arXiv subjects

Toshiyuki Kobayashi

Publications and source records attributed to Toshiyuki Kobayashi.

At least 19 recordsLinked to original sources

arXiv2026

Rankin--Cohen brackets in Representation Theory

The Rankin--Cohen brackets provide a basic example of ``non-elementary" differential symmetry breaking operators. They can be interpreted as bi-differential operators remarkable for reflecting the structure of fusion rules for holomorphic discrete series representations of the Lie group $SL(2,\mathbb R)$ and are intimately connected to classical special polynomials. In this introductory article, we explore the combinatorial structure of these operators and discuss a general framework for constructing their higher-dimensional analogues from the representation-theoretic perspective on branching problems. The exposition is based on lectures delivered by the authors during the thematic semester ``Representation Theory and Noncommutative Geometry", held in Spring 2025 at the Henri Poincaré Institute in Paris.

math.RT↗

arXiv2026

Stability of Multiplicities in Symmetry Breaking: The sl_2 Case

This expository paper explains, in the case of $\mathfrak{sl}_2$, the ideas introduced in the preprints (arXiv:2509.17007, 2604.22262), which develop a new framework for the study of multiplicities in branching laws of representations, with particular emphasis on their dependence on representation parameters. Taking the Lie algebra $\mathfrak{sl}_2$ as a guiding example, we show that multiplicities, which are often computed via ad hoc, case-by-case arguments, are in fact governed by universal systems of linear inequalities. To describe these inequalities, we introduce the notion of \emph{fences}, which encode the piecewise-linear boundaries of regions in parameter space on which multiplicities remain constant. Within this framework, we give an explicit description of how multiplicities vary as parameters move inside reduced coherent families of representations. Our approach applies uniformly both to finite-dimensional representations and to admissible smooth Fréchet representations of real reductive Lie groups, and reveals a subtle and intrinsic interplay between the parameters of a group and those of its subgroup. As an application of the general theory, we establish stability results and explicit formulas that clarify and unify a variety of classical phenomena, including the Pieri rule, $K$-type formulas, fusion rules, and tensor products of Verma modules. In particular, the stability of fusion multiplicities provides a concrete manifestation of the theory. More broadly, this framework suggests a unified approach to branching multiplicities extending beyond the $\mathfrak{sl}_2$ case.

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arXiv2026

Stability of Branching Multiplicities for Orthogonal Gelfand Pairs

We propose a structural framework for branching multiplicities in representation theory, emphasizing their behavior under variation of infinitesimal characters. For the orthogonal reductive pairs $(G,G')$ with complexified Lie algebras $(\mathfrak{o}(n+1,\mathbb{C}), \mathfrak{o}(n,\mathbb{C}))$, we show that branching multiplicities are governed by universal systems of linear inequalities on the parameter space of reduced coherent families introduced in this paper. To describe the loci where multiplicities may change, we introduce \emph{fences}: piecewise-linear hypersurfaces that divide the parameter space into convex regions. We prove that the multiplicity function is locally constant on each such region bounded by these fences. The framework applies uniformly to finite-dimensional representations and to admissible smooth Fréchet representations of real reductive groups. It accounts for classical results such as the Weyl branching law and provides a unified explanation for a range of phenomena, including the Gross--Prasad conjecture, sporadic symmetry breaking operators, and fusion rules for Verma modules. These results establish a general paradigm for branching multiplicities in orthogonal Gelfand pairs.

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arXiv2025

Harish-Chandra's admissibility theorem and beyond

This article is a record of the lecture at the centennial conference for Harish-Chandra. The admissibility theorem of Harish-Chandra concerns the restrictions of irreducible representations to maximal compact subgroups. In this article, we begin with a brief explanation of two directions for generalizing his pioneering work to {\it{non-compact}} reductive subgroups: one emphasizes discrete decomposability with the finite multiplicity property, while the other focuses on finite/uniformly bounded multiplicity properties. We discuss how the recent representation-theoretic developments in these directions collectively offer a powerful method for the new spectral analysis of standard locally symmetric spaces, extending beyond the classical Riemannian setting.

