arXiv subjects

Toshiyuki Kobayashi

Publications and source records attributed to Toshiyuki Kobayashi.

At least 55 records · Page 3Linked to original sources

arXiv2016

Conformal symmetry breaking operators for differential forms on spheres

We give a complete classification of conformally covariant differential operators between the spaces of $i$-forms on the sphere $S^n$ and $j$-forms on the totally geodesic hypersphere $S^{n-1}$. Moreover, we find explicit formulæ for these new matrix-valued operators in the flat coordinates in terms of basic operators in differential geometry and classical orthogonal polynomials. We also establish matrix-valued factorization identities among all possible combinations of conformally covariant differential operators. The main machinery of the proof is the "F-method" based on the "algebraic Fourier transform of Verma modules" (Kobayashi-Pevzner [Selecta Math. 2016]) and its extension to matrix-valued case developed here. A short summary of the main results was announced in [C. R. Acad. Sci. Paris, 2016].

math.DG↗

arXiv2016

Global analysis by hidden symmetry

Hidden symmetry of a G'-space X is defined by an extension of the G'-action on X to that of a group G containing G' as a subgroup. In this setting, we study the relationship between the three objects: (A) global analysis on X by using representations of G (hidden symmetry); (B) global analysis on X by using representations of G'; (C) branching laws of representations of G when restricted to the subgroup G'. We explain a trick which transfers results for finite-dimensional representations in the compact setting to those for infinite-dimensional representations in the noncompact setting when $X_C$ is $G_C$-spherical. Applications to branching problems of unitary representations, and to spectral analysis on pseudo-Riemannian locally symmetric spaces are also discussed.

math.RT↗

arXiv2016

Intrinsic sound of anti-de Sitter manifolds

As is well-known for compact Riemann surfaces, eigenvalues of the Laplacianbare distributed discretely and most of eigenvalues vary viewed as functions on the Teichmuller space. We discuss a new feature in the Lorentzian geometry, or more generally, in pseudo-Riemannian geometry. One of the distinguished features is that $L^2$-eigenvalues of the Laplacian may be distributed densely in R in pseudo-Riemannian geometry. For three-dimensional anti-de Sitter manifolds, we also explain another feature proved in joint with F. Kassel [Adv. Math. 2016] that there exist countably many $L^2$-eigenvalues of the Laplacian that are stable under any small deformation of anti-de Sitter structure.

math.DG↗

arXiv2016

Classification of differential symmetry breaking operators for differential forms

We give a complete classification of conformally covariant differential operators between the spaces of differential $i$-forms on the sphere $S^n$ and $j$-forms on the totally geodesic hypersphere $S^{n-1}$ by analyzing the restriction of principal series representations of the Lie group $O(n+1,1)$. Further, we provide explicit formulæ for these matrix-valued operators in the flat coordinates and find factorization identities for them.

math.DG↗

arXiv2015

A program for branching problems in the representation theory of real reductive groups

We wish to understand how irreducible representations of a group G behave when restricted to a subgroup G' (the branching problem). Our primary concern is with representations of reductive Lie groups, which involve both algebraic and analytic approaches. We divide branching problems into three stages: (A) abstract features of the restriction; (B) branching laws (irreducible decompositions of the restriction); and (C) construction of symmetry breaking operators on geometric models. We could expect a simple and detailed study of branching problems in Stages B and C in the settings that are {\it{a priori}} known to be "nice" in Stage A, and conversely, new results and methods in Stage C that might open another fruitful direction of branching problems including Stage A. The aim of this article is to give new perspectives on the subjects, to explain the methods based on some recent progress, and to raise some conjectures and open questions.

