Descent Construction for GSpin groups

Hundley, Joseph; Sayag, Eitan

doi:10.1090/memo/1148

Cited by 25 publications

(29 citation statements)

References 62 publications

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“…The group GSpin 2n can be defined as the Levi subgroup of the maximal parabolic subgroup of Spin 2(n+1) corresponding to the subset of simple roots obtained by removing the first root ([49] Section 2.3). For a definition using a based root datum see [5,6,36].…”

Section: Contentsmentioning

confidence: 99%

The Double Cover of Odd General Spin Groups, Small Representations, and Applications

Kaplan¹

2015

J. Inst. Math. Jussieu

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Abstract. We construct local and global metaplectic double covers of odd general spin groups, using the cover of Matsumoto of spin groups. Following Kazhdan and Patterson, a local exceptional representation is the unique irreducible quotient of a principal series representation, induced from a certain exceptional character. The global exceptional representation is obtained as the multi-residue of an Eisenstein series, it is an automorphic representation and decomposes as the restricted tensor product of local exceptional representations. As in the case of the small representation of SO 2n+1 of Bump, Friedberg and Ginzburg, exceptional representations enjoy the vanishing of a large class of twisted Jacquet modules (locally), or Fourier coefficients (globally). Consequently they are useful in many settings, including lifting problems and Rankin-Selberg integrals. We describe one application, to a calculation of a co-period integral.Let G n = GSpin 2n+1 be the split odd general spin group of rank n + 1. Its derived group is G ′ n = Spin 2n+1 , the simple split simply-connected algebraic group of type B n . The group G n occurs as a Levi subgroup of G ′ n+1 . For a local field F of characteristic 0, letG ′ n+1 (F ) be the metaplectic double cover of G ′ n+1 (F ) defined by Matsumoto [58]. We can obtain a double coverG n (F ) of G n (F ) by restriction.Following Banks, Levi and Sepanski [10] we define a section s and a 2-cocycle σ of G ′ n+1 (F ), representing the cohomology class in H 2 (G ′ n+1 (F ), {±1}) ofG ′ n+1 (F ). We show that the restriction of σ to G n (F ) × G n (F ) satisfies a block-compatibility relation, with respect to standard Levi subgroups. This is a useful condition for studying parabolically induced representations.Fix a Borel subgroup in G n (F ). The preimageT n+1 (F ) of the maximal torus T n+1 (F ) in the cover is a two step nilpotent subgroup, its irreducible genuine representations are parameterized by genuine characters of its center CT n+1 (F ) . The analogous theory for covers of GL n was developed by Kazhdan and Patterson [48], who studied a special class of genuine characters which they called "exceptional". In our setting these are characters χ of CT n+1 (F ) satisfying χ(α ∨ * (x l(α) )) = x for all simple roots α of G n and x ∈ F * , where α ∨ * is a certain lift of the coroot α ∨ to the cover and l(α) is the length of α.We use an exceptional character χ to construct a genuine principal series representation ofG n (F ), which has a unique irreducible quotient denoted Θ = Θ Gn,χ . The quotient Θ is an exceptional representation, or a small representation in the terminology of Bump, Friedberg and Ginzburg [16]. Our purpose is to develop a theory of these representations.Let O be a unipotent class of G n , it corresponds to a partition of 2n + 1 for which an even number appears with an even multiplicity. Let V O be the corresponding unipotent subgroup. We consider certain characters of V O (F ), called "generic". Roughly, a character ψ of V O (F ) is generic if it is in general p...

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Section: Contentsmentioning

confidence: 99%

The Double Cover of Odd General Spin Groups, Small Representations, and Applications

Kaplan¹

2015

J. Inst. Math. Jussieu

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“…At any rate, Arthur's work is not a prerequisite for the formulation of Conjecture 1.2. In fact, one can easily modify Conjecture 1.2 for Gspin groups using the work of Asgari-Shahidi [AS06,AS11] and Hundley-Sayag [HS09,HS11].…”

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confidence: 99%

A conjecture on Whittaker–Fourier coefficients of cusp forms

Lapid

Mao

2015

Journal of Number Theory

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“…The group G n has been the focus of study of a few recent works, among which are the works of Asgari [1,2] on local L-functions, Asgari and Shahidi [3,4] on functoriality and Hundley and Sayag [31] on the descent construction.…”

Section: Theorem 11 the Space Of θ ⊗ θ As A Representation Of G N (Fmentioning

confidence: 99%

“…The group G n = GSpin 2n+1 is an F-split connected reductive algebraic group, which can be defined using a based root datum as in [2,3,31]. It is also embedded in G n+1 = Spin 2n+3 as the Levi part of the parabolic subgroup corresponding to Δ G n+1 −{α 1 } (see [52]).…”

Section: The Group Gspin 2n+1mentioning

confidence: 99%

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Theta distinguished representations, inflation and the symmetric square L-function

Kaplan

2016

Math. Z.

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A theta distinguished representation is a quotient of a tensor of exceptional representations, where "exceptional" is in the sense of Kazhdan and Patterson. We study relations between theta distinguished representations of GL n and GSpin 2n+1 . In the case of GSpin 2n+1 (or SO 2n+1 ) exceptional (or small) representations were constructed by Bump, Friedberg and Ginzburg. We prove a Rodier-type hereditary property: a tempered representation τ is distinguished if and only if the representation I(τ ) induced to GSpin 2n+1 is distinguished, and the multiplicity of both quotients is at most one. If τ is supercuspidal and distinguished, then so is the Langlands quotient of I(τ ). As a corollary, we characterize supercuspidal distinguished representations, in terms of the pole of the symmetric square L-function at s = 0.

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Descent Construction for GSpin groups

Cited by 25 publications

References 62 publications

The Double Cover of Odd General Spin Groups, Small Representations, and Applications

The Double Cover of Odd General Spin Groups, Small Representations, and Applications

A conjecture on Whittaker–Fourier coefficients of cusp forms

Theta distinguished representations, inflation and the symmetric square L-function

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