  In 2000, Ariki and Mathas showed that the simple modules of the Ariki--Koike
algebras $\mathcal{H}_{\mathbb{C},q;Q_1,\ldots, Q_m}\big(G(m, 1, n)\big)$ (when
the parameters are roots of unity and $q\neq 1$) are labeled by the so-called
Kleshchev multipartitions. This together with Ariki's categorification theorem
enabled Ariki and Mathas to obtain the generating function for the number of
Kleshchev multipartitions by making use of the Weyl--Kac character formula. In
this paper, we revisit this generating function for the $q=-1$ case. This
$q=-1$ case is particularly interesting, for the corresponding Kleshchev
multipartitions have a very close connection to generalized Rogers--Ramanujan
type partitions when $Q_1=\cdots=Q_a=-1$ and $Q_{a+1}=\cdots =Q_m =1$. Based on
this connection, we provide an analytic proof of the result of Ariki and Mathas
for $q=Q_1=\cdots Q_a=-1$ and $Q_{a+1}=\cdots =Q_m =1$. Our second objective is
to investigate simple modules of the Ariki--Koike algebra in a fixed block. It
is known that these simple modules in a fixed block are labeled by the
Kleshchev multiparitions with a fixed partition residue statistic. This
partition statistic is also studied in the works of Berkovich, Garvan, and
Uncu. Employing their results, we provide two bivariate generating function
identities when $m=2$.

Combinatorics;Group Theory;Number Theory