Spectrum Generating Operators and Intertwining Operators for Representations Induced from a Maximal Parabolic Subgroup

Branson, Thomas; Ólafsson, Gestur; Ørsted, Bent

doi:10.1006/jfan.1996.0008

Cited by 61 publications

(93 citation statements)

References 12 publications

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“…In particular, P θ γ agrees with the intertwining operators on the CR sphere calculated in [9,36,8]. Moreover, we can write explicitly…”

Section: Scattering Theory and The Conformal Fractional Sub-laplaciansupporting

confidence: 71%

“…In the particular case of the Heisenberg group H n , they are simply the intertwining operators on the CR sphere calculated in [8,51,9] using classical representation theory tools. It is precisely the article by Branson, Fontana and Morpurgo [8] that underlined the importance and nice PDE properties of these operators.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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An extension problem for the CR fractional Laplacian

Frank

González

Monticelli

et al. 2015

Advances in Mathematics

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“…In particular, P θ γ agrees with the intertwining operators on the CR sphere calculated in [9,36,8]. Moreover, we can write explicitly…”

Section: Scattering Theory and The Conformal Fractional Sub-laplaciansupporting

confidence: 71%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

An extension problem for the CR fractional Laplacian

Frank

González

Monticelli

et al. 2015

Advances in Mathematics

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“…These are differential operators D 2l,k , on forms of all even orders 2l, on differential forms of all orders k, on the double cover of the n-dimensional compactified Minkowski space. Tom's early ideas were put into a representation theoretic framework and generalised in the significant work [37], withÓlafsson and Ørsted. There they used a spectrum generating operator, constructed from a combination of quadratic Casimirs and the development of ideas as above to construct intertwinors for representations induced from a maximal parabolic subgroup.…”

Section: Spectrum Generating Functions and Intertwining Operatorsmentioning

confidence: 99%

The Research of Thomas P. Branson

Eastwood¹

2008

SIGMA

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Abstract. The Midwest Geometry Conference 2007 was devoted to the substantial mathematical legacy of Thomas P. Branson who passed away unexpectedly the previous year. This contribution to the Proceedings briefly introduces this legacy. We also take the opportunity of recording his bibliography. Thomas Branson was on the Editorial Board of SIGMA and we are pleased that SIGMA is able to publish the Proceedings.Key words: conformal differential geometry; Q-curvature; spectrum geometry; intertwining operators; heat kernel asymptotics; global geometric invariants 2000 Mathematics Subject Classification: 01A70; 22E46; 22E70; 53A30; 53A55; 53C21; 58J52; 58J60; 58J70In memory of Thomas P. Branson (1953 Tom's research interests ranged broadly; this is perhaps best indicated by the diverse group of collaborators listed implicitly in his bibliography, which follows this section. Tom was also known for his general knowledge in many areas of mathematics, and many of his colleagues will remember well his prompt and in-depth replies to email questions.In the 1980s Tom's research on conformal invariance was significantly ahead of its time. An enduring theme of his research was the natural interplay between invariance and the underlying symmetry groups. Tom's work continues to motivate and inspire a thriving and impressive international research effort. Many of the conference speakers have contributed to the various research trends that Tom Branson started.It is impossible in a written summary to do justice to Tom's research career. Here we shall outline just a few directions that we feel to be especially significant. Q-curvature and extremal problemsTom Branson is perhaps most well known for his definition, development, and application of a new curvature quantity in Riemannian geometry. For dimension 4 this first entered the public arena in his joint work [22] with Ørsted, but it was extended to all even dimensions and developed significantly in [29] and [36]; these days it is usually termed "Branson's Q-curvature". A key feature is its conformal transformation law in dimension n, e −nω Q = Q + P ω,

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“…In [3], a method was developed for finding the spectrum of intertwining operators for certain representations of semisimple groups. In this calculation scheme, one first inputs the spectral data of a differential spectrum generating operator.…”

Section: Introductionmentioning

confidence: 99%

“…The spectra of the other operators is less well known, but nevertheless can be obtained by specializing formulas in the literature (for example [3]). However, the philosophy is avoid all heavy machinery, and derive the spectra just from a couple of elementary operator commutation relations.…”

Section: Introductionmentioning

confidence: 99%

Spontaneous generation of eigenvalues

Branson¹,

Ørsted²

2006

Journal of Geometry and Physics

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We show that the action of conformal vector fields on functions on the sphere determines the spectrum of the Laplacian (or the conformal Laplacian), without further input of information. The spectra of intertwining operators (both differential and non-local) with principal part a power of the Laplacian follows as a corollary. An application of the method is the sharp form of Gross' entropy inequality on the sphere. The same method gives the spectrum of the Dirac operator on the sphere, as well as of a continuous family of nonlocal intertwinors, and an infinite family of odd-order differential intertwinors.

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Spectrum Generating Operators and Intertwining Operators for Representations Induced from a Maximal Parabolic Subgroup

Cited by 61 publications

References 12 publications

An extension problem for the CR fractional Laplacian

An extension problem for the CR fractional Laplacian

The Research of Thomas P. Branson

Spontaneous generation of eigenvalues

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