Multiplicity-free representations and visible actions on complex manifolds.

*(English)*Zbl 1085.22010Multiplicity-free representations appear in various contexts of mathematics, and numerous multiplicity-free theorems have been found over several decades. It seems that no single known technique could be applied to all multiplicity-free representations, in both finite and infinite dimensional cases. This paper presents a simple principle based on complex geometry to explain various kinds of multiplicity-free representations in a general framework. The idea is focused on the non-standard geometric perspectives of multiplicity-free representations in both finite and infinite dimensional cases, and new proofs of classical multiplicity-free theorems in various contexts are obtained along with an investigation of complex geometry in which a totally real submanifold meets generic orbits of a group. Thus also a number of new theorems are proved. The main machinery is an abstract multiplicity-free theorem for the section of equivariant holomorphic vector bundles. The notion of visible action on a complex manifold is introduced and discussed in the broad context of symplectic geometry and Riemannian geometry, such as coisotropic actions and polar actions. Some examples of visible actions are given.Various examples of multiplicity-free representations in both finite and infinite dimensional cases are explained. At last, the results on the multiplicity-free theorem of representations reflected by the geometry of coadjoint orbits are briefly considered.

Reviewer: Na Jisheng (Beijing)

##### MSC:

22E46 | Semisimple Lie groups and their representations |

32A37 | Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) |

05E15 | Combinatorial aspects of groups and algebras (MSC2010) |

20G05 | Representation theory for linear algebraic groups |

##### Keywords:

multiplicity-free representation; branching law; visible action; semisimple Lie group; Hermitian symmetric space; flag variety; totally real submanifold
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\textit{T. Kobayashi}, Publ. Res. Inst. Math. Sci. 41, No. 3, 497--549 (2005; Zbl 1085.22010)

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##### References:

[1] | Alikawa, H., Multiplicity-free branching rules for outer automorphisms of simple Lie algebras, preprint. · Zbl 1136.17005 |

[2] | Barbasch, D., Spherical unitary dual for split classical groups, preprint. · Zbl 0692.22006 |

[3] | Ben Saïd, S., Weighted Bergman spaces on bounded symmetric domains, Pacific J. Math., 206 (2002), 39-68. · Zbl 1055.32004 |

[4] | Benson, C. and Ratcliff, G., On multiplicity free actions, Representations of Real and p-adic Groups, E.-C. Tan and C.-B. Zhu eds., Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., (2004), 221-304. · Zbl 1061.22017 |

[5] | Bertram, W. and Hilgert, J., Hardy spaces and analytic continuation of Bergman spaces, Bull. Soc. Math. France, 126 (1998), 435-482. · Zbl 0920.22006 |

[6] | Brion, M., Sur l’image de l’application moment, Lecture Notes in Math., 1296 (1987), 177-192, Springer. · Zbl 0667.58012 |

[7] | Cartan, É., Sur la détermination d’un syst‘ eme orthogonal complet dans un espace de Riemann symetrique clos, Rend. Circ. Mat. Palermo, 53 (1929), 217-252. · JFM 55.1029.01 |

[8] | Deitmar, A., Invariant operators on higher K-types, J. Reine Angew. Math., 412 (1990), 97-107. · Zbl 0712.43006 |

[9] | van Dijk, G. and Hille, S. C., Canonical representations related to hyperbolic spaces, J. Funct. Anal., 147 (1997), 109-139. 547 · Zbl 0882.22017 |

[10] | Enright, T., Howe, R. and Wallach, N., A classification of unitary highest weight mod- ules, Representation theory of reductive groups, Progr. in Math., Birkhäuser, 40 (1983), 97-143. · Zbl 0535.22012 |

[11] | Enright, T. and Joseph, A., An intrinsic classification of unitary highest weight modules, Math. Ann., 288 (1990), 571-594. · Zbl 0725.17009 |

[12] | Faraut, J. and Thomas, E. G. F., Invariant Hilbert spaces of holomorphic functions, J. Lie Theory, 9 (1999), 383-402. · Zbl 1014.32005 |

