Simple modules for quiver Hecke algebras and the Robinson-Schensted-Knuth correspondence

TL;DR

This paper develops a categorical framework for smooth representations of p-adic general linear groups using quiver Hecke algebras, introduces graded RSK-standard modules, and establishes their properties and classification methods.

Contribution

It formalizes categorical equivalences, constructs graded RSK-standard modules, and links these to p-adic group representations, extending the Specht construction.

Findings

RSK-standard modules have simple heads

A method for classifying simple modules as quotients of induced modules

Properties of the decomposition matrix of RSK-standard modules

Abstract

We formalize some known categorical equivalences to give a rigorous treatment of smooth representations of p-adic general linear groups, as ungraded modules over quiver Hecke algebras of type A. Graded variants of RSK-standard modules are constructed for quiver Hecke algebras. Exporting recent results from the p-adic setting, we describe an effective method for construction and classification of all simple modules as quotients of modules induced from maximal homogenous data. It is established that the products involved in the RSK construction fit the Kashiwara-Kim notion of normal sequences of real modules. We deduce that RSK-standard modules have simple heads, devise a formula for the shift of grading between RSK-standard and simple self-dual modules, and establish properties of their decomposition matrix, thus confirming expectations for p-adic groups raised in a work of the author with Lapid. Subsequent work will exhibit how the presently introduced RSK construction generalizes the much-studied Specht construction, when inflated from cyclotomic quotient algebras.