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

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arxiv 1902.00880 v2 pith:YHUYF2ZQ submitted 2019-02-03 math.NT math.RT

Eyal Kaplan This is my paper · ORCID
classification math.NTmath.RT
keywords coveringgroupgroupsrepresentationsarbitrarydoublingintegrallocal
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abstract

In this work we develop an integral representation for the partial $L$-function of a pair $\pi\times\tau$ of genuine irreducible cuspidal automorphic representations, $\pi$ of the $m$-fold covering of Matsumoto of the symplectic group $Sp_{2n}$, and $\tau$ 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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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Gelfand--Graev representation as a Hecke algebra module of simple types of a finite central cover of $\mathrm{GL}(r)$

    math.RT 2025-02 conditional novelty 7.0 of 10

    For tame Kazhdan-Patterson and Savin covers of GL_r, the Gelfand-Graev representation decomposes as explicit simple-type Hecke algebra modules, yielding the Whittaker dimension of discrete series as |X(λ)/S_k|.

  2. Twisted doubling integrals for classical groups

    math.NT 2019-08 conditional novelty 6.0 of 10

    The paper reformulates twisted doubling integrals in a unified framework, proves a uniform unfolding identity, and extends the construction to quaternionic unitary groups.

  3. Tensor Product $L$-Functions On Metaplectic Covering Groups of $GL_r$

    math.RT 2019-08 conditional novelty 5.0 of 10

    Theorem 1 proves Suzuki's local integral (2) equals L(π(n) × τ(n), ns − (n−1)/2) for all r<nm, and Section 5 gives conditional global integral representations.

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