Character value estimates for non-semisimple elements
This paper continues the author's study on upper bounds for \(| \chi(x)/\chi(1)|\) where \(\chi\) is a non-trivial irreducible character of a group \(G\) of Lie type and \(x \not\in Z(G)\) [cf. the author, Pac. J. Math. 150, 279-307 (1991; Zbl 0756.20002)]. If \(x_ s\) resp. \(x_ u\) denotes the semisimple resp. unipotent part of \(x\), then the bounds given here are \[ | \chi(x)/\chi(1)| \leq \begin{cases} C/q& \text{if \(x_ s \not\in Z(G)\)}\\(C/q)^{1/2}& \text{if \(x_ s \in Z(G)\) and \(x_ u \neq 1\),}\end{cases} \] with an absolute constant \(C\). The order of magnitude is reached for unipotent elements of the symplectic group over \(\mathbb{F}_ q\). As one of the applications the author proves a problem of Zalesskij, namely that \(\mathbb{C}\overline{G(p)}\) where \(\overline{G(p)}\) is a Chevalley group over the algebraic closure of \(\mathbb{F}_ p\) (\(p\) a prime) has only 3 two-sided ideals. The proof depends on the following interesting fact, which is a consequence of the bounds given above. If \(m\) is large enough, then for all nontrivial \(\chi \in \text{Irr}(G(q^ m))\), the restriction \(\chi_{G(q)}\) contains all irreducible characters of \(G(q)\) as constituents.
scientific article; zbMATH DE number 168188
scientific article