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arXiv2025

How does the restriction of representations change under translations? A story for the general linear groups and the unitary groups

We present a new approach to symmetry breaking for pairs of real forms of $(GL(n, \mathbb{C}), GL(n-1, \mathbb{C}))$. Translation functors are powerful tools for studying families of representations of a single reductive group $G$. However, when applied to a pair of groups $G \supset G'$, they can significantly alter the nature of symmetry breaking between the representations of $G$ and $G'$, even within the same Weyl chamber of the direct product group $G \times G'$. We introduce the concept of "fences for the interleaving pattern", which provides a refinement of the usual notion of walls of Weyl chambers. We then establish a theorem stating that the multiplicity remains constant unless these "fences" are crossed, together with a new general vanishing theorem for symmetry breaking. These general results are illustrated with examples involving both tempered and non-tempered representations. In addition, we present a new non-vanishing theorem for period integrals for pairs of reductive symmetric spaces, which is further strengthened by this approach.

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arXiv2025

Translation functors, branching problems, and applications to the restriction of coherent cohomology of Shimura varieties

We study properties of the restriction of discrete series representations of $G=U(p,q)$ to $G'= U(p-1,q)$ and the corresponding symmetry breaking operators in $\operatorname{Hom}_{G'}(π|_{G'}, π')$. This leads to the introduction of elementary and coherent pairs of discrete series representations and their classification. Translations of symmetry breaking operators are defined via tensor products with finite-dimensional representations, which leads to the study of the coherent cohomology of discrete series representations under restriction and translations. This is applied to the study of cup products of coherent cohomology of associated Shimura varieties, and to the arithmetic of central values of certain Rankin--Selberg $L$-functions of $GL(n+1)\times GL(n)$.

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arXiv2025

Zariski-dense deformations of standard discontinuous groups for pseudo-Riemannian homogeneous spaces

Let $X=G/H$ be a homogeneous space of a Lie group $G$. When the isotropy subgroup $H$ is non-compact, a discrete subgroup $Γ$ may fail to act properly discontinuously on $X$. In this article, we address the following question: in the setting where $G$ and $H$ are reductive Lie groups and $Γ\backslash X$ is a standard quotient, to what extent can one deform the discrete subgroup $Γ$ while preserving the proper discontinuity of the action on $X$? We provide several classification results, including conditions under which local rigidity holds for compact standard quotients $Γ\backslash X$, when a standard quotient can be deformed into a non-standard quotient, a characterization of the largest Zariski-closure of discontinuous groups under small deformations, and conditions under which Zariski-dense deformations occur.

math.DG↗

arXiv2025

Deformations of Standard Locally Homogeneous Spaces

Let $X=G/H$ be a homogeneous space, where $G \supset H$ are reductive Lie groups. We ask: in the setting where $Γ\backslash G/H$ is a standard quotient, to what extent can the discrete subgroup $Γ$ be deformed while preserving the proper discontinuity of the $Γ$-action on $X$? We provide several classification results, including: conditions under which local rigidity holds for compact standard quotients $Γ\backslash X$; criteria for when a standard quotient can be deformed into a nonstandard one; a characterization of the maximal Zariski-closure of discontinuous groups under small deformations; and conditions under which Zariski-dense deformations occur. Proofs of the results stated in this paper are provided in detail in arXiv:2507.03476.

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arXiv2025

Proper Actions and Representation Theory

This exposition presents recent developments on proper actions, highlighting their connections to representation theory. It begins with geometric aspects, including criteria for the properness of homogeneous spaces in the setting of reductive groups. We then explore the interplay between the properness of group actions and the discrete decomposability of unitary representations realized on function spaces. Furthermore, two contrasting new approaches to quantifying proper actions are examined: one based on the notion of sharpness, which measures how strongly a given action satisfies properness; and another based on dynamical volume estimates, which measure deviations from properness. The latter quantitative estimates have proven especially fruitful in establishing temperedness criterion for regular unitary representations on $G$-spaces. Throughout, key concepts are illustrated with concrete geometric and representation-theoretic examples.

