On the degrees and rationality of certain characters of finite Chevalley groups

Abstract:

Let $\mathcal {S}$ be a system of finite groups with (B, N)-pairs, with Coxeter system (W, R) and set of characteristic powers $\{ q\}$ (see [4]). Let A be the generic algebra of the system, over the polynomial ring $\mathfrak {o} = Q[u]$. Let K be $Q(u)$, K an algebraic closure of K, and ${\mathfrak {o}^ \ast }$ the integral closure of $\mathfrak {o}$ in K. For the specialization $f:u \to q$ mapping $\mathfrak {o} \to Q$, let ${f^ \ast }:{\mathfrak {o}^ \ast } \to \bar Q$ be a fixed extension of f. For each irreducible character $\chi$ of the algebra ${A^{\bar K}}$, there exists an irreducible character ${\zeta _{\chi ,{f^ \ast }}}$ of the group $G(q)$ in the system corresponding to q, such that $({\zeta _{\chi ,{f^ \ast }}},1_{B(q)}^{G(q)}) > 0$, and $\chi \to {\zeta _{\chi ,{f^ \ast }}}$ is a bijective correspondence between the irreducible characters of ${A^{\bar K}}$ and the irreducible constituents of $1_{B(q)}^{G(q)}$. Assume almost all primes occur among the characteristic powers $\{ q\}$. The first main result is that, for each $\chi$, there exists a polynomial ${d_\chi }(t) \in Q[t]$ such that, for each specialization $f:u \to q$, the degree ${\zeta _{\chi ,{f^ \ast }}}(1)$ is given by ${d_\chi }(q)$. The second result is that, with two possible exceptions in type ${E_7}$, the characters ${\zeta _{\chi ,{f^ \ast }}}$ are afforded by rational representations of $G(q)$.