Analysis on the minimal representation of O(p,q) I. Realization via conformal geometry

Kobayashi, Toshiyuki; Ørsted, Bent

doi:10.1016/s0001-8708(03)00012-4

Cited by 74 publications

(80 citation statements)

References 23 publications

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“…The differential operators D X on C ∞ (R m−2|2n ) for X ∈ osp(m|2n) defined in equation (23) and calculated explicitly in Theorem 9, satisfy the relation…”

Section: This Is Zero Sincementioning

confidence: 98%

Joseph Ideals and Harmonic Analysis for

Coulembier

Somberg

Souček

2013

International Mathematics Research Notices

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The Joseph ideal in the universal enveloping algebra U(so(m)) is the annihilator ideal of the so(m)-representation on the harmonic functions on R m−2 . The Joseph ideal for sp(2n) is the annihilator ideal of the Segal-Shale-Weil (metaplectic) representation. Both ideals can be constructed in a unified way from a quadratic relation in the tensor algebra ⊗g for g equal to so(m) or sp(2n). In this paper we construct two analogous ideals in ⊗g and U(g) for g the orthosymplectic Lie superalgebra osp(m|2n) = spo(2n|m) and prove that they have unique characterizations that naturally extend the classical case. Then we show that these two ideals are the annihilator ideals of respectively the osp(m|2n)-representation on the spherical harmonics on R m−2|2n and a generalization of the metaplectic representation to spo(2n|m). This proves that these ideals are reasonable candidates to establish the theory of Joseph-like ideals for Lie superalgebras. We also discuss the relation between the Joseph ideal of osp(m|2n) and the algebra of symmetries of the super conformal Laplace operator, regarded as an intertwining operator between principal series representations for osp(m|2n). As a side result we obtain the proof of a conjecture of M. Eastwood about the Cartan product of irreducible representations of semisimple Lie algebras made in [Bull.

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“…The differential operators D X on C ∞ (R m−2|2n ) for X ∈ osp(m|2n) defined in equation (23) and calculated explicitly in Theorem 9, satisfy the relation…”

Section: This Is Zero Sincementioning

confidence: 98%

Joseph Ideals and Harmonic Analysis for

Coulembier

Somberg

Souček

2013

International Mathematics Research Notices

View full text Add to dashboard Cite

show abstract

“…The study of minimal representations of other reductive groups is relatively new, and it is only in the last two decades that various models of minimal representations have been proposed and studied extensively (see [6,7,13,17,21,27,39,40,46,47,48,49,50,52,63,73,80] for instance).…”

Section: Minimal Representations Of Simple Lie Groupsmentioning

confidence: 99%

“…This book is a continuation of a series of our research projects [43,44,46,47,48,49]. Our motif is to open up and develop geometric analysis of a single infinite dimensional representation, namely, the minimal representation π of the indefinite (even) orthogonal group.…”

Section: Introductionmentioning

confidence: 99%

“…The problem of finding the explicit inner product was solved in the previous paper [47] as the theory of conserved quantities for ultra-hyperbolic equations, such as the energy for the wave equation.…”

Section: Introductionmentioning

confidence: 99%

“…The understanding of the whole group action was a missing piece of [47]. This problem is reduced to finding the generalization of the Fourier-Hankel transform on the isotropic cone C, namely, the above mentioned operator F C .…”

Section: Introductionmentioning

confidence: 99%

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Conformal Symmetry Breaking on Differential Forms and Some Applications

Kobayashi

2019

Trends in Mathematics

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Rapid progress has been made recently on symmetry breaking operators for real reductive groups. Based on Program A-C for branching problems (T. Kobayashi [Progr. Math. 2015]), we illustrate a scheme of the classification of (local and nonlocal) symmetry breaking operators by an example of conformal representations on differential forms on the model space (X, Y ) = (S n , S n−1 ), which generalizes the scalar case (Kobayashi-Speh [Mem. Amer. Math. Soc. 2015]) and the case of local operators (Kobayashi-Kubo-Pevzner [Lect. Notes Math. 2016]). Some applications to automorphic form theory, motivations from conformal geometry, and the methods of proof are also discussed.

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Analysis on the minimal representation of O(p,q) I. Realization via conformal geometry

Cited by 74 publications

References 23 publications

Joseph Ideals and Harmonic Analysis for

Joseph Ideals and Harmonic Analysis for

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

Conformal Symmetry Breaking on Differential Forms and Some Applications

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