Metaplectic tensor products for irreducible representations

Mezo, Paul

doi:10.2140/pjm.2004.215.85

Cited by 19 publications

(29 citation statements)

References 11 publications

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“…We start with a list of representations, one for each of GL r 1 , · · · , GL rn , and then form their tensor product to obtain a representation of M. However, since M is not simply the amalgamated direct product of the various GL r i , this construction cannot be generalized directly to the metaplectic case. Fortunately, we have a replacement, which is defined in Mezo [25]. We review the construction in this section.…”

Section: Notice That Glmentioning

confidence: 99%

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Fourier coefficients for theta representations on covers of general linear groups

Cai¹

2019

Trans. Amer. Math. Soc.

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Abstract. We show that the theta representations on certain covers of general linear groups support certain types of unique functionals. The proof involves two types of Fourier coefficients. The first are semi-Whittaker coefficients, which generalize coefficients introduced by Bump and Ginzburg for the double cover. The covers for which these coefficients vanish identically (resp. do not vanish for some choice of data) are determined in full. The second are the Fourier coefficients associated with general unipotent orbits. In particular, we determine the unipotent orbit attached, in the sense of Ginzburg, to the theta representations.

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Section: Notice That Glmentioning

confidence: 99%

“…To overcome this difficulty, a construction called the metaplectic tensor product has been introduced (see Section 3.4 and 4.4). The local version is developed in Mezo [25] and the global version is given in Takeda [33,34]. Roughly speaking, the construction goes as follows (both locally and globally).…”

Section: Introductionmentioning

confidence: 99%

Fourier coefficients for theta representations on covers of general linear groups

Cai¹

2019

Trans. Amer. Math. Soc.

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“…Recall that by [48] (Proposition 0.1.1), 2 only when n is even. This causes technical difficulties when trying to adapt definitions of metaplectic tensor product from GL n ( [42,61,80]) to G n , see…”

Section: 14mentioning

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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“…Namely, the support of χ π is contained in Z GLr M (n) . (Indeed, this argument by the distribution character is crucially used in [Me,Lemma 4.2]. ) This explains why π is essentially determined by the restriction π| Z GLr M (n) .…”

Section: Mezo's Metaplectic Tensor Productmentioning

confidence: 99%

Metaplectic Tensor Products for Automorphic Representation of (r)

Takeda¹

2016

Can. j. math.

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Abstract. Let M = GL r1 × · · · × GL r k ⊆ GL r be a Levi subgroup of GL r , where r = r 1 + · · · + r k , and M its metaplectic preimage in the n-fold metaplectic cover GL r of GL r . For automorphic representations π 1 , . . . , π k of GL r1 (A), . . . , GL r k (A), we construct (under a certain technical assumption, which is always satisfied when n = 2) an automorphic representation π of M(A) which can be considered as the "tensor product" of the representations π 1 , . . . , π k . This is the global analogue of the metaplectic tensor product defined by P. Mezo in the sense that locally at each place v, π v is equivalent to the local metaplectic tensor product of π 1,v , . . . , π k,v defined by Mezo. Then we show that if all of π i are cuspidal (resp. square-integrable modulo center), then the metaplectic tensor product is cuspidal (resp. square-integrable modulo center). We also show that (both locally and globally) the metaplectic tensor product behaves in the expected way under the action of a Weyl group element, and show the compatibility with parabolic inductions.

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Metaplectic tensor products for irreducible representations

Cited by 19 publications

References 11 publications

Fourier coefficients for theta representations on covers of general linear groups

Fourier coefficients for theta representations on covers of general linear groups

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

Metaplectic Tensor Products for Automorphic Representation of (r)

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