On character values of $G{L}_n({\mathbb{F}}_q)$

In this paper, we use vertex operator techniques to compute character values on unipotent classes of ${\text{GL}}_n({\mathbb{F}}_q)$. By realizing the Grothendieck ring $R_G={⨁}_{n\ge 0}^{\mathrm{\infty }}R({\text{GL}}_n({\mathbb{F}}_q))$ as Fock spaces, we formulate the Murnanghan-Nakayama rule of ${\text{GL}}_n({\mathbb{F}}_q)$ between Schur functions colored by an orbit $\phi$ of linear characters of ${\bar{\mathbb{F}}}_q$ under the Frobenius automorphism on and modified Hall-Littlewood functions colored by $f_1=t-1$, which provides detailed information on the character table of ${\text{GL}}_n({\mathbb{F}}_q)$. As applications, we use vertex algebraic methods to determine the Steinberg characters of ${\text{GL}}_n({\mathbb{F}}_q)$, which were previously determined by Curtis-Lehrer-Tits via geometry of homology groups of spherical buildings and Springer-Zelevinsky utilizing Hopf algebras.