Symmetry in Geometry and Analysis is a Festschrift honoring Toshiyuki Kobayashi. The three volumes feature 35 selected contributions from invited speakers of twin conferences held in June 2022 in Reims, France, and in September 2022 in Tokyo, Japan. These contributions highlight the profound impact of Prof. Kobayashi’s pioneering ideas, groundbreaking discoveries, and significant achievements in the development of analytic representation theory, noncommutative harmonic analysis, and the geometry of discontinuous groups beyond the Riemannian context, among other areas, over the past four decades.
This third volume of the Festschrift contains original articles on branching problems in representation theory of reductive Lie groups and related topics.
Contributions are by Ali Baklouti, Hidenori Fujiwara, Dmitry Gourevitch, Masatoshi Kitagawa, Salma Nasrin, Yoshiki Oshima, and Petr Somberg.

lgrsnf/Pevzner2025c.pdf

Springer Basel AG

SPRINGER NATURE

S.l, 2024

Contents
Contents of Volume 1
Contents of Volume 2
A Solution to Duflo's Polynomial Problem for Nilpotent Lie Groups Restricted Representations
1 Introduction
2 Backgrounds and Notations
2.1 Saturation of Coadjoint Orbits
2.2 Saturation with Respect to a Normal Subgroup of Codimension 1
2.3 Orbit Layers and Kernels
2.4 Corwin-Greenleaf e-Central Elements
3 Generating Families of C(π|K) and Z(C[Ω]K)
4 Duflo's Polynomial Problem
4.1 Case of Finite Multiplicity Restrictions
4.2 The General Case
References
Multiplicities and Associated Varieties in Representation Theory of Reductive Groups
1 Introduction
1.1 Notation
2 Finite Multiplicities Beyond Spherical Pairs
2.1 Homogeneous Case
2.2 Branching Problems
2.3 Related Results
2.4 Twisted Case
2.5 Symplectic Sphericity
2.6 Open Questions
2.6.1 Geometric Questions
3 Sufficient Conditions for Vanishing
4 Non-Archimedean Case
5 Bibliographic Note
References
Kobayashi's Conjectures on the Discrete Decomposability
1 Introduction
1.1 Notation and Convention
2 Preliminaries
2.1 Compact Operator
2.2 Continuous Representation and Smooth Vectors
2.3 (g, K)-Module
2.4 Infinitesimal Character
2.5 Moderate Growth and Rapid Decrease
2.6 Null Cone
3 Growth and Infinitesimal Character
3.1 Heat Kernel
3.2 Estimate of Infinitesimal Characters
4 Discrete Decomposability
4.1 Discrete Decomposability
4.2 Non-compact Center and Algebraicity
4.3 Z(g)-Action
5 H to Hinfty
5.1 Existence of Irreducible Quotient
5.2 Compactness and Discrete Decomposability
5.3 Smoothness and Discrete Decomposability
5.4 Unitary to Algebraic
6 Wave Front Set and Hinfty
6.1 Wave Front Set and Estimate