math.RT↗

arXiv2015

Branching laws for Verma modules and applications in parabolic geometry. I

We initiate a new study of differential operators with symmetries and combine this with the study of branching laws for Verma modules of reductive Lie algebras. By the criterion for discretely decomposable and multiplicity-free restrictions of generalized Verma modules [T. Kobayashi, http://dx.doi.org/10.1007/s00031-012-9180-y {Transf. Groups (2012)}], we are brought to natural settings of parabolic geometries for which there exist unique equivariant differential operators to submanifolds. Then we apply a new method (F-method) relying on the Fourier transform to find singular vectors in generalized Verma modules, which significantly simplifies and generalizes many preceding works. In certain cases, it also determines the Jordan--Hölder series of the restriction for singular parameters. The F-method yields an explicit formula of such unique operators, for example, giving an intrinsic and new proof of Juhl's conformally invariant differential operators [Juhl, http://dx.doi.org/10.1007/978-3-7643-9900-9 {Progr. Math. 2009}] and its generalizations. This article is the first in the series, and the next ones include their extension to curved cases together with more applications of the F-method to various settings in parabolic geometries.

math.RT↗

arXiv2015

Global uniqueness of small representations

We prove that automorphic representations whose local components are certain small representations have multiplicity one. The proof is based on the multiplicity-one theorem for certain functionals of small representations, also proved in this paper.

math.RT↗

arXiv2014

Differential symmetry breaking operators. I-Genreal theory and F-method. II-Rankin-Cohen Operators for Symmetric Pairs

Part I. We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method (F-method) based on the algebraic Fourier transform for generalized Verma modules, which characterizes differential symmetry breaking operators by means of certain systems of partial differential equations. In contrast to the setting of real flag varieties, continuous symmetry breaking operators of Hermitian symmetric spaces are proved to be differential operators in the holomorphic setting. In this case symmetry breaking operators are characterized by differential equations of second order via the F-method. Part II. Rankin--Cohen brackets are symmetry breaking operators for the tensor product of two holomorphic discrete series representations of SL(2,R). We address a general problem to find explicit formulae for such intertwining operatorsin the setting of multiplicity-free branching laws for reductive symmetric pairs. We discover explicit formulae of new differential symmetry breaking operators for all the six different complex geometries arising from semisimple symmetric pairs of split rank one, and reveal an intrinsic reason why the coefficients of orthogonal polynomials appear in these operators (Rankin--Cohen type) in the three geometries and why normal derivatives are symmetry breaking operators in the other three cases. Further, we analyze a new phenomenon that the multiplicities in the branching laws of Verma modules may jump up at singular parameters.

math.RT↗

arXiv2014

Vector-valued covariant differential operators for the Möbius transformation

We obtain a family of functional identities satisfied by vector-valued functions of two variables and their geometric inversions. For this we introduce particular differential operators of arbitrary order attached to Gegenbauer polynomials. These differential operators are symmetry breaking for the pair of Lie groups $(SL(2,\mathbb C), SL(2,\mathbb R))$ that arise from conformal geometry.

math.RT↗

arXiv2014

Shintani functions, real spherical manifolds, and symmetry breaking operators

For a pair of reductive groups $G \supset G'$, we prove a geometric criterion for the space $Sh(λ, ν)$ of Shintani functions to be finite-dimensional in the Archimedean case. This criterion leads us to a complete classification of the symmetric pairs $(G,G')$ having finite-dimensional Shintani spaces. A geometric criterion for uniform boundedness of $dim Sh(λ, ν)$ is also obtained. Furthermore, we prove that symmetry breaking operators of the restriction of smooth admissible representations yield Shintani functions of moderate growth, of which the dimension is determined for $(G, G') = (O(n+1,1), O(n,1))$.

math.RT↗

arXiv2014

Classification of finite-multiplicity symmetric pairs

We give a complete classification of the reductive symmetric pairs (G,H) for which the homogeneous space $(G \times H)/diag(H)$ is real spherical in the sense that a minimal parabolic subgroup has an open orbit. Combining with a criterion established in [T. Kobayashi--T. Oshima, Adv. Math. 2013], we give a necessary and sufficient condition for a reductive symmetric pair $(G,H)$ such that the multiplicities for the branching law of the restriction any admissible smooth representation of $G$ to $H$ have finiteness/boundedness property.

