Semisimplicity of Parabolic Hecke Algebras

The semisimplicity of Iwahori–Hecke algebras has been studied by several authors. A. Gyoja (J. Algebra174, 