Extensions of tori in SL(2)

Jeffrey Adams

Vol. 200 (2001), No. 2, 257–271

DOI: 10.2140/pjm.2001.200.257

Abstract

Let SL(2, 𝔽) be the metaplectic two-fold cover of SL(2, 𝔽), the special linear group in two variables over a local field 𝔽 of characteristic 0. The inverse image T of a maximal torus T in SL(2, 𝔽) is an abelian extension of T by ±1. We consider the question of whether this extension is trivial. More generally we find the minimal subgroup A of the circle for which the extension is split when considered with coefficients in A. We see that |A| = 2,4 or 8 in the p-adic case. We also find an explicit splitting function for the cocycle.

Milestones

Received: 29 April 1999

Published: 1 October 2001