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

We formalize some known categorical equivalences to give a rigorous treatment of smooth representations of (Formula presented.) -adic general linear groups, as ungraded modules over quiver Hecke algebras of type (Formula presented.). Graded variants of RSK-standard modules are constructed as a new basis for Grothendieck groups of quiver Hecke algebras. Exporting recent results from the (Formula presented.) -adic setting, we describe an effective method for construction and classification of simple modules as quotients of modules induced from maximal homogenous data. It is established that the products involved in the Robinson–Schensted–Knuth 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 (Formula presented.) -adic groups raised in a previous work of the author with Lapid. We lay the ground for a subsequent work that exhibits the RSK construction as a generalization the better explored Specht construction, when inflated from cyclotomic quotient algebras.