Resumen de Spherical and Whittaker functions via DAHA I

Ivan Cherednik, Xiaoguang Ma

• This paper begins with an exposition of the classical p-adic theory of the Macdonald, Matsumoto, and Whittaker functions aimed at the affine generalizations. The major directions are as follows: (1) extending the theory of DAHA to arbitrary levels and (2) the affine Satake map and Hall functions via DAHA. The key result is the proportionality of the two different formulas for the affine symmetrizer, the Satake-type formula and that based on the polynomial representation of DAHA. The latter approach results in two important formulas for the affine symmetrizer generalizing the relations between the Kac–Moody characters and Demazure characters.