On the degrees and rationality of certain characters of finite Chevalley groups

Let a'' be a system of finite groups with (5, N)-pairs, with Coxeter system (W, R) and set of characteristic powers {q} (see [4]). Let A be the generic algebra of the system, over the polynomial ring o—Q[u]. Let K be ß(w), K an algebraic closure of K, and o* the integral closure of o in K. For the specialization /: «->-<? mapping o -*■ Q, let/* : o* -> Q be a fixed extension of/. For each irreducible character x of the algebra As, there exists an irreducible character £,,,. of the group G{q) in the system corresponding to q, such that (£»,/•, lf<S))>0, and x -*■ ix.r is a bijective correspondence between the irreducible characters of A* and the irreducible constituents of lg$. Assume almost all primes occur among the characteristic powers {q}. The first main result is that, for each x, there exists a polynomial dx(t) e Q [/] such that, for each specialization /: u^-q, the degree {,,/•(!) is given by dx{q). The second result is that, with two possible exceptions in type E7, the characters £/t/. are afforded by rational representations of G(q).