In this paper, we prove certain multiplicity one theorems and define GL-twisted gamma factors for irreducible generic cuspidal representations of quasi-split classical groups ${\mathrm{G}}_r={\mathrm{Sp}}_{2r},{\mathrm{U}}_{2r}$, ${\mathrm{U}}_{2r+1},{\mathrm{SO}}_{2r+1}$ over finite fields of odd characteristic, using Rankin-Selberg method. As applications, we prove converse theorems for these groups, namely, ${\mathrm{GL}}_n$-twisted gamma factors, $n=1,2,\dots ,r$, will uniquely determine irreducible generic cuspidal representations of ${\mathrm{G}}_r({\mathbb{F}}_q)$.

中文翻译：

---

有限域上经典群的 Gamma 因子和逆定理

在本文中，我们证明了某些多重一定理，并为准分裂经典群的不可约泛尖表示定义了 GL 扭曲伽马因子 ${\mathrm{G}}_r={\mathrm{Sp}}_{2r},{\mathrm{ü}}_{2r}$,${\mathrm{ü}}_{2r+1},{\mathrm{所以}}_{2r+1}$使用 Rankin-Selberg 方法在奇特征的有限域上。作为应用，我们证明了这些组的逆定理，即，${\mathrm{总帐}}_n$-扭曲的伽马因子，$n=1,2,\dots ,r$, 将唯一地确定不可约的通用 cuspidal 表示${\mathrm{G}}_r({\mathbb{F}}_q)$.