Maximal tori determining the algebraic group

Abstract

Let k be a finite field, a global field or a local non-archimedean field. Let H_1 and H_2 be two split, connected, semisimple algebraic groups defined 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. We remark that due to a recent result of Philippe Gille, above result holds for fields which admit arbitrary cyclic extensions.