math.RT↗

arXiv2013

Poincaré series for non-Riemannian locally symmetric spaces

The discrete spectrum of the Laplacian has been extensively studied on reductive symmetric spaces and on Riemannian locally symmetric spaces. Here we examine it for the first time in the general setting of non-Riemannian, reductive, locally symmetric spaces. For any non-Riemannian, reductive symmetric space X on which the discrete spectrum of the Laplacian is nonempty, and for any discrete group of isometries Gamma whose action on X is sufficiently proper, we construct L^2-eigenfunctions of the Laplacian on X_{Gamma}:=Gamma\X for an infinite set of eigenvalues. These eigenfunctions are obtained as generalized Poincaré series, i.e. as projections to X_{Gamma} of sums, over the Gamma-orbits, of eigenfunctions of the Laplacian on X. We prove that the Poincaré series we construct still converge, and define nonzero L^2-functions, after any small deformation of Gamma, for a large class of groups Gamma. In other words, the infinite set of eigenvalues we construct is stable under small deformations. This contrasts with the classical setting where the nonzero discrete spectrum varies on the Teichmüller space of a compact Riemann surface. We actually construct joint L^2-eigenfunctions for the whole commutative algebra of invariant differential operators on X_{Gamma}.

math.RT↗

arXiv2013

Temperedness of reductive homogeneous spaces

Let G be a semisimple algebraic Lie group and H a reductive subgroup. We find geometrically the best even integer p for which the representation of G in L^2(G/H) is almost L^{p}. As an application, we give a criterion which detects whether this representation is tempered.

math.RT↗

arXiv2013

Symmetry breaking for representations of rank one orthogonal groups

We give a complete classification of intertwining operators (symmetry breaking operators) between spherical principal series representations of G=O(n+1,1) and G'=O(n,1). We construct three meromorphic families of the symmetry breaking operators, and find their distribution kernels and their residues at all poles explicitly.Symmetry breaking operators at exceptional discrete parameters are thoroughly studied. We obtain closed formulae for the functional equations which the composition of the the symmetry breaking operators with the Knapp-Stein intertwining operators of $G$ and G' satisfy, and use them to determine the symmetry breaking operators between irreducible composition factors of the spherical principal series representations of G and G'. Some applications are included.

math.RT↗

arXiv2013

F-method for symmetry breaking operators

We provide some insights in the study of branching problems of reductive groups, and a method of investigations into symmetry breaking operators. First, we give geometric criteria for finiteness property of linearly independent continuous (respectively, differential) operators that intertwine two induced representations of reductive Lie groups and their reductive subgroups. Second, we extend the F-method known for local operators to non-local operators. We then illustrate the idea by concrete examples in conformal geometry, and explain how the F-method works for detailed analysis of symmetry breaking operators, e.g., finding functional equations and explicit residue formulae of regular symmetry breaking operators with meromorphic parameters.

math.RT↗

arXiv2013

Finite multiplicity theorems for induction and restriction

We find upper and lower bounds of the multiplicities of irreducible admissible representations $π$ of a semisimple Lie group $G$ occurring in the induced representations $Ind_H^Gτ$ from irreducible representations $τ$ of a closed subgroup $H$. As corollaries, we establish geometric criteria for finiteness of the dimension of $Hom_G(π,Ind_H^G τ)$ (induction) and of $Hom_H(π|_H,τ)$ (restriction) by means of the real flag variety $G/P$, and discover that uniform boundedness property of these multiplicities is independent of real forms and characterized by means of the complex flag variety.

math.RT↗

arXiv2013

Transmission-phase measurement of the 0.7 anomaly in a quantum point contact

We measure the transmission phase of a quantum point contact (QPC) at a low carrier density in which electron interaction is expected to play an important role and anomalous behaviors are observed. In the first conductance plateau, the transmission phase shifts monotonically as the carrier density is decreased by the gate voltage. When the conductance starts to decrease, in what is often called the 0.7 regime, the phase exhibits an anomalous increase compared with the noninteracting model. The observation implies an increase in the wave vector as the carrier density is decreased, suggesting a transition to a spin-incoherent Luttinger liquid.

cond-mat.mes-hall↗

arXiv2013

F-method for constructing equivariant differential operators

Using an algebraic Fourier transform of operators, we develop a method (F-method) to obtain explicit highest weight vectors in the branching laws by differential equations. This article gives a brief explanation of the F-method and its applications to a concrete construction of some natural equivariant operators that arise in parabolic geometry and in automorphic forms.

math.RT↗