Quantum wreath products and $p$-adic general linear group

We study the pro-$p$ Iwahori-Hecke algebra and its Gelfand-Graev modules for the $p$-adic general linear group and its metaplectic covers. We develop the theory of quantum wreath products of skew polynomial type and use it to provide transparent descriptions of these Hecke algebras and their modules that were previously inaccessible through standard $p$-adic methods. We introduce the notion of (anti)spherical and Kashiwara-Miwa-Stern modules for these quantum wreath products for the first time and interpret the $p$-adic Gelfand-Graev modules in terms of these new modules. As an application, we study the structure theory for the corresponding $p$-adic pro-$p$ Schur algebras and obtain an explicit basis and multiplication rules. Moreover, we give algebraic (re)proofs of several results of $p$-adic interest including the existence of PBW basis for the pro-$p$ metaplectic and Iwahori-Hecke algebras, identification of the Iwahori-Schur algebra with the quantum affine Schur algebra, and failure of the local Shimura correspondence at the pro-$p$ level.