knapp,*Representation theory of semisimple groups*, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR**855239****[Le]**Eric Y. Leung,*On resolving the multiplicity of tensor products of irreducible representations of symplectic groups*, J. Phys. A**26**(1993), no. 21, 5851–5866. MR**1252792****[Li]**Peter Littelmann,*A generalization of the Littlewood-Richardson rule*, J. Algebra**130**(1990), no. 2, 328–368. MR**1051307**, 10.1016/0021-8693(90)90086-4**[MQ]**M. Moshinsky and C. Quesne,*Noninvariance groups in the second-quantization picture and their applications*, J. Mathematical Phys.**11**(1970), 1630–1639. MR**0261878****[Sc]**Wilfried Schmid,*On the characters of the discrete series. The Hermitian symmetric case*, Invent. Math.**30**(1975), no. 1, 47–144. MR**0396854****[TT1]**Tuong Ton That,*Symplectic Stiefel harmonics and holomorphic representations of symplectic groups*, Trans. Amer. Math. Soc.**232**(1977), 265–277. MR**0476926**, 10.1090/S0002-9947-1977-0476926-1**[TT2]**Tuong Ton That,*Explicit decomposition of tensor products of certain analytic representations of symplectic groups*, J. Math. Phys.**19**(1978), no. 12, 2514–2519. MR**512972**, 10.1063/1.523634**[We]**Hermann Weyl,*The classical groups*, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR**1488158****[Ze]**D. P. Želobenko,*Compact Lie groups and their representations*, American Mathematical Society, Providence, R.I., 1973. Translated from the Russian by Israel Program for Scientific Translations; Translations of Mathematical Monographs, Vol. 40. MR**0473098**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
22E46,
22E45

Retrieve articles in all journals with MSC: 22E46, 22E45

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1995-1227520-2

Keywords:
Weights of representations,
reciprocity theorems,
holomorphic discrete series,
holomorphically induced representations,
dual pairs

Article copyright:
© Copyright 1995
American Mathematical Society