Characters of symmetric groups and of the general linear group are tied together by Schur–Weyl duality. I will explain this classical picture and show how it survives the quantum affine generalization.

For quantum affine algebras there is a natural notion of character, the q-character of Frenkel-Reshetikhin, which records the dimensions of certain weight spaces. Together with my advisor Maxim Gurevich, we construct a new invariant on the affine Hecke algebra side, which we call the reciprocal character. Under quantum affine Schur–Weyl duality it corresponds to the dominant part of the q-character. It also admits a natural interpretation in the representation theory of p-adic groups.

Additionally, we discovered that the dominant part of the q-character corresponds to monomial bases in Lusztig's algebra, through Ariki's categorification.