A Hecke algebra on the double cover of a Chevalley group over ℚ2

Abstract

We prove that a certain genuine Hecke algebra $\mathscr{H}$ 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 $\mathscr{H}$-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 $G∕Z_2$.