#### Vol. 15, No. 7, 2021

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A Hecke algebra on the double cover of a Chevalley group over $\mathbb{Q}_{2}$

### Edmund Karasiewicz

Vol. 15 (2021), No. 7, 1729–1753
DOI: 10.2140/ant.2021.15.1729
##### Abstract

We prove that a certain genuine Hecke algebra $\mathsc{ℋ}$ on the nonlinear double cover of a simple, simply laced, simply connected, Chevalley group $G$ over ${ℚ}_{2}$ admits a Bernstein presentation. This presentation has two consequences. First, the Bernstein component containing the genuine unramified principal series is equivalent to $\mathsc{ℋ}$-mod. Second, $\mathsc{ℋ}$ 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 $G∕{Z}_{2}$.

##### Keywords
Bernstein components, Hecke algebra, $p$-adic groups, metaplectic group
##### Mathematical Subject Classification
Primary: 11F70, 22E50
##### Milestones
Received: 2 June 2020
Revised: 5 November 2020
Accepted: 5 February 2021
Published: 1 November 2021
##### Authors
 Edmund Karasiewicz Department of Mathematics Ben-Gurion University of the Negev Be’er Sheva Israel