A new way integral for the $n$-fold cover of $S{L}_2$

Abstract

Let $\pi$ be a genuine irreducible cuspidal automorphic representation of the $n$-fold cover $S{L}_2^{(n)}(\mathbb{A})$ of $S{L}_2(\mathbb{A})$, $n\ge 3$ odd, and let $\tau$ be the genuine representation of $G{L}_1^{(n)}(\mathbb{A})$ obtained from a unitary Hecke character of ${\mathbb{A}}^{\times }$. We prove that, for a certain choice of data, a new way integral for $S{L}_2^{(n)}(\mathbb{A})$ represents the partial $L$-function $L^S(ns+\frac{1}{2},\pi \times \tau )$. The integral is derived from the generalized doubling integral of Cai, Friedberg, Ginzburg and Kaplan, following the idea of Ginzburg and Soudry.