Ever since the classical theory of a highest-weight classification, one ever-present strength of Lie theory, was its ability to produce 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 GL_n. 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. Yet, throughout these settings, a systematic understanding of the nature of reducible finite-length representations remains largely out of grasp.
Recently, joint with Erez Lapid, we have revisited the 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 algorithm, 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 the new 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.
[ L’exposé aura lieu en ligne, sur Zoom. Pour recevoir le lien de connection, contacter l’organisateur LP]