Wave front sets of reductive Lie group representations III

Harris, Brian; Weich, Tobias

doi:10.1016/j.aim.2017.03.025

Cited by 10 publications

(10 citation statements)

References 30 publications

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“…In the second part, the wave front set of representations plays a central role. Our argument is partly similar to [HHO16,Har18,HW17], but requires some new ingredients. It was proved in [HW17, Theorem 2.1] that the wave front set of L 2 (G R /H 0 ) equals the image of moment map.…”

Section: Note Thatmentioning

confidence: 64%

On the asymptotic support of Plancherel measures for homogeneous spaces

Harris¹,

Oshima²

2022

Preprint

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Let G be a real linear reductive group and let H be a unimodular, locally algebraic subgroup. Let supp L 2 (G/H) be the set of irreducible unitary representations of G contributing to the decomposition of L 2 (G/H), namely the support of the Plancherel measure. In this paper, we will relate supp L 2 (G/H) with the image of moment map from the cotangent bundleFor the homogeneous space X = G/H, we attach a complex Levi subgroup L X of the complexification of G and we show that in some sense "most" of representations in supp L 2 (G/H) are obtained as quantizations of coadjoint orbits O such that O ≃ G/L and that the complexification of L is conjugate to L X . Moreover, the union of such coadjoint orbits O coincides asymptotically with the moment map image. As a corollary, we show that L 2 (G/H) has a discrete series if the moment map image contains a nonempty subset of elliptic elements.2010 Mathematics Subject Classification. 22E46.

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Section: Note Thatmentioning

confidence: 64%

On the asymptotic support of Plancherel measures for homogeneous spaces

Harris¹,

Oshima²

2022

Preprint

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“…-The Kirillov-Kostant orbit methods works fairly well for tempered representations, see [11] for example.…”

Section: Tempered Representationsmentioning

confidence: 99%

Tempered homogeneous spaces II

Benoist¹,

Kobayashi²

2017

Preprint

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“…The first is to clarify and generalize the relationship between distinction and genericity studied in [PrSa19] to arbitrary G-spaces over local fields, Archimedean or not. The second is a search for a non-Archimedean analogue of the qualitative study of the Plancherel decomposition provided by [HW17].…”

Section: Introductionmentioning

confidence: 99%

“…Understanding Whittaker support allows, in the non-Archimedean case, to deduce exact information on wave front of individual distinguished representations leading to our answer to the first question. The study of WO(S(X)), which is our smooth replacement to the problem studied in [HW17], is carried out using the theory of invariant distributions and in particular the orbitwise technique introduced by Gelfand-Kazhdan [GK75] in the non-Archimedean case (and its various extensions and ramifications), that in some cases reduces the study of invariant distributions on a space to the study of invariant distributions on each of the orbits separately.…”

Section: Introductionmentioning

confidence: 99%

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Generalized Whittaker quotients of Schwartz functions on G-spaces

Gourevitch¹,

Sayag²

2020

Preprint

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Let G be a reductive group over a local field F of characteristic zero, Archimedean or not. Let X be a G-space. In this paper we study the existence of generalized Whittaker quotients for the space of Schwartz functions on X, considered as a representation of G. We show that the set of nilpotent elements of the dual space to the Lie algebra such that the corresponding generalized Whittaker quotient does not vanish contains the nilpotent part of the image of the moment map, and lies in the closure of this image. This generalizes recent results of Prasad and Sakellaridis.Applying our theorems to symmetric pairs (G, H) we show that there exists an infinitedimensional H-distinguished representation of G if and only if the real reductive group corresponding to the pair (G, H) is non-compact. For quasi-split G we also extend to the Archimedean case the theorem of Prasad stating that there exists a generic H-distinguished representation of G if and only if the real reductive group corresponding to the pair (G, H) is quasi-split.In the non-Archimedean case our result also gives bounds on the wave-front sets of distinguished representations.

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Wave front sets of reductive Lie group representations III

Cited by 10 publications

References 30 publications

On the asymptotic support of Plancherel measures for homogeneous spaces

On the asymptotic support of Plancherel measures for homogeneous spaces

Tempered homogeneous spaces II

Generalized Whittaker quotients of Schwartz functions on G-spaces

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