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

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

2011-01-25

arxiv.org/abs/1101.4912v2

Mathematics

Representation Theory

Scientific paper

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.

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