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 on the nonlinear double cover of a simple, simply laced, simply connected, Chevalley group G over 2 admits a Bernstein presentation. This presentation has two consequences. First, the Bernstein component containing the genuine unramified principal series is equivalent to -mod. Second, is isomorphic to the Iwahori–Hecke algebra of the linear group GZ2, where Z2 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 GZ2.

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