On a multiplicity formula of weights of representations of 𝑆𝑂*(2𝑛) and reciprocity theorems for symplectic groups

Leung, Eric Y.; Ton-That, Tuong

doi:10.1090/S0002-9939-1995-1227520-2

On a multiplicity formula of weights of representations of $\textrm {SO}^ *(2n)$ and reciprocity theorems for symplectic groups
HTML articles powered by AMS MathViewer

by Eric Y. Leung and Tuong Ton-That

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

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

PDF | Request permission

Abstract:

A formula relating the multiplicity of a weight of a holomorphic discrete series of the group $S{O^ \ast }(2n)$ to the frequency of occurrence of an irreducible holomorphically induced representation of the group $Sp(2n,\mathbb {C})$ in an n-fold tensor product of irreducible symmetric representations of $Sp(2n,\mathbb {C})$ is given. Reciprocity theorems relating holomorphic discrete series of $S{O^ \ast }(2n)$ (resp. $Sp(2n,\mathbb {R})$) to holomorphically induced representations of $Sp(2k,\mathbb {C})$ (resp. $SO(N,\mathbb {C})$) are also derived.

References

R. M. Delaney and B. Gruber, Inner and restriction multiplicity for classical groups, J. Mathematical Phys. 10 (1969), 252–265. MR 243780, DOI 10.1063/1.1664841
Dudley E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, Oxford University Press, New York, 1940. MR 0002127
Henryk Hecht and Wilfried Schmid, A proof of Blattner’s conjecture, Invent. Math. 31 (1975), no. 2, 129–154. MR 396855, DOI 10.1007/BF01404112
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
Roger Howe, $(\textrm {GL}_n,\textrm {GL}_m)$-duality and symmetric plethysm, Proc. Indian Acad. Sci. Math. Sci. 97 (1987), no. 1-3, 85–109 (1988). MR 983608, DOI 10.1007/BF02837817
James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
W. H. Klink and Tuong Ton-That, $n$-fold tensor products of $\textrm {GL}(N,\textbf {C})$ and decomposition of Fock spaces, J. Funct. Anal. 84 (1989), no. 1, 1–18. MR 999487, DOI 10.1016/0022-1236(89)90109-2
Kazuhiko Koike, On new multiplicity formulas of weights of representations for the classical groups, J. Algebra 107 (1987), no. 2, 512–533. MR 885808, DOI 10.1016/0021-8693(87)90100-1
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, DOI 10.1515/9781400883974
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
Peter Littelmann, A generalization of the Littlewood-Richardson rule, J. Algebra 130 (1990), no. 2, 328–368. MR 1051307, DOI 10.1016/0021-8693(90)90086-4
M. Moshinsky and C. Quesne, Noninvariance groups in the second-quantization picture and their applications, J. Mathematical Phys. 11 (1970), 1630–1639. MR 261878, DOI 10.1063/1.1665304
Wilfried Schmid, On the characters of the discrete series. The Hermitian symmetric case, Invent. Math. 30 (1975), no. 1, 47–144. MR 396854, DOI 10.1007/BF01389847
Tuong Ton That, Symplectic Stiefel harmonics and holomorphic representations of symplectic groups, Trans. Amer. Math. Soc. 232 (1977), 265–277. MR 476926, DOI 10.1090/S0002-9947-1977-0476926-1
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, DOI 10.1063/1.523634
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
D. P. Zhelobenko, Kompaktnye gruppy Li i ikh predstavleniya, Izdat. “Nauka”, Moscow, 1970 (Russian). MR 0473097