The Schrödinger model for the minimal representation of the indefinite orthogonal group 𝑂(𝑝,𝑞)

Kobayashi, Toshiyuki; Mano, Gen

doi:10.1090/s0065-9266-2011-00592-7

Cited by 80 publications

(119 citation statements)

References 68 publications

(72 reference statements)

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“…This was first done in the context of minimal representations (see, e.g. [19][20][21]) and subsequently generalized to the so-called radial deformations (see, e.g. [2,8]).…”

Section: Introductionmentioning

confidence: 99%

On the Clifford-Fourier Transform

Bie

2011

International Mathematics Research Notices

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“…This was first done in the context of minimal representations (see, e.g. [19][20][21]) and subsequently generalized to the so-called radial deformations (see, e.g. [2,8]).…”

Section: Introductionmentioning

confidence: 99%

On the Clifford-Fourier Transform

Bie

2011

International Mathematics Research Notices

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“…Representation-theoretic considerations play a guiding role in formalizing problems there. In fact, we have seen in Theorem 2.6 that x brought us to the construction of conservative quantities for ultra-hyperbolic equations, whereas y leads us to the notion of a Fourier transform on the isotropic cone Ξ [3,35,36], as discussed below.…”

Section: §4 Deformation Of Fourier Transformsmentioning

confidence: 99%

“…Algebraically, F Ξ intertwines multiplication by coordinate functions ξ j (1 ≤ j ≤ p + q) with the Bargmann-Todorov operators R j (1 ≤ j ≤ p + q) which are mutually commuting differential operators of second order on Ξ (see [2], [36,Chapter 1]). …”

Section: §4 Deformation Of Fourier Transformsmentioning

confidence: 99%

“…We will discuss those operators in this generality in Section 4.4. An alternative approach was taken in [36] based on the Barnes-Mellin integral to find the kernel function of F Ξ for general p and q. §4.…”

Section: §4 Deformation Of Fourier Transformsmentioning

confidence: 99%

“…The aim of this article is to highlight the prominent role of Sato's idea of hyperfunctions and D-modules in the new developments in the analysis of minimal representations [4,35,36,40].…”

Section: §1 Introductionmentioning

confidence: 99%

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Algebraic Analysis of Minimal Representations

Kobayashi¹

2011

Publ. Res. Inst. Math. Sci.

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Small representations of a group yield large symmetries in the representation space. Analysis of minimal representations utilizes large symmetries in their geometric models, and serves as a driving force in creating new interesting problems that interact with other branches of mathematics.This article discusses the following three topics that arise from minimal representations of the indefinite orthogonal group:1. construction of conservative quantities for ultra-hyperbolic equations, 2. quantitative discrete branching laws, 3. deformation of the Fourier transform, with emphasis on the prominent role of Sato's ideas in algebraic analysis.

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

Kobayashi

2015

Progress in Mathematics

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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 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.

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The Schrödinger model for the minimal representation of the indefinite orthogonal group 𝑂(𝑝,𝑞)

Cited by 80 publications

References 68 publications

On the Clifford-Fourier Transform

On the Clifford-Fourier Transform

Algebraic Analysis of Minimal Representations

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

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