A Hecke Algebra on the Double Cover of a Chevalley Group Over ${\mathbb{Q}}_2$

We prove that a certain genuine Hecke algebra $\mathcal{H}$ on the non-linear 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 $\mathcal{H}$-mod. Second, $\mathcal{H}$ is isomorphic to the Iwahori-Hecke algebra of the linear group $G\mathrm{/}{Z}_2$, where $Z_2$ is the $2$-torsion of the center of $G$. This isomorphism of Hecke algebras provides a correspondence between certain genuine unramified principal series of the double cover of $G$ and the Iwahori-unramified representations of the group $G\mathrm{/}{Z}_2$.