Quantum Affine Algebras, Canonical Bases, And Q-Deformation Of Arithmetical Functions

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