Homomorphisms of the Alternating Group a 5 into Reductive Groups 3

0.1. Let A5 be the group defined by generators x2, x3, x5 and relations x 2 2 = x 3 3 = x5 = x2x3x5 = 1. It is well known that A5 is isomorphic to the alternating group in 5 letters. Let k be an algebraically closed field of characteristic p where p = 0 or a prime ≥ 7. Let G be a connected reductive algebraic group over k. We shall be interested in the classification of homomorphisms A5 −→ G up to conjugation by G. When G = GLn this is the same as the classification of n-dimensional representations of A5 over k up to isomorphism; this can be reduced to the classification of irreducible representations of A5 which is classical (there are five of them up to isomorphism, of degrees 1, 3, 3, 4, 5). In the general case, there is again something analogous to the notion of irreducible representation, which we call a regular homomorphism A5 −→ G. This is, by definition, a homomorphism A5 −→ G whose image is not contained in a Levi subgroup of a proper parabolic subgroup of G. Again the classification of homomorphisms A5 −→ G up to conjugacy can be more or less reduced to the analogous question for regular homomorphisms (for G and also for smaller groups). D.D.Frey [F1],[F2],[F3] has classified up to conjugacy the homomorphisms (resp. non-regular homomorphisms) A5 −→ G for G of type E6, E7 (resp. E8) over C. For G of type E8 he showed that there is at least one regular homomorphism A5 −→ G, but the problem of classifying up to conjugacy the regular homomorphisms A5 −→ G remained open. Serre [S] has suggested that this problem could be attacked using the complex representation theory of E8(Fq), Fq a finite field. He showed that the number d of conjugacy classes of regular homomorphisms A5 −→ E8(k) can be extracted from a certain sum over all irreducible characters of E8(Fq), if this sum can be computed. The purpose of this paper is to show that this sum can be computed with enough precision so that a solution of the above problem is obtained. (See Theorem 4.5.) In fact, we show that d = 1. Thus, our results, in conjunction with the earlier