Bounded multiplicity theorems for induction and restriction

We prove a geometric criterion for the bounded multiplicity property of “small” infinite-dimensional representations of real reductive Lie groups in both induction and restrictions. Applying the criterion to symmetric pairs, we give a full description of the triples H ⊂ G ⊃ G such that any irreducible admissible representations of G with H-distinguished vectors have the bounded multiplicity property when restricted to the subgroup G. This article also completes the proof of the general results announced in the previous paper [Adv. Math. 2021, Section 7]. MSC 2020: Primary 22E46; Secondary 22E45, 53D50, 58J42, 53C50.

[1]  J. Wolf,et al.  The action of a real semisimple group on a complex flag manifold , 1969 .

[2]  B. Orsted,et al.  Branching laws for discrete series of some affine symmetric spaces , 2019, 1907.07544.

[3]  M. Kashiwara,et al.  On Holonomic Systems of Micro-differential Equations. III-Systems with Regular Singularities- , 1981 .

[4]  Multiple Flag Varieties of Finite Type , 1998, math/9805067.

[5]  P. Littelmann.  On Spherical Double Cones , 1994 .

[6]  È. Vinberg,et al.  Complexity of action of reductive groups , 1986 .

[7]  柏原 正樹.  Systems of microdifferential equations , 1983 .

[8]  Mikio Sato.  Theory of Hyperfunctions, II. , 1960 .

[9]  J. Möllers,et al.  Restriction of Most Degenerate Representations of O(1 ,N ) with Respect to Symmetric Pairs , 2015 .

[10]  Classification of discretely decomposable Aq(λ) with respect to reductive symmetric pairs , 2011, 1104.4400.

[11]  Toshiyuki Kobayashi,et al.  Symmetry breaking for representations of rank one orthogonal groups II. , 2013, 1310.3213.

[12]  R. Strichartz.  Analysis of the Laplacian on the Complete Riemannian Manifold , 1983 .

[13]  Toshiyuki Kobayashi,et al.  Geometric analysis on small unitary representations of GL(N,R) , 2010, 1002.3006.

[14]  M. Brion.  Quelques proprietes des espaces homogenes spheriques , 1986 .

[15]  T. Oshima,et al.  Finite multiplicity theorems for induction and restriction , 2011, 1108.3477.

[16]  Toshiyuki Kobayashi.  Discrete decomposability of the restriction ofAq(λ) with respect to reductive subgroups and its applications , 1994 .

[17]  T. Tauchi.  A generalization of the Kobayashi–Oshima uniformly bounded multiplicity theorem , 2021, International Journal of Mathematics.

[18]  Toshiyuki Kobayashi.  Visible actions on symmetric spaces , 2006, math/0607005.

[19]  R. Harrington.  Part II , 2004 .

[20]  Pierre Clare.  Invariant trilinear forms for spherical degenerate principal series of complex symplectic groups , 2015, 1509.00505.

[21]  A. Borel.  Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes , 1961 .

[22]  Toshiyuki Kobayashi,et al.  CLASSIFICATION OF FINITE-MULTIPLICITY SYMMETRIC PAIRS , 2013, 1312.4246.

[23]  Sigurdur Helgason,et al.  Geometric Analysis on Symmetric Spaces , 1994 .

[24]  I. Satake,et al.  ON REPRESENTATIONS AND COMPACTIFICATIONS OF SYMMETRIC RIEMANNIAN SPACES , 1960 .

[25]  Dusa McDuff,et al.  HARMONIC ANALYSIS ON SEMI‐SIMPLE LIE GROUPS—I , 1974 .

[26]  M. Krämer.  Multiplicity free subgroups of compact connected Lie groups , 1976 .

[27]  N. Wallach.  Real reductive groups , 1988 .

[28]  Toshiyuki Kobayashi.  Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups III. Restriction of Harish-Chandra modules and associated varieties , 1998 .

[29]  Toshiyuki Kobayashi.  A program for branching problems in the representation theory of real reductive groups , 2015, 1509.08861.

[30]  T. Oshima.  A Realization of Semisimple Symmetric Spaces and Construction of Boundary Value Maps , 1988 .

[31]  Binyong Sun,et al.  Multiplicity one theorems: the Archimedean case , 2009, 0903.1413.

[32]  Toshiyuki Kobayashi.  重複度１の表現と複素多様体上の可視的な作用Multiplicity-free representations and visible actions on complex manifolds , 2005 .

[33]  Toshiyuki Kobayashi.  Branching laws of unitary representations associated to minimal elliptic orbits for indefinite orthogonal group O(p,q) , 2019, 1907.07994.

[34]  Eckhard Meinrenken,et al.  LIE GROUPS AND LIE ALGEBRAS , 2021, Lie Groups, Lie Algebras, and Cohomology. (MN-34), Volume 34.

[35]  Real double flag varieties for the symplectic group , 2017, 1703.06852.

[36]  Hiroyuki Ochiai,et al.  On Orbits in Double Flag Varieties for Symmetric Pairs , 2012, 1208.2084.

[37]  Toshiyuki Kobayashi,et al.  Analysis on the minimal representation of O(p,q) II. Branching laws , 2001, math/0111085.

[38]  Toshiyuki Kobayashi,et al.  Classification of symmetric pairs with discretely decomposable restrictions of (g, K)-modules , 2012, 1202.5743.

[39]  Toshiyuki Kobayashi.  Shintani Functions, Real Spherical Manifolds, and Symmetry Breaking Operators , 2013, 1401.0117.

[40]  Toshiyuki Kobayashi.  Discrete decomposability of the restriction of A_q(λ)with respect to reductive subgroups II-micro-local analysis and asymptotic K-support , 1998 .

[41]  Michael E. Taylor,et al.  Differential Geometry I , 1994 .

[42]  Y. Benoist,et al.  Tempered homogeneous spaces III , 2020, 2009.10389.

[43]  Multiplicity-free Theorems of the Restrictions of Unitary Highest Weight Modules with respect to Reductive Symmetric Pairs , 2006, math/0607002.