On the Character Degrees of a Sylow p-Subgroup of a Finite Chevalley Group G(pf) Over a Bad Prime

摘要

Let $q$ be a power of a prime $p$ and let $U(q)$ be a Sylow $p$-subgroup of afinite Chevalley group $G(q)$ defined over the field with $q$ elements. Wefirst give a parametrization of the set $ext{Irr}(U(q))$ of irreduciblecharacters of $U(q)$ when $G(q)$ is of type $mathrm{G}_2$. This is uniform forprimes $p ge 5$, while the bad primes $p=2$ and $p=3$ have to be consideredseparately. We then use this result and the contribution of several authors toshow a general result, namely that if $G(q)$ is any finite Chevalley group with$p$ a bad prime, then there exists a character $chi in ext{Irr}(U(q))$ suchthat $chi(1)=q^n/p$ for some $n in mathbb{Z}_{ge_0}$. In particular, foreach $G(q)$ and every bad prime $p$, we construct a family of characters ofsuch degree as inflation followed by an induction of linear characters of anabelian subquotient $V(q)$ of $U(q)$.