We prove that a certain genuine Hecke algebra $\mathcal{\mathscr{H}}$ on the nonlinear double cover of a simple, simply laced, simply connected, Chevalley group $G$ over ${\mathbb{Q}}_2$ admits a Bernstein presentation. This presentation has two consequences. First, the Bernstein component containing the genuine unramified principal series is equivalent to $\mathcal{\mathscr{H}}$-mod. Second, $\mathcal{\mathscr{H}}$ is isomorphic to the Iwahori–Hecke algebra of the linear group $G∕Z_2$, where $Z_2$ is the $2$-torsion of the center of $G$. This isomorphism of Hecke algebras provides a correspondence between genuine unramified principal series of the double cover of $G$ and the Iwahori-unramified representations of the group $G∕Z_2$.

我们证明某个真正的 Hecke 代数 $\mathcal{\mathscr{H}}$ 在一个简单的、简单的、简单的、简单连接的 Chevalley 群的非线性双盖上 $G$ 超过 ${\mathbb{Q}}_2$承认伯恩斯坦的演讲。这种介绍有两个后果。首先，包含真正的无分支主级数的伯恩斯坦分量等价于$\mathcal{\mathscr{H}}$-模式。第二，$\mathcal{\mathscr{H}}$ 与线性群的 Iwahori-Hecke 代数同构 $G∕Z_2$， 在哪里 $Z_2$ 是个 $2$- 中心的扭转 $G$. Hecke 代数的这种同构提供了$G$ 和 Iwahori 无分支的代表 $G∕Z_2$.