[13] | Gelfand, I. M., Spherical functions on symmetric spaces, Dokl. Akad. Nauk. SSSR, 70 (1950), 5-8. |

[14] | Gross, B., Some applications of Gelfand pairs to number theory, Bull. Amer. Math. Soc., 24 (1991), 277-301. · Zbl 0733.11018 |

[15] | Guillemin, V. and Sternberg, S., Multiplicity-free spaces, J. Differential Geom., 19 (1984), 31-56. · Zbl 0548.58017 |

[16] | Gutkin, E., Coefficients of Clebsch-Gordan for the holomorphic discrete series, Lett. Math. Phys., 3 (1979), 185-192. · Zbl 0423.22020 |

[17] | Heckman, G. and Schlichtkrull, H., Harmonic Analysis and special functions on sym- metric spaces, Perspectives in Mathematics, 16, Academic Press, San Diego, 1994. · Zbl 0836.43001 |

[18] | Helgason, S., Groups and Geometric Analysis, Academic Press, New York, 1984. · Zbl 0543.58001 |

[19] | Howe, R., Reciprocity laws in the theory of dual pairs, Representation Theory of Re- ductive Groups (P. C. Trombi ed.), Progr. in Math., Birkhäuser, 40 (1983), 159-175. · Zbl 0543.22009 |

[20] | , Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures 1992, Israel Math. Conf. Proc., 8 (1995), 1-182. · Zbl 0844.20027 |

[21] | Howe, R. E. and Tan, E.-C., Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series representations, Bull. Amer. Math. Soc., 28 (1993), 1-74. · Zbl 0794.22012 |

[22] | Huckleberry, A. T. and Wurzbacher, T., Multiplicity-free complex manifolds, Math. Ann., 286 (1990), 261-280. · Zbl 0765.32016 |

[23] | Jaffee, H., Real forms of Hermitian symmetric spaces, Bull. Amer. Math. Soc., 81 (1975), 456-458. · Zbl 0312.32018 |

[24] | Jakobsen, H. P., Tensor products, reproducing kernels, and power series, J. Funct. Anal., 31 (1979), 293-305. · Zbl 0403.22011 |

[25] | , Hermitian symmetric spaces and their unitary highest weight modules, J. Funct. Anal., 52 (1983), 385-412. · Zbl 0517.22014 |

[26] | Jakobsen, H. P. and Vergne, M., Restrictions and expansions of holomorphic represen- tations, J. Funct. Anal., 34 (1979), 29-53. · Zbl 0433.22011 |

[27] | Johnson, K., On a ring of invariant polynomials on a Hermitian symmetric space, J. Algebra, 67 (1980), 72-81. · Zbl 0491.22007 |

[28] | , Degenerate principal series and compact groups, Math. Ann., 287 (1990), 703-718. · Zbl 0687.22003 |

[29] | Kac, V., Some remarks on nilpotent orbits, J. Algebra, 64 (1980), 190-213. · Zbl 0431.17007 |

[30] | Klimyk, A. U. and Gruber, B., Matrix elements for infinitesimal operators of the groups U (p + q) and U (p, q) in a U (p) \times U(q) basis I, II, J. Math. Phys., 20 (1979), 1995-2010, 2011-2013. · Zbl 0427.22016 |

[31] | Kobayashi, S., Irreducibility of certain unitary representations, J. Math. Soc. Japan, 20 (1968), 638-642. · Zbl 0165.40504 |

[32] | Kobayashi, T., Multiplicity-free theorem in branching problems of unitary highest weight modules, Proceedings of the Symposium on Representation Theory held at Saga, Kyushu 1997 (ed. K. Mimachi), (1997), 9-17. |

[33] | , Discrete decomposability of the restriction of Aq(\lambda ) with respect to reductive subgroups and its applications, Part I, Invent. Math., 117 (1994), 181-205; Part II, Ann. of Math., 147 (1998), 709-729; Part III, Invent. Math., 131 (1998), 229-256. · Zbl 0826.22015 |

[34] | , Discrete series representations for the orbit spaces arising from two involutions of real reductive Lie groups, J. Funct. Anal., 152 (1998), 100-135. · Zbl 0937.22008 |

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