On a geometric comparison of representations of complex and $p$-adic ${\mathbf{G}\mathbf{L}}_n$

In this paper, we use geometric methods to study the relations between admissible representations of ${\mathbf{G}\mathbf{L}}_n(\mathbb{C})$ and unramified representations of ${\mathbf{G}\mathbf{L}}_m({\mathbb{Q}}_p)$. We show that the geometric relationship between Langlands parameter spaces of ${\mathbf{G}\mathbf{L}}_n(\mathbb{C})$ and ${\mathbf{G}\mathbf{L}}_m({\mathbb{Q}}_p)$ constructed by the first named author is compatible with the functor recently defined algebraically by Chan-Wong. We then show that the said relationship intertwines translation functors on representations of ${\mathbf{G}\mathbf{L}}_n(\mathbb{C})$ and partial Bernstein-Zelevinskii derivatives on representations of ${\mathbf{G}\mathbf{L}}_m({\mathbb{Q}}_p)$, providing purely geometric counterparts to some results of Chan-Wong. In the sequels, the techniques of this work will be extended to real and $p$-adic classical groups and used to study their Arthur packets.