# zbMATH — the first resource for mathematics

Finite multiplicity theorems for induction and restriction. (English) Zbl 1317.22010
Summary: We find upper and lower bounds of the multiplicities of irreducible admissible representations $${\pi}$$ of a semisimple Lie group $$G$$ occurring in the induced representations $$\mathrm{Ind}_H^G{\tau}$$ from irreducible representations $${\tau}$$ of a closed subgroup $$H$$. As corollaries, we establish geometric criteria for finiteness of the dimension of $$\mathrm{Hom}_G({\pi}, \mathrm{Ind}_H^G{\tau})$$ (induction) and of $$\mathrm{Hom}_H({\pi}|_H,{\tau})$$ (restriction) by means of the real flag variety $$G/P$$, and discover that uniform boundedness property of these multiplicities is independent of real forms and characterized by means of the complex flag variety.

##### MSC:
 22E46 Semisimple Lie groups and their representations 14M27 Compactifications; symmetric and spherical varieties
Full Text:
##### References:
 [1] Aizenbud, A.; Gourevitch, D.; Rallis, S.; Schiffmann, G., Multiplicity one theorems, Ann. of Math. (2), 172, 1407-1434, (2010) · Zbl 1202.22012 [2] van den Ban, E. P., Invariant differential operators on a semisimple symmetric space and finite multiplicities in a Plancherel formula, Ark. Mat., 25, 175-187, (1987) · Zbl 0645.43009 [3] Bernstein, J.; Reznikov, A., Estimates of automorphic functions, Mosc. Math. J., 4, 19-37, (2004), p. 310 · Zbl 1081.11037 [4] Bien, F., Orbit, multiplicities, and differential operators, Contemp. Math., vol. 145, 199-227, (1993), Amer. Math. Soc. · Zbl 0799.14030 [5] Brion, M., Classification des espaces homogènes sphériques, Compos. Math., 63, 189-208, (1986) · Zbl 0642.14011 [6] Casselman, W., Jacquet modules for real reductive groups, (Proceedings of the International Congress of Mathematicians, Helsinki, 1978, (1980), Acad. Sci. Fennica Helsinki), 557-563 [7] Clerc, J.-L.; Kobayashi, T.; Ørsted, B.; Pevzner, M., Generalized Bernstein-reznikov integrals, Math. Ann., 349, 395-431, (2011) · Zbl 1207.42021 [8] Harish-Chandra, Representations of semisimple Lie groups on a Banach space, Proc. Natl. Acad. Sci. USA, 37, 170-173, (1951) · Zbl 0042.12602 [9] Kashiwara, M.; Kawai, T.; Kimura, T., Foundations of algebraic analysis, Princeton Math. Ser., vol. 37, (1986) [10] Kashiwara, M.; Oshima, T., Systems of differential equations with regular singularities and their boundary value problems, Ann. of Math., 106, 145-200, (1977) · Zbl 0358.35073 [11] Kimelfeld, B., Homogeneous domains in flag manifolds of rank 1, J. Math. Anal. Appl., 121, 506-588, (1987) · Zbl 0632.53047 [12] Knapp, A.; Vogan, D., Cohomological induction and unitary representations, (1995), Princeton University Press · Zbl 0863.22011 [13] Kobayashi, T., Discrete decomposability of the restriction of $$A_{\mathfrak{q}}(\lambda)$$ with respect to reductive subgroups and its applications, Invent. Math., 117, 181-205, (1994) · Zbl 0826.22015 [14] Kobayashi, T., Introduction to harmonic analysis on real spherical homogeneous spaces, (Sato, F., Proceedings of the 3rd Summer School on Number Theory “Homogeneous Spaces and Automorphic Forms” in Nagano, (1995)), 22-41, (in Japanese) [15] Kobayashi, T., Invariant measures on homogeneous manifolds of reductive type, J. Reine Angew. Math., 490, 37-53, (1997) · Zbl 0881.22013 [16] Kobayashi, T., Discrete decomposability of the restriction of $$A_{\mathfrak{q}}(\lambda)$$ with respect to reductive subgroups II—micro-local analysis and asymptotic K-support, Ann. of Math. (2), 147, 709-729, (1998) · Zbl 0910.22016 [17] Kobayashi, T., Discrete decomposability of the restriction of $$A_{\mathfrak{q}}(\lambda)$$ with respect to reductive subgroups III—restriction of harish-chandra modules and associated varieties, Invent. Math., 131, 229-256, (1998) · Zbl 0907.22016 [18] Kobayashi, T., Restrictions of generalized Verma modules to symmetric pairs, Transform. Groups, 17, 523-546, (2012) · Zbl 1257.22014 [19] Kostant, B., On the existence and irreducibility of certain series of representation, Bull. Amer. Soc. Math., 75, 627-642, (1969) · Zbl 0229.22026 [20] Kostant, B., On Whittaker vectors and representation theory, Invent. Math., 48, 101-184, (1978) · Zbl 0405.22013 [21] Krämer, M., Multiplicity free subgroups of compact connected Lie groups, Arch. Math. (Basel), 27, 28-36, (1976) · Zbl 0322.22011 [22] Krämer, M., Sphärishche untergruppen in kompakten zusammenhängenden liegruppen, Compos. Math., 38, 129-153, (1979) · Zbl 0402.22006 [23] Lipsman, R. L., Orbital symmetric spaces and finite multiplicity, J. Funct. Anal., 135, 1-38, (1996) · Zbl 0843.22018 [24] Lynch, T. E., Generalized Whittaker vectors and representation theory, (1979), M.I.T, Thesis [25] Matsuki, T., Orbits on flag manifolds, (Proceedings of the International Congress of Mathematicians, vol. II, Kyoto, 1990, (1991), Springer-Verlag), 807-813 · Zbl 0745.22010 [26] Oshima, T., Boundary value problems for systems of linear partial differential equations with regular singularities, Adv. Stud. Pure Math., 4, 391-432, (1984) [27] Oshima, T., A realization of semisimple symmetric spaces and construction of boundary value maps, Adv. Stud. Pure Math., 14, 603-650, (1988) [28] Oshima, T.; Sekiguchi, J., Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math., 57, 1-81, (1980) · Zbl 0434.58020 [29] Sun, B.; Zhu, C.-B., Multiplicity one theorems: the Archimedean case, Ann. of Math. (2), 175, 23-44, (2012) · Zbl 1239.22014 [30] Vinberg, È. B., Commutative homogeneous spaces and co-isotropic symplectic actions, Russian Math. Surveys, 56, 1-60, (2001) · Zbl 0996.53034 [31] Vinberg, È. B.; Kimelfeld, B. N., Homogeneous domains on flag manifolds and spherical subgroups of semisimple Lie groups, Funct. Anal. Appl., 12, 168-174, (1978) · Zbl 0439.53055 [32] Wallach, N. R., Real reductive groups I, II, Pure Appl. Math., vol. 132, (1988), Academic Press, 1992 · Zbl 0666.22002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.