Maximal tori determining the algebraic groups

Shripad M. Garge

Vol. 220 (2005), No. 1, 69–85

DOI: 10.2140/pjm.2005.220.69

Abstract

Let k be a finite field, a global field, or a local non-archimedean field, and let H₁ and H₂ be split, connected, semisimple algebraic groups over k. We prove that if H₁ and H₂ share the same set of maximal k-tori, up to k-isomorphism, then the Weyl groups W(H₁) and W(H₂) are isomorphic, and hence the algebraic groups modulo their centers are isomorphic except for a switch of a certain number of factors of type B_n and C_n.

(Due to a recent result of Philippe Gille, this result also holds for fields which admit arbitrary cyclic extensions.)

Keywords

maximal tori, algebraic groups

Mathematical Subject Classification 2000

Primary: 20G15

Milestones

Received: 7 July 2003

Revised: 14 January 2004

Accepted: 14 January 2004

Published: 1 May 2005