Title: On the character degrees of a Sylow $p$-subgroup of a finite Chevalley group $G(p^f)$ over a bad prime,
Abstract: Let $q$ be a power of a prime $p$ and let $U(q)$ be a Sylow $p$-subgroup of a
finite Chevalley group $G(q)$ defined over the field with $q$ elements. We
first give a parametrization of the set $\text{Irr}(U(q))$ of irreducible
characters of $U(q)$ when $G(q)$ is of type $\mathrm{G}_2$. This is uniform for
primes $p \ge 5$, while the bad primes $p=2$ and $p=3$ have to be considered
separately. We then use this result and the contribution of several authors to
show a general result, namely that if $G(q)$ is any finite Chevalley group with
$p$ a bad prime, then there exists a character $\chi \in \text{Irr}(U(q))$ such
that $\chi(1)=q^n/p$ for some $n \in \mathbb{Z}_{\ge_0}$. In particular, for
each $G(q)$ and every bad prime $p$, we construct a family of characters of
such degree as inflation followed by an induction of linear characters of an
abelian subquotient $V(q)$ of $U(q)$.