Metaplectic tori over local fields

Weissman, Martin H.

doi:10.2140/pjm.2009.241.169

Cited by 24 publications

(20 citation statements)

References 15 publications

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“…By the Stone von-Neumann theorem (cf. [We1,Theorem 3.1] and [Mc1,Theorem 3]), the construction χ → i(χ) gives a bijection between isomorphism classes of genuine representations of Z(T ) and T . Since we consider an unramified covering group G in this paper, we take A to be Z(T ) · (K ∩ T ) from now.…”

Section: Resultsmentioning

confidence: 99%

Distinguished theta representations for certain covering groups

Gao

2017

Pacific J. Math.

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Abstract. For Brylinski-Deligne covering groups of an arbitrary split reductive group, we consider theta representations attached to certain exceptional genuine characters. The goal of the paper is to study the dimension of the space of Whittaker functionals of a theta representation. In particular, we investigate when the dimension is exactly one, in which case the theta representation is called distinguished. For this purpose, we first give effective lower and upper bounds for the dimension of Whittaker functionals for general theta representations. As a consequence, the dimension in many cases can be reduced to simple combinatorial computations, e.g., the Kazhdan-Patterson covering groups of the general linear groups, or covering groups whose complex dual groups (à la Finkelberg-Lysenko-McNamara-Reich) are of adjoint type. In the second part of the paper, we consider coverings of certain semisimple simply-connected groups and give necessary and sufficient condition for the theta representation to be distinguished. There are subtleties arising from the relation between the rank and the degree of the covering group. However, in each case we will determine the exceptional character such that its associated theta representation is distinguished.

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Section: Resultsmentioning

confidence: 99%

Distinguished theta representations for certain covering groups

Gao

2017

Pacific J. Math.

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“…Brylinski and Deligne [BD01] provided a general framework, functorial in nature, of a covering group for any split reductive group. Their construction inspired numerous works on extending the Langlands program to those coverings, including [Sav04,Wei09,Wei11,McN12,Li14a,Wei14,GG18,Wei18a]. The work here can be regarded as a first attempt to launch the analytic counterpart of those efforts, namely a theory of integral representations and local factors for covering groups.…”

Section: Note That Spmentioning

confidence: 98%

Doubling Constructions and Tensor Product $L$-Functions: coverings of the symplectic group

Kaplan

2019

Preprint

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In this work we develop an integral representation for the partial L-function of a pair π × τ of genuine irreducible cuspidal automorphic representations, π of the m-fold covering of Matsumoto of the symplectic group Sp 2n , and τ of a certain covering group of GL k , with arbitrary m, n and k. Our construction is based on the recent extension by Cai, Friedberg, Ginzburg and the author, of the classical doubling method of Piatetski-Shapiro and Rallis, from rank-1 twists to arbitrary rank twists. We prove a basic global identity for the integral and compute the local integrals with unramified data. Possible applications include an analytic definition of local factors for representations of covering groups, and a Shimura type lift of representations from covering groups to general linear groups.

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“…On démontrera dans §3.5 que les K 2 -torseurs multiplicatifs de Brylinski-Deligne [13], qui généralisent la construction de Steinberg, Moore et Matsumoto [22], fournissent des revêtements vérifiant nos hypothèses. La démonstration est basé sur un résultat de Weissman [31]. Notons en passant que la démonstration sera beaucoup plus simple si l'on considère un revêtement tel que deuxéléments dansG commutent si et seulement si leurs images par p commutent.…”

Section: Introductionunclassified

La formule des traces pour les revêtements de groupes réductifs connexes. I. Le développement géométrique fin

2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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RésuméOnétudie la partie spécifique de la formule des traces d'Arthur-Selberg pour certains revêtements des groupes réductifs connexes. Comme un premier pas vers la formules des traces invariante, on exprime le côté géométrique en termes des intégrales orbitales pondérées. Les résultats s'appliquent, en particulier, aux revêtements construits par Brylinski et Deligne.MSC classification (2010) : 11F72, 11F70.

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Metaplectic tori over local fields

Cited by 24 publications

References 15 publications

Distinguished theta representations for certain covering groups

Distinguished theta representations for certain covering groups

Doubling Constructions and Tensor Product $L$-Functions: coverings of the symplectic group

La formule des traces pour les revêtements de groupes réductifs connexes. I. Le développement géométrique fin

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