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

Karasiewicz, Edmund

doi:10.2140/ant.2021.15.1729

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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 $ℋ$ 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 $ℋ$ -mod. Second, $ℋ$ 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

Vol. 15, No. 7, 2021