  In this paper, we present a vertex operator approach to construct and compute
all complex irreducible characters of the general linear group $\GL_n(\mathbb
F_q)$. Green's theory of $\GL_n(\mathbb F_q)$ is recovered and enhanced under
the realization of the Grothendieck ring of representations
$R_G=\bigoplus_{n\geq 0}R(\GL_n(\mathbb F_q))$ as two isomorphic Fock spaces
associated to two infinite-dimensional $F$-equivariant Heisenberg Lie algebras
$\widehat{\mathfrak{h}}_{\hat{\overline{\mathbb F}}_q}$ and
$\widehat{\mathfrak{h}}_{\overline{\mathbb F}_q}$, where $F$ is the Frobenius
automorphism of the algebraically closed field $\overline{\mathbb F}_q$. Under
this picture, the irreducible characters are realized by the Bernstein vertex
operators for Schur functions, the characteristic functions of the conjugacy
classes are realized by the vertex operators for the Hall-Littlewood functions,
and the character table is completely given by matrix coefficients of vertex
operators of these two types. One of the features of the current approach is a
simpler identification of the Fock space $R_G$ as the Hall algebra of symmetric
functions via vertex operator calculus, and another is that we are able to
compute in general the character table, where Green's degree formula is
demonstrated as an example.
