A description of the depth-$r$ Bernstein center for rational depths

Let $G$ be a split connected reductive over a non-archimedean local field $k$. In this paper we give a description of the depth-$r$ Bernstein center of $G(k)$ for rational depths as a limit of depth-$r$ standard parahoric Hecke algebras, extending our previous work in the integral depths case (arXiv:2407.15128). Using this description, we construct maps from the space of stable functions on depth-$r$ Moy-Prasad quotients to the depth-$r$ center, and attach depth-$r$ Deligne-Lusztig parameters to smooth irreducible representations, with the assignment of parameters to irreducible representations shown to be consistent with restricted Langlands parameters for Moy-Prasad types described Chen-Debacker-Tsai (arXiv:2509.07780). As an application, we give a decomposition of the category of smooth representations into a product of full subcategories indexed by restricted depth-$r$ Langlands parameters.