Affine Lie group approach to a derivative nonlinear Schr\"odinger equatoin and its similarity reduction

The generalized Drinfel'd-Sokolov hierarchies studied by de Groot-Hollowood-Miramontes are extended from the viewpoint of Sato-Wilson dressing method. In the A_1^(1) case, we obtain the hierarchy that include the derivative nonlinear Schr\"odinger equation. We give two types of affine Weyl group symmetry of the hierarchy based on the Gauss decomposition of the A_1^(1) affine Lie group. The fourth Painlev\'e equation and their Weyl group symmetry are obtained as a similarity reduction. We also clarify the connection between these systems and monodromy preserving deformations.