6.2 Wave Front Set of a Representation
6.3 Wave Front Set and Discrete Decomposability
6.4 Nonreductive Case
7 Criteria for the Discrete Decomposability
7.1 Associated Variety
7.2 Asymptotic K-Support
7.3 Sekiguchi–Kostant Correspondence
7.4 Finite Orbits and Admissibility
References
Kobayashi's Multiplicity-One Theorems in Branching Laws and Orbit Philosophy Beyond Tempered Representations
1 Introduction
2 Kobayashi's ABC Program on Branching Problems
2.1 General Results in Stage A: Pioneered by Kobayashi
2.2 Bounded Multiplicity Condition for the Restriction O(p,q)↓O(p1,q1)×O(p2,q2)
2.3 Restriction of ``Small'' πIrr(G) to Reductive Subgroups
2.4 Specific Results in Stage A
3 Orbit Philosophy
3.1 Geometric Quantization and the Classical Limit
3.2 Functorial Property for Geometric Quantization
3.3 [Q, R] = 0 for Reductive Lie Groups in the Non-compact Setting
3.4 Elliptic Orbit and Its Quantization
3.5 [Q, R] = 0 for Holomorphic Discrete Series Representations
4 Discrete Series Representations
4.1 Generalized Hyperboloid O(p,q)/O(p-1,q)
4.2 Parameterization of Elliptic Coadjoint Orbits that Meet h
4.3 Geometric Quantization of Elliptic Coadjoint Orbits
5 Statement of Main Results
5.1 What Will Be the Counterpart of Kobayashi's Theorems in the Geometry of Coadjoint Orbits?
5.2 Coadjoint Geometry Corresponding to Kobayashi's Uniformly Bounded Multiplicity Theorem
5.3 O(p,q)↓O(p1,q1)×O(p2,q2)
5.4 G=O(p,q)G' = G1 ×G2 = O(p1, q1)×O(p2, q2)
5.5 Discrete Decomposability and Coadjoint Orbits
5.6 Explicit Branching Laws and Coadjoint Orbits
References
Discrete Branching Laws of Derived Functor Modules
1 Introduction
2 Cohomological Induction
3 Discretely Decomposable (g,K)-Modules
4 Localization of Cohomological Induction
5 Decomposition of Aq(λ)
6 Setting-up and Outline for Branching Laws
7 Branching Laws for Isolated Type
7.1 su(2m,2n)↓sp(m,n)
7.1.1
7.1.2
7.1.3
7.1.4
7.2 so(2m+1,2n)↓so(2m+1,k)⊕so(2n-k)
7.3 so(2m+1,2n+1)↓so(2m+1,k)⊕so(2n-k+1)
7.4 so(2m,2n)↓u(m,n)
7.5 so*(2n)↓so*(2n-2)⊕so(2)
7.6 so*(2n)↓u(n-1,1)
7.7 sp(m,n)↓sp(k,l)⊕sp(m-k,n-l)
7.8 sl(2n,C)↓sp(n,C)
7.9 sl(2n,C)↓su*(2n)⊕R
7.10 so(2n,C)↓so(2n-1,C)
7.11 so(2n,C)↓so(2n-1,1)
7.12 f4(-20)↓so(8,1)
7.12.1
7.12.2
7.13 e6(2)↓so*(10)⊕so(2)
7.14 e6(-14)↓so(8,2)⊕so(2)
7.15 e6(-14)↓f4(-20)
7.15.1
7.15.2
7.15.3
7.16 e7(-5)↓e6(-14)⊕so(2)
8 Branching Laws for Non-holomorphic Discrete Series Type
8.1 u(m,n)↓u(m,k)⊕u(n-k)
8.1.1
8.1.2
8.1.3
8.2 so(2m,n)↓so(2m,k)⊕so(n-k)
8.2.1
8.2.2
8.2.3
8.3 sp(m,n)↓sp(m,k)⊕sp(n-k)
8.3.1
8.3.2
8.3.3
8.4 sp(m,n)↓sp(m,k)⊕sp(n-k) (Isolated Type)
8.5 su(2,2n)↓sp(1,n)
8.6 so(4,2n)↓u(2,n)
8.6.1
