On Representations of Reductive $p$--adic Groups over $\mathbb{Q}$--algebras

In this paper we study certain category of smooth modules for reductive $p$–adic groups analogous to the usual smooth complex representations but with the field of complex numbers replaced by a $\mathbb{Q}$–algebra. We prove some fundamental results in these settings, and as an example we give a classification of admissible unramified irreducible representations proving by reduction to the complex case that if the space of $K$–invariants is finite dimensional in an irreducible smooth unramified representation that the representation is admissible.