# zbMATH — the first resource for mathematics

Relationship between orbit decomposition on the flag varieties and multiplicities of induced representations. (English) Zbl 1439.22026
Summary: Let $$G$$ be a real reductive Lie group and $$H$$ a closed subgroup. T. Kobayashi and T. Oshima established a finiteness criterion of multiplicities of irreducible $$G$$-modules occurring in the regular representation $$C^{\infty}(G/H)$$ by a geometric condition, referred to as real sphericity, namely, $$H$$ has an open orbit on the real flag variety $$G/P$$. This note discusses a refinement of their theorem by replacing a minimal parabolic subgroup $$P$$ with a general parabolic subgroup $$Q$$ of $$G$$, where a careful analysis is required because the finiteness of the number of $$H$$-orbits on the partial flag variety $$G/Q$$ is not equivalent to the existence of $$H$$-open orbit on $$G/Q$$.
##### MSC:
 22E46 Semisimple Lie groups and their representations 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 20G05 Representation theory for linear algebraic groups 53C30 Differential geometry of homogeneous manifolds 14M15 Grassmannians, Schubert varieties, flag manifolds 14M27 Compactifications; symmetric and spherical varieties
Full Text:
##### References:
 [1] A. Aizenbud, D. Gourevitch and A. Minchenko, Holonomicity of relative characters and applications to multiplicity bounds for spherical pairs, Selecta Math. (N.S.) 22 (2016), no. 4, 2325-2345. · Zbl 1392.22006 [2] Y. Benoist and T. Kobayashi, Tempered reductive homogeneous spaces, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 12, 3015-3036. · Zbl 1332.22015 [3] F. Bien, Orbits, multiplicities and differential operators, in Representation theory of groups and algebras, 199-227, Contemp. Math., 145, Amer. Math. Soc., Providence, RI, 1993. · Zbl 0799.14030 [4] B. Kimelfeld, Homogeneous domains on flag manifolds, J. Math. Anal. Appl. 121 (1987), no. 2, 506-588. · Zbl 0632.53047 [5] Harish-Chandra, Representations of semisimple Lie groups. II, Trans. Amer. Math. Soc. 76 (1954), 26-65. · Zbl 0056.25902 [6] M. E. Herrera, Integration on a semianalytic set, Bull. Soc. Math. France 94 (1966), 141-180. · Zbl 0158.20601 [7] M. Kashiwara, Systems of microdifferential equations, based on lecture notes by Teresa Monteiro Fernandes translated from the French, Progress in Mathematics, 34, Birkhäuser Boston, Inc., Boston, MA, 1983. · Zbl 0521.58057 [8] T. Kobayashi, Introduction to harmonic analysis on real spherical homogeneous spaces, Proceedings of the 3rd Summer School on Number Theory “Homogeneous Spaces and Automorphic Forms” in Nagano (F. Sato, ed.), 1995, 22-41 (in Japanese). [9] T. Kobayashi, Shintani functions, real spherical manifolds, and symmetry breaking operators, in Developments and retrospectives in Lie theory, 127-159, Dev. Math., 37, Springer, Cham, 2014. · Zbl 1362.22018 [10] T. Kobayashi and T. Oshima, Finite multiplicity theorems for induction and restriction, Adv. Math. 248 (2013), 921-944. · Zbl 1317.22010 [11] T. Kobayashi and B. Speh, Symmetry breaking for representations of rank one orthogonal groups, Mem. Amer. Math. Soc. 238 (2015), no. 1126. · Zbl 1334.22015 [12] T. Matsuki, Orbits on flag manifolds, in Proceedings of the International Congress of Mathematicians, Vol. II (Kyoto, 1990), 807-813, Math. Soc. Japan, Springer-Verlag, Tokyo, 1991. · Zbl 0745.22010
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.