Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$L$-functions for $\mathrm{Sp}(2n)\times\mathrm{GL}(k)$ via non-unique models

Yubo Jin, Pan Yan

TL;DR

The paper develops new global integrals for $\mathrm{Sp}_{2n}\times\mathrm{GL}_k$ using the generalized doubling method, valid for even $n$, and shows they unfold to non-unique $\mathrm{Sp}_{2n}$-models. Employing Piatetski-Shapiro–Rallis’ New Way, it proves these integrals represent the L-function $L^S(s+\tfrac12,$ $\pi\times\tau)$, extending known results from $n=k=2$ and from $k=1$. The construction relies on a detailed calculus of theta-series, $(k,c)$-representations, Speh representations, and Eisenstein series, together with a careful global unfolding and a robust unramified local computation. The results yield a broad New Way-type framework for symplectic-GL pair L-functions, with potential applications to pole-pole criteria and CAP-detection in the spirit of Kudla–Rallis–Soudry. These integrals thus provide new avenues for understanding and accessing $L$-functions attached to $\mathrm{Sp}_{2n}\times\mathrm{GL}_k$ via Fourier-analytic models, Theta correspondences, and Weil representations.

Abstract

Let $n$ and $k$ be positive integers such that $n$ is even. We derive new global integrals for $\mathrm{Sp}_{2n}\times\mathrm{GL}_k$ from the generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan, following a strategy and extending a previous result of Ginzburg and Soudry on the case $n=k=2$. We show that these new integrals unfold to non-unique models on $\mathrm{Sp}_{2n}$. Using the New Way method of Piatetski-Shapiro and Rallis, we show that these new global integrals represent the $L$-functions for $\mathrm{Sp}_{2n}\times\mathrm{GL}_k$, generalizing a previous result of the second-named author on $\mathrm{Sp}_{4}\times\mathrm{GL}_2$ and a previous work of Piatetski-Shapiro and Rallis on $\mathrm{Sp}_{2n}\times\mathrm{GL}_1$.

$L$-functions for $\mathrm{Sp}(2n)\times\mathrm{GL}(k)$ via non-unique models

TL;DR

The paper develops new global integrals for using the generalized doubling method, valid for even , and shows they unfold to non-unique -models. Employing Piatetski-Shapiro–Rallis’ New Way, it proves these integrals represent the L-function , extending known results from and from . The construction relies on a detailed calculus of theta-series, -representations, Speh representations, and Eisenstein series, together with a careful global unfolding and a robust unramified local computation. The results yield a broad New Way-type framework for symplectic-GL pair L-functions, with potential applications to pole-pole criteria and CAP-detection in the spirit of Kudla–Rallis–Soudry. These integrals thus provide new avenues for understanding and accessing -functions attached to via Fourier-analytic models, Theta correspondences, and Weil representations.

Abstract

Let and be positive integers such that is even. We derive new global integrals for from the generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan, following a strategy and extending a previous result of Ginzburg and Soudry on the case . We show that these new integrals unfold to non-unique models on . Using the New Way method of Piatetski-Shapiro and Rallis, we show that these new global integrals represent the -functions for , generalizing a previous result of the second-named author on and a previous work of Piatetski-Shapiro and Rallis on .
Paper Structure (16 sections, 32 theorems, 269 equations)

This paper contains 16 sections, 32 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Theta series
  5. Representations of $(k,c)$ type
  6. Eisenstein series
  7. New integrals derived from the generalized doubling method
  8. The generalized doubling construction
  9. New integrals derived from the generalized doubling method
  10. The global zeta integral and statement of theorems
  11. The basic identity
  12. The unramified computation
  13. Relation between unramified sections
  14. Reformulating the unramified integral from the generalized doubling method
  15. Reformulating the unramified New Way integral
  16. ...and 1 more sections

Key Result

Theorem 1.1

(Theorem thm 3.3) Let $n$ be an even positive integer. Given a Schwartz function $\Phi\in\mathcal{S}(\mathrm{Mat}_{n}(\mathbb{A}))$, there are nontrivial choices of sections such that $\mathcal{L}(\phi,f_{2n,k,s})$ is equal to Here: (1) $\phi\in V_\pi$ is a non-zero cusp form; (2) $N_{n^{k-1},kn}$ is a certain unipotent subgroup of $\mathrm{Sp}_{2kn}$ and $\psi_k$ is a character on $N_{n^{k-1},k

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma 3.4
  • ...and 41 more