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

Abstract

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.

Twisted doubling integrals for Brylinski-Deligne extensions of classical groups

“…In the second factor we only taking a linear group instead of a cover of GL k . This is different from the L-functions obtained in [Gao18] and [Kap19]. It it an interesting question to relate these L-functions.…”

“…A consequence of this calculation is the meromorphic continuation of many interesting L-functions. Another example is a generalization of the doubling method ([PSR87, CFGK19]) to covers of symplectic groups [Kap19] (see also [CFGK16] for a brief description of the method). In [Kap19], the unfolding argument and unramified calculation are carried out.…”

“…Another example is a generalization of the doubling method ([PSR87, CFGK19]) to covers of symplectic groups [Kap19] (see also [CFGK16] for a brief description of the method). In [Kap19], the unfolding argument and unramified calculation are carried out. As a consequence, the global integral represents the tensor product Lfunction for a cover of a symplectic group and a cover of a general linear group under certain assumptions.…”

“…This gives a global integral for the standard L-function of a classical group. As a global integral, the doubling integral [PSR87] and its generalizations [CFGK19,Kap19] enjoy the following nice properties:…”

“…Main results. The purpose of this article is to explain how to develop the twisted doubling integrals (see [CFGK19,Kap19]) for Brylinski-Deligne extensions of connected classical groups. We summarize our results here:…”