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arXiv2024

Spectral analysis on standard locally homogeneous spaces

Let $X=G/H$ be a reductive homogeneous space with $H$ noncompact, endowed with a $G$-invariant pseudo-Riemannian structure. Let $L$ be a reductive subgroup of $G$ acting properly on $X$ and $Γ$ a torsion-free discrete subgroup of $L$. Under the assumption that the complexification $X_{\mathbb C}$ is $L_{\mathbb C}$-spherical, we prove an explicit correspondence between spectral analysis on the standard locally homogeneous space $X_Γ=Γ\backslash X$ and on $Γ\backslash L$ via branching laws for the restriction to $L$ of irreducible representations of $G$. In particular, we prove that the pseudo-Riemannian Laplacian on $X_Γ$ is essentially self-adjoint, and that it admits an infinite point spectrum when $X_Γ$ is compact or $Γ\subset L$ is arithmetic. The proof builds on structural results for invariant differential operators on spherical homogeneous spaces with overgroups.

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arXiv2024

Recent advances in branching problems of representations

How does an irreducible representation of a group behave when restricted to a subgroup? This is part of branching problems, which are one of the fundamental problems in representation theory, and also interact naturally with other fields of mathematics. This expository paper is an up-to-date account on some new directions in representation theory highlighting the branching problems for real reductive groups and their related topics ranging from global analysis of manifolds via group actions to the theory of discontinuous groups beyond the classical Riemannian setting. This article is an outgrowth of the invited lecture that the author delivered at the commemorative event for the 70th anniversary of the re-establishment of the Mathematical Society of Japan, and originally appeared in Japanese in Sugaku 71 (2019).

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arXiv2024

A short proof for Rankin--Cohen brackets and generating operators

Motivated by the concept of "generating operators" for a countable family of operators introduced in the recent paper (arXiv:2306.16800), we find a method to reconstruct the Rankin--Cohen brackets from a very simple multivariable contour integral, and obtain a new proof of their covariance. We also establish a closed formula of the "generating operator" for the Rankin--Cohen brackets in full generality.

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arXiv2024

Generating operators of symmetry breaking -- from discrete to continuous

Based on the "generating operator" of the Rankin--Cohen brackets introduced in Kobayashi-Pevzner [arXiv:2306.16800], we present a method to construct various fundamental operators with continuous parameters such as invariant trilinear forms on infinite-dimensional representations, the Fourier and the Poisson transforms on the anti-de Sitter space, and integral symmetry breaking operators for the fusion rules, among others, out of a countable set of differential symmetry breaking operators.

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arXiv2023

A generating operator for Rankin-Cohen brackets

Motivated by the classical ideas of generating functions for orthogonal polynomials, we initiate a new line of investigation on "generating operators" for a family of differential operators between two manifolds. We prove a novel formula of the generating operators for the Rankin-Cohen brackets by using higher-dimensional residue calculus. Various results on the generating operators are also explored from the perspective of infinite-dimensional representation theory.

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arXiv2022

Temperedness criterion of the tensor product of parabolic induction for $GL_n$

We give a necessary and sufficient condition for a pair of parabolic subgroups $P$ and $Q$ of $G=GL_n(\mathbb{R})$ such that the tensor product of any two unitarily induced representations from $P$ and $Q$ are tempered. We also give an $L^p$-estimate of matrix coefficients of the regular representations on $L^2(G/L)$ when $L$ is a Levi subgroup of $G$.

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arXiv2022

Bounded multiplicity branching for symmetric pairs

We prove that any simply connected non-compact semisimple Lie group $G$ admits an infinite-dimensional irreducible representation $Π$ with bounded multiplicity property of the restriction $Π|_{G'}$ for all symmetric pairs $(G, G')$. We also discuss which irreducible representations $Π$ satisfy the bounded multiplicity property.

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arXiv2022

Conjectures on reductive homogeneous spaces

We address some conjectures and open problems in "analysis of symmetries" which include the study of non-commutative harmonic analysis and discontinuous groups for reductive homogeneous spaces beyond the classical framework: (1) discrete series for non-symmetric homogeneous spaces $G/H$; (2) discontinuous group $Γ$ for $G/H$ beyond the Riemannian setting; (3) analysis on pseudo-Riemannian locally homogeneous spaces.

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arXiv2022

Multiplicity in restricting minimal representations

We discuss the action of a subgroup on small nilpotent orbits, and prove a bounded multiplicity property for the restriction of minimal representations of real reductive Lie groups with respect to arbitrary reductive symmetric pairs.

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