Periods of automorphic forms, poles of L-functions and functorial lifting

Ginzburg, David; Jiang, Di Hua; Soudry, David

doi:10.1007/s11425-010-4020-9

Cited by 11 publications

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“…Among the examples given in [29] is the case of G = SO 2n+1 and M = GL n . Let A + be the subgroup of idèles of F whose finite components are trivial, and Archimedean components are equal, real, and positive.…”

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

Kaplan¹

2015

J. Inst. Math. Jussieu

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“…The residual representation E π is the space spanned by the residues E 1/2 (·; ρ) of E(g; ρ, s) at s = 1/2. The following result formulated in [20,(Theorem 3.3)] was proved in a series of works [7,9,20,40]: the following conditions are equivalent.…”

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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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“…This idea has been used in multiple works of Ginzburg, Jiang, Rallis and Soudry, see [Jia98,GJR04,GJR05,GJR09] to cite a few. The reference [GJS10] contains a comprehensive account of the method and results that can and have been obtained through it. See also [IY,GL07] for different approaches.…”

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A G2-period of a Fourier coefficient of an Eisenstein series on E6

2019

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We calculate a G2-period of a Fourier coefficient of a cuspidal Eisenstein series on the split simply-connected group E6, and relate this period to the Ginzburg-Rallis period of cusp forms on GL6. This gives us a relation between the Ginzburg-Rallis period and the central value of the exterior cube L-function of GL6.1 2 AARON POLLACK, CHEN WAN, AND MICHA L ZYDOR Let us describe the method in more detail. Let Q be the parabolic subgroup of the split, simply connected E 6 whose Levi subgroup is of D 4 type. In section 1.6 we define a certain generic character ξ of the unipotent radical N of Q and we define a subgroup H of Q to which the character extends. This group turns out to be isomorphic to N ⋊ G 2 , where G 2 is the split exceptional group of type G 2 . We can thus define the periodwhere f is an automorphic form on E 6 (A). Let now P denote the maximal parabolic subgroup of E 6 with Levi factor of type A 5 . Given a cuspidal representation π of GL 6 (A) we can consider the Eisenstein series E(φ, s) which realizes the induction from π ⊗ | det | s to E 6 (A) (see 3.4 for unexplained notation). By the Langlands-Shahidi theory, the non-vanishing of the residue at s = 1/2 of E(φ, s) implies that L( 1 2 , π, Λ 3 ) = 0. The idea is to study P G 2 (Res s=1/2 E(φ, s)) and relate it to the Ginzburg-Rallis period. In fact, we essentially show the two are equal.More specifically, one starts by computing the orbits P (F )\ E 6 (F )/H(F ). This problem is challenging when working with exceptional groups and we do this using the explicit realization of E 6 as determinant preserving linear transformations of the 27-dimensional exceptional Jordan algebra. These computations allow us to compute P G 2 (Λ T E(φ, s)) explicitly. Here Λ T is the Arthur-Langlands truncation operator [Art80]. The truncation operator brings us to the next difficulty in this approach, namely convergence issues. In fact, these can be quite daunting, especially when dealing with non-reductive periods. Indeed, unless we're dealing with compact periods, then even for cusp forms, convergence of periods requires a non-trivial argument, see for example [JS90]. In [IY], Ichino and Yamana handle this using a truncation procedure adapted to their period. However, their case is reductive. Here, we do not introduce any new truncation. Instead, we extend the analysis introduced in the appendix of [BP16] which is based on norms on adelic points of linear algebraic groups and their quotients. In fact, as noticed in loc. cit., the key property is for the quotient to be quasi-affine. We believe this approach should generalize to other periods.The computation of P G 2 (Λ T E(φ, s)) allows us to easily deduce the following theorem.Theorem 0.2. Let π be a cuspidal automorphic representation of GL 6 (A) with trivial central character. Suppose that the Ginzburg-Rallis period P GR (ϕ) is non-zero on the space of π. Then, there exists φ in the induced space of π such that Res s= 1 2 E(φ, s) = 0.Corollary 0.3. Theorem 0.2 implies Theorem 0.1.Proof. By Theorem 0.2 the intertwi...

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Periods of automorphic forms, poles of L-functions and functorial lifting

Abstract: In this paper, we survey our work on period, poles or special values of certain automorphic Lfunctions and their relations to certain types of Langlands functorial transfers. We outline proofs for some results and leave others as conjectures.

Cited by 11 publications

References 58 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

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

A G2-period of a Fourier coefficient of an Eisenstein series on E6

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