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A Hecke Algebra on the Double Cover of a Chevalley Group Over $\mathbb{Q}_{2}$ (2003.13248v2)
Published 30 Mar 2020 in math.NT and math.RT
Abstract: 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 $G/Z{2}$, 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 $G/Z_{2}$.