8.6.2
8.7 f4(4)↓sp(1,2)⊕su(2)
8.8 f4(4)↓so(4,5)
8.8.1
8.8.2
8.9 e6(2)↓so(4,6)⊕so(2)
8.9.1
8.9.2
8.9.3
8.10 e6(2)↓su(2,4)⊕su(2)
8.11 e6(2)↓sp(1,3)
8.12 e6(2)↓f4(4)
8.12.1
8.12.2
8.12.3
8.12.4
8.13 e7(-5)↓so(4,8)⊕su(2)
8.14 e7(-5)↓su(2,6)
8.15 e7(-5)↓e6(2)⊕so(2)
8.15.1
8.15.2
8.15.3
8.16 e8(-24)↓so(4,12)
8.17 e8(-24)↓e7(-5)⊕su(2)
9 Branching Laws for Holomorphic Type
9.1 u(m,n)↓u(k,l)⊕u(m-k,n-l)
9.1.1
9.1.2
9.2 u(n,n)↓so*(2n)
9.3 u(n,n)↓sp(n,R)
9.4 so(2,n)↓so(2,k)⊕so(n-k)
9.5 so(2,2n)↓u(1,n)
9.5.1
9.5.2
9.6 so*(2n)↓u(m,n-m)
9.6.1
9.6.2
9.7 so*(2n)↓so*(2m)⊕so*(2n-2m)
9.7.1
9.7.2
9.7.3
9.8 sp(n,R)↓u(m,n-m)
9.9 sp(n,R)↓sp(m,R)⊕sp(n-m,R)
9.9.1
9.9.2
9.10 e6(-14)↓so(2,8)⊕so(2)
9.10.1
9.10.2
9.10.3
9.11 e6(-14)↓su(2,4)⊕su(2)
9.12 e6(-14)↓so*(10)⊕so(2)
9.12.1
9.12.2
9.12.3
9.13 e6(-14)↓su(1,5)⊕sl(2,R)
9.14 e7(-25)↓e6(-14)⊕so(2)
9.14.1
9.14.2
9.15 e7(-25)↓so(2,10)⊕sl(2,R)
9.16 e7(-25)↓su(6,2)
9.17 e7(-25)↓so*(12)⊕su(2)
10 Tensor Product of Holomorphic Representations
10.1 Tensor Product for u(m,n)
10.1.1
10.1.2
10.2 Tensor Product for so(2,2n+1)
10.3 Tensor Product for so(2,2n)
10.3.1
10.3.2
10.4 Tensor Product for so*(2n)
10.4.1
10.4.2
10.4.3
10.4.4
10.5 Tensor Product for sp(n,R)
10.5.1
10.5.2
10.6 Tensor Product for e6(-14)
10.6.1
10.6.2
10.6.3
10.7 Tensor Product for e7(-25)
11 Orbit Decomposition of Partial Flag Varieties
11.1 (GL(m+n),GL(m)×GL(n))
11.1.1
11.1.2
11.1.3
11.2 (GL(2n),Sp(n))
11.3 (SO(2m+2n),SO(2m)×SO(2n))
11.3.1
11.3.2
11.3.3
11.3.4
11.4 (SO(2m+2n),SO(2m+1)×SO(2n-1))
11.4.1
11.4.2
11.5 (SO(2n+1),SO(2n))
11.6 (SO(2n),GL(n))
11.6.1
11.6.2
11.6.3
11.7 (Sp(m+n),Sp(m)×Sp(n))
11.7.1
11.7.2
11.8 (E6, Spin (10)×SO(2))
11.9 (E6,SL(6)×Sp(1))
12 Formulas of Finite-Dimensional Representations
13 Some Equations of (g,K)-Modules
13.1 u(m,n) (Holomorphic)
13.2 so(2,2n)
13.3 so(2,2n-1)
13.4 so*(2n)
13.5 sp(n,R)
13.6 u(m,n) (Non-holomorphic)
13.7 so(2m,n)
13.8 sp(m,n)
13.9 Dual Pair Correspondence
References
Rankin-Cohen Brackets for Orthogonal Lie Algebras and Bilinear Conformally Equivariant Differential Operators
1 Introduction and Motivation
2 F-Method and Diagonal Branching Problem for Generalized Verma Modules
3 The Construction of Singular Vectors for Diagonal Branching Rules Applied to Scalar Generalized Verma Modules for so(n+1,1,R)
3.1 Description of the Representation
3.2 Reduction to Scalar Differential Equation in Two Variables
3.3 Polynomial Solutions of the Differential Equation in Two Variables Produced by the F-Method
4 Application: The Classification of Bilinear Conformally Equivariant Differential Operators on Line Bundles
References

2025-03-26