On reciprocal characters and the quantum affine Schur-Weyl duality

Schur-Weyl duality is a cornerstone of representation theory. It has a modern analogue, where modules over the quantum affine algebras (objects of intense study in mathematical physics) are placed in correspondence with modules over affine Hecke algebras, and ultimately, with smooth representations of p-adic groups.

Within this picture, one may wonder about the notion of a character, that is, which invariants are good enough to 'characterize' a representation?
For quantum affine algebras, much success was attained by analyzing the so-called q-character, devised by Frenkel-Reshetikhin.

In a recent work with PhD candidate Angelina Vargulevich, we show that under Schur-Weyl duality, the dominant part of the q-character admits a natural, and yet surprising and underexplored, interpretation on the p-adic side. It is nicely expressible in terms of the Local Langlands Reciprocity.

I will try to convey the main phenomena to a broad audience with minimal specialized background.