On the character degrees of Sylow p-subgroups of Chevalley groups G(pf) of type E.

Let q be a field of characteristic p with q elements. It is known that the degrees of the irreducible characters of the Sylow p-subgroup of GL(픽q) are powers of q. On the other hand Sangroniz (2003) showed that this is true for a Sylow p-subgroup of a classical group defined over 픽q if and only if p is odd. For the classical groups of Lie type B, C and D the only bad prime is 2. For the exceptional groups there are others. In this paper we construct irreducible characters for the Sylow p-subgroups of the Chevalley groups D4(q) with q = 2f of degree q3/2. Then we use an analogous construction for E6(q) with q = 3f to obtain characters of degree q7/3, and for E8(q) with q = 5f to obtain characters of degree q16/5. This helps to explain why the primes 2, 3 and 5 are bad for the Chevalley groups of type E in terms of the representation theory of the Sylow p-subgroup.