Symmetries of the Kac-Peterson Modular Matrices of Affine Algebras

One of the more fruitful observations in recent years was the discovery [19] that the (normalized) character χμ of the integrable highest weight g-module L(μ) of a nontwisted Kac-Moody algebra g = X (1) r is a modular form. Thanks to the structure of the affine Weyl group, these χμ can be written as ratios of theta functions, so on them exists a natural action of SL2(Z). In fact, this action defines a representation