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

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 ${\mathrm{Sp}}_{2n}\times {\mathrm{GL}}_1$.

37 pages, final version, to appear in International Mathematics Research Notices. Previous title is "On New Way integrals for Sp(2n)xGL(k)"