On the Local Coefficients Matrix for Coverings of $$\mathrm{SL}_2$$SL2

Gao, Fan; Shahidi, Freydoon; Szpruch, Dani

doi:10.1007/978-3-319-97379-1_10

Cited by 7 publications

(2 citation statements)

References 39 publications

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“…al. [GSS18] and Szpruch [Szp] have recently developed a reinterpretation of the coefficients of (1.77): the coefficients were presented as linear combinations of Tate γfactors, or "metaplectic" γ-factors (defined in [Szp11]), where the scalars were computed via harmonic analysis on F * r F * . The determinant of their local coefficients matrix was related to the Plancherel measure and may shed new light on the Plancherel formula for covering groups.…”

Section: The Casselman-shalika Formula For Glmentioning

confidence: 99%

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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Section: The Casselman-shalika Formula For Glmentioning

confidence: 99%

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

Kaplan

2019

Preprint

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“…al. [GSS18,GSS] and Szpruch [Szp] replaced the local coefficient with a proportionality matrix, for generic representations. The determinant of this matrix becomes an invariant of the representation, and for unramified representations (and ramified principal series in low rank cases) they expressed this determinant in terms of Plancherel measures and Tate γ-factors.…”

Section: Introductionmentioning

confidence: 99%

Doubling Constructions: the complete L-function for coverings of the symplectic group

Kaplan¹

2020

Preprint

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We develop the local theory of the generalized doubling method for the m-fold central extension Sp (m) 2n of Matsumoto of the symplectic group. We define local γ-, L-and ǫ-factors for pairs of genuine representations of Sp (m)2n × GL k and prove their fundamental properties, in the sense of Shahidi. Here GL k is the central extension of GL k arising in the context of the Langlands-Shahidi method for covering groups of Sp 2n × GL k . We then construct the complete L-function for cuspidal representations and prove its global functional equation. Possible applications include classification results and a Shimura type lift of representations from covering groups to general linear groups (a global lift is sketched here for m = 2).

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Kazhdan–Lusztig representations and Whittaker space of some genuine representations

Gao

2019

Math. Ann.

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We prove a formula for the dimension of Whittaker functionals of irreducible constituents of a regular unramified genuine principal series for covering groups. The formula explicitly relates such dimension to the Kazhdan-Lusztig representations associated with certain right cells of the Weyl group. We also state a refined version of the formula, which is proved under some natural assumption. The refined formula is also verified unconditionally in several important cases.

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On the Local Coefficients Matrix for Coverings of $$\mathrm{SL}_2$$SL2

Cited by 7 publications

References 39 publications

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

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

Doubling Constructions: the complete L-function for coverings of the symplectic group

Kazhdan–Lusztig representations and Whittaker space of some genuine representations

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