One ever-recurring goal of Lie theory is the quest for effective and elegant descriptions of collections of simple objects in categories of interest. A cornerstone feat achieved by Zelevinsky in that regard, was the combinatorial explication of the Langlands classification for smooth irreducible representations of p-adic GLn. It was a forerunner for an exploration of similar classifications for various categories of similar nature, such as modules over affine Hecke algebras or quantum affine algebras, to name a few. A next step - reaching an effective understanding of all reducible finite-length representations remains largely a difficult task throughout these settings.
Recently, joint with Erez Lapid, we have revisited the original Zelevinsky setting by suggesting a refined construction of all irreducible representations, with the hope of shedding light on standing decomposition problems. This construction applies the Robinson-Schensted-Knuth transform, while categorifying the determinantal Doubilet-Rota-Stein basis for matrix polynomial rings appearing in invariant theory. In this talk, I would like to introduce the new construction into the setting of modules over quiver Hecke (KLR) algebras. In type A, this category may be viewed as a quantization/gradation of the category of representations of p-adic groups. I will explain how adopting that point of view and exploiting recent developments in the subject (such as the normal sequence notion of Kashiwara-Kim) brings some conjectural properties of the RSK construction (back in the p-adic setting) into resolution. Time permits, I will discuss the relevance of the RSK construction to the representation theory of cyclotomic Hecke algebras.
