Let ${\operatorname {GL}}_c^{(m)}$ be the covering group of ${\operatorname {GL}}_c$, obtained by restriction from the $m$-fold central extension of Matsumoto of the symplectic group. We introduce a new family of Rankin–Selberg integrals for representations of ${\operatorname {GL}}_c^{(m)}\times {\operatorname {GL}}_k^{(m)}$. The construction is based on certain assumptions, which we prove here for $k=1$. Using the integrals, we define local $\gamma $-, $L$-, and $\epsilon $-factors. Globally, our construction is strong in the sense that the integrals are truly Eulerian. This enables us to define the completed $L$-function for cuspidal representations and prove its standard functional equation.

中文翻译：

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用于覆盖一般线性群的群的 Rankin-Selberg 积分和 L-函数

令${\operatorname {GL}}_c^{(m)}$ 为${\operatorname {GL}}_c$ 的覆盖群，由辛的Matsumoto 的$m$-fold 中心扩展限制得到团体。我们引入了一个新的 Rankin-Selberg 积分族来表示 ${\operatorname {GL}}_c^{(m)}\times {\operatorname {GL}}_k^{(m)}$。该构造基于某些假设，我们在此证明 $k=1$。使用积分，我们定义了局部 $\gamma $-、$L$- 和 $\epsilon $-因子。在全球范围内，我们的构造是强大的，因为积分是真正的欧拉积分。这使我们能够为尖瓣表示定义完整的 $L$-函数并证明其标准函数方程。