Abstract. A Gelfand model for a finite group G is a complex linear representation of G that contains each of its irreducible representations with multiplicity one. For a finite group G with a faithful representation V , one constructs a representation which we call the polynomial model for G associated to V . Araujo and others have proved that the polynomial models for certain irreducible Weyl groups associated to their canonical representations are Gelfand models.In this paper, we give an easier and uniform treatment for the study of the polynomial model for a general finite Coxeter group associated to its canonical representation. Our final result is that such a polynomial model for a finite Coxeter group G is a Gelfand model if and only if G has no direct factor of the type W ðD 2n Þ, W ðE 7 Þ or W ðE 8 Þ.

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 kisomorphism, 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.)

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Let G be an algebraic group. For d ≥ 1, we define the commuting prob-is the variety of commuting d-tuples in G. We prove that for a reductive group G when d is large, cp d (G) ∼ α n where n = dim(G), and α is the maximal dimension of an Abelian subgroup of G. For a finite reductive group G defined over the field Fq, we show that cp d+1 (G(Fq)) ∼ q (α−n)d , and give several examples.

The aim of this paper is to investigate the order coincidences among the finite semisimple groups and to give a reasoning of such order coincidences through the transitive actions of compact Lie groups.It is a theorem of Artin and Tits that a finite simple group is determined by its order, with the exception of the groups (A 3 (2), A 2 (4)) and (B n (q),C n (q)) for n ≥ 3, q odd. We investigate the situation for finite semisimple groups of Lie type. It turns out that the order of the finite group H(F q ) for a split semisimple algebraic group H defined over F q , does not determine the group H up to isomorphism, but it determines the field F q under some mild conditions. We then put a group structure on the pairs (H 1 , H 2 ) of split semisimple groups defined over a fixed field F q such that the orders of the finite groups H 1 (F q ) and H 2 (F q ) are the same and the groups H i have no common simple direct factors. We obtain an explicit set of generators for this abelian, torsion-free group. We finally show that the order coincidences for some of these generators can be understood by the inclusions of transitive actions of compact Lie groups.