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.
This video was produced by the Okinawa Institute of Science and Technology, as part of their OIST Representation Theory Seminar series.
