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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

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 · doi:10.1007/s00029-016-0276-4
[2] Y. Benoist and T. Kobayashi, Tempered reductive homogeneous spaces, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 12, 3015-3036. · Zbl 1332.22015 · doi:10.4171/JEMS/578
[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 · doi:10.1016/0022-247X(87)90258-7
[5] Harish-Chandra, Representations of semisimple Lie groups. II, Trans. Amer. Math. Soc. 76 (1954), 26-65. · Zbl 0056.25902 · doi:10.1073/pnas.40.11.1078
[6] M. E. Herrera, Integration on a semianalytic set, Bull. Soc. Math. France 94 (1966), 141-180. · Zbl 0158.20601 · doi:10.24033/bsmf.1637
[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 · doi:10.1016/j.aim.2013.07.015
[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
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