We introduce the concept of a triangular decomposition for Banach and Fr\'echet-Stein algebras over $p$-adic fields, which allows us to define a category $\mathcal{ O}$ for a wide array of topological algebras. In particular, we apply this concept to $p$-adic rational Cherednik algebras, which allows us to obtain an analytic version of the category $\mathcal{ O}$ developed by Ginzburg, Guay, Opdam and Rouquier. Along the way, we study the global sections of $p$-adic Cherednik algebras on smooth Stein spaces, and determine their behavior with respect to the rigid analytic GAGA functor.
Categories: math.NT math.QA math.RT