Quantum affine algebras, canonical bases and $q$-deformation of arithmetical functions

TL;DR

This paper develops affine analogues of key formulas in representation theory, introduces q-deformations of arithmetical functions, and establishes identities and limits connecting these deformations to classical functions.

Contribution

It provides new affine analogues of classical formulas and introduces natural q-deformations of arithmetical functions with proven identities.

Findings

Affine Gindikin-Karpelevich and Casselman-Shalika formulas derived

q-deformations of partition and Ramanujan tau functions defined and related identities proved

Classical identities recovered through limits of q-deformed functions

Abstract

In this paper, we obtain affine analogues of Gindikin-Karpelevich formula and Casselman-Shalika formula as sums over Kashiwara-Lusztig's canonical bases. Suggested by these formulas, we define natural $q$-deformation of arithmetical functions such as (multi-)partition function and Ramanujan $\tau$-function, and prove various identities among them. In some examples, we recover classical identities by taking limits. We also consider $q$-deformation of Kostant's function and study certain $q$-polynomials whose special values are weight multiplicities.