We study the fields of values of the irreducible characters of a finite group of degree not divisible by a prime p. In the case where $p=2$, we fully characterise these fields. In order to accomplish this, we generalise the main result of [ILNT] to higher irrationalities. We do the same for odd primes, except that in this case the analogous results hold modulo a simple-to-state conjecture on the character values of quasi-simple groups.

We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig label and in terms of its degree. Then we prove explicit upper bounds for character values at elements with not-too-large centralizers and derive upper bounds on the covering number and mixing time of random walks corresponding to these conjugacy classes. We also characterize the level of the character in terms of certain dual pairs and prove explicit exponential character bounds for the character values, provided that the level is not too large. Several further applications are also provided. Related results for other finite classical groups are obtained in the sequel [Guralnick et al. ‘Character levels and character bounds for finite classical groups’, Preprint, 2019, arXiv:1904.08070] by different methods.

The classical Itô-Michler theorem on character degrees of finite groups asserts that if the degree of every complex irreducible character of a finite group G is coprime to a given prime p, then G has a normal Sylow p-subgroup. We propose a new direction to generalize this theorem by introducing an invariant concerning character degrees. We show that if the average degree of linear and even-degree irreducible characters of G is less than 4/3 then G has a normal Sylow 2-subgroup, as well as corresponding analogues for real-valued characters and strongly real characters. These results improve on several earlier results concerning the Itô-Michler theorem.

In 1981, Thompson proved that, if $n\geqslant 1$ is any integer and $G$ is any finite subgroup of $\text{GL}_{n}(\mathbb{C})$ , then $G$ has a semi-invariant of degree at most $4n^{2}$ . He conjectured that, in fact, there is a universal constant $C$ such that for any $n\in \mathbb{N}$ and any finite subgroup $G<\text{GL}_{n}(\mathbb{C})$ , $G$ has a semi-invariant of degree at most $Cn$ . This conjecture would imply that the ${\it\alpha}$ -invariant ${\it\alpha}_{G}(\mathbb{P}^{n-1})$ , as introduced by Tian in 1987, is at most $C$ . We prove Thompson’s conjecture in this paper.

Let $w_{1}$ and $w_{2}$ be nontrivial words in free groups $F_{n_{1}}$ and $F_{n_{2}}$, respectively. We prove that, for all sufficiently large finite nonabelian simple groups $G$, there exist subsets $C_{1}\subseteq w_{1}(G)$ and $C_{2}\subseteq w_{2}(G)$ such that $|C_{i}|=O(|G|^{1/2}\log ^{1/2}|G|)$ and $C_{1}C_{2}=G$. In particular, if $w$ is any nontrivial word and $G$ is a sufficiently large finite nonabelian simple group, then $w(G)$ contains a thin base of order $2$. This is a nonabelian analog of a result of Van Vu [‘On a refinement of Waring’s problem’, Duke Math. J. 105(1) (2000), 107–134.] for the classical Waring problem. Further results concerning thin bases of $G$ of order $2$ are established for any finite group and for any compact Lie group $G$.

Perhaps unexpectedly, there is a rich and deep connection between field of values of characters, their degrees and the structure of a finite group. Some of the fundamental results on the degrees of characters of finite groups, as the Ito–Michler and Thompson's theorems, admit a version involving only characters with certain fixed field of values ([DNT, NS, NST2, NT1, NT3]).

Dedicated to the memory of Professor A. I. Kostrikin

The main problem under discussion is to determine, for quasi-simple groups of Lie type G, irreducible representations $\phi$ of G that remain irreducible under reduction modulo the natural prime p. The method is new. It works only for p >3 and for representations $\phi$ that can be realized over an unramified extension of ${\mathbb Q}_p$, the field of p -adic numbers. Under these assumptions, the main result says that the trivial and the Steinberg representations of G are the only representations in question provided G is not of type A1. This is not true for G=SL(2, p). The paper contains a result of independent interest on infinitesimally irrreducible representations $\rho$ of G over an algebraically closed field of characteristic p. Assuming that G is not of rank 1 and $G\neq G_2(5)$, it is proved that either the Jordan normal form of a root element contains a block of size d with 1<d<p -1 or the highest weight of $\rho$ is equal to p -1 times the sum of the fundamental weights.

2000 Mathematical Subject Classification: 20C33, 20G15.

The low-dimensional projective irreducible representations in cross characteristics of the projective special linear group $\mbox{PSL}_{n}(q)$ are investigated.

If $n \geq 3$ and $(n,q) \neq (3,2)$, $(3,4)$, $(4,2)$, $(4,3)$, all such representations of the first degree (which is $(q^{n}-q)/(q-1) - \kappa$ with $\kappa = 0$ or $1$) and the second degree (which is $(q^{n}-1)/(q-1)$) come from Weil representations. We show that the gap between the second and the third degree is roughly $q^{2n-4}$.