Tempered reductive homogeneous spaces

Benoist, Yves; Kobayashi, Toshiyuki

doi:10.4171/jems/578

Cited by 25 publications

(61 citation statements)

References 27 publications

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“…Putting together Theorems 1.1 and 1.2, we obtain the following Corollary. From Example 5.6 of [3], we see that if G = SO(p, q) and H = r i=1 SO(p i , q i ) with p = r i=1 p i , q = r i=1 q i , and 2(p i + q i ) ≤ p + q + 2 whenever p i q i = 0, then L 2 (G/H) is weakly contained in the regular representation. To the best of the authors' knowledge, Plancherel formulas are not known for the vast majority of these cases.…”

Section: Introductionmentioning

confidence: 99%

“…First, suppose G is a real, semisimple algebraic group and H ⊂ G is a reductive subgroup. In Theorem 4.1 of [3], Benoist and Kobayashi give a concrete and computable necessary and sufficient condition for Ind G H ½ = L 2 (G/H) to be weakly contained in the regular representation. Putting together Theorems 1.1 and 1.2, we obtain the following Corollary.…”

Section: Introductionmentioning

confidence: 99%

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Wave front sets of reductive Lie group representations

Harris¹,

He²,

Ólafsson³

2016

Duke Math. J.

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If G is a Lie group, H ⊂ G is a closed subgroup, and τ is a unitary representation of H, then the authors give a sufficient condition on ξ ∈ ig * to be in the wave front set of Ind G H τ . In the special case where τ is the trivial representation, this result was conjectured by Howe. If G is a real, reductive algebraic group and π is a unitary representation of G that is weakly contained in the regular representation, then the authors give a geometric description of WF(π) in terms of the direct integral decomposition of π into irreducibles. Special cases of this result were previously obtained by Kashiwara-Vergne, Howe, and Rossmann. The authors give applications to harmonic analysis problems and branching problems.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Wave front sets of reductive Lie group representations

Harris¹,

He²,

Ólafsson³

2016

Duke Math. J.

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“…We conclude this section by remarking that the recent work of Benoist and Kobayashi [BK15] provides a wealth of additional examples where the right hand side of Corollary 1.3 is nonempty.…”

Section: On Integrals Of Characters and Contoursmentioning

confidence: 89%

Wave front sets of reductive Lie group representations II

Harris

2017

Trans. Amer. Math. Soc.

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In this paper it is shown that the wave front set of a direct integral of singular, irreducible representations of a real, reductive algebraic group is contained in the singular set. Combining this result with the results of the first paper in this series, the author obtains asymptotic results on the occurrence of tempered representations in induction and restriction problems for real, reductive algebraic groups.temp Date: March 25, 2017. 2010 Mathematics Subject Classification. 22E46, 22E45, 43A85.

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“…Theorem 7.8 implies that the smooth representation Π ∞ of a tempered representation Π with nonzero (g, K)-cohomologies (see Example 7.4) occurs in C ∞ (G/G (k) ) if k ≤ n 2 + 1. On the other hand, for a reductive homogeneous space G/H, a general criterion for the unitary representation L 2 (G/H) to be tempered was proved in a joint work [1] with Y. Benoist by a geometric method. In particular, the unitary representation L 2 (G/G (k) ) is tempered if and only if k ≤ n 2 + 1, see [1,Ex.…”

Section: Application To Periods and Automorphic Form Theorymentioning

confidence: 99%

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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Tempered reductive homogeneous spaces

Abstract: 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.

Cited by 25 publications

References 27 publications

Wave front sets of reductive Lie group representations

Wave front sets of reductive Lie group representations

Wave front sets of reductive Lie group representations II

Conformal Symmetry Breaking on Differential Forms and Some Applications

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