The study of finite dimensional representations of quantum affine algebras goes
back several decades. Many important results have been proved and deep techniques have
been developed over the years; but many interesting natural questions remain unanswered.
More recently the work of Hernandez and Leclerc on monoidal categorification has led to a
renewed interest in the subject. The influential work of Kashiwara and his collaborators has
led to further substantial developments in the subject.

In this talk, we will be guided by another remarkable connection, via affine Hecke algebras,
with the category of smooth representations of \(\mathrm{GL}_n(F)\) where \(F\) is a non–Archimedean field.
We will discuss this connection, especially with the work of Lapid–Mínguez. Motivated in
part by their work, we introduce a family of modules for the quantum affine algebra and give
an explicit determinantal formula for its character. As an application of our results we explain
how to compute certain Kazhdan–Lusztig coefficients in the Bernstein–Gelfand–Gelfand
category \(\mathcal{O}\).