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

TL;DR

This paper introduces quantum wreath products to analyze the pro-$p$ Iwahori-Hecke algebra and its modules for the $p$-adic general linear group, providing new descriptions and structural insights.

Contribution

It develops the theory of quantum wreath products of skew polynomial type and applies it to describe $p$-adic Hecke algebras and modules, including new notions like (anti)spherical modules.

Findings

Explicit basis and multiplication rules for $p$-adic pro-$p$ Schur algebras.

Algebraic proofs of PBW basis existence for pro-$p$ Hecke algebras.

Identification of Iwahori-Schur algebra with quantum affine Schur algebra.

Abstract

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.