A Godement–Jacquet type integral and the metaplectic Shalika model

Abstract

We present a novel integral representation for a quotient of global automorphic L-functions, the symmetric square over the exterior square. The pole of this integral characterizes a period of a residual representation of an Eisenstein series. As such, the integral itself constitutes a period, of an arithmetic nature. The construction involves the study of local and global aspects of a new model for double covers of general linear groups, the metaplectic Shalika model. In particular, we prove uniqueness results over p-adic and archimedean fields, and a new Casselman-Shalika type formula.

Hecke $L$-functions and Fourier coefficients of covering Eisenstein series

“…Furthermore, in a series of works by Kaplan [Kap15, Kap16a, Kap16b, Kap17a, Kap17b], the theory was further studied and the author also found applications to the problems of computing certain periods. Notably, determining unipotent orbits of theta representations has also important applications, which is already clear in the work [BG92,Tak14,FK19] mentioned above, and is also one of the foci in the study by Friedberg and Ginzburg [FG18,FG17], Y.-Q. Cai [Cai19] and Leslie [Les19].…”

“…The case of twisted symmetric square L-functions was treated by Takeda [Tak14]. See yet another recent work [FK19] of obtaining (quotient of) L-functions of π by using a Godement-Jacquet type integral involving theta representations. In another direction, starting with the work of Savin [Sav92] on representations distinguished by theta representations, the investigation was continued in [Kab01,Kab02].…”

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

“…and ϕ N GL 2 2 vanish unless k = 1 (they factor through V (1,rk−1) and V (rk−1,1) , respectively), in which case the constant terms can be computed as in [FK19,Theorem 4.4] (see also [KP84, § II.1]). Then…”