Maximal tori determining the algebraic groups

Abstract

Let k be a finite field, a global field, or a local non-archimedean field, and let H 1 and H 2 be split, connected, semisimple algebraic groups over k. We prove that if H 1 and H 2 share the same set of maximal k-tori, up to k-isomorphism, then the Weyl groups W(H 1 ) and W(H 2 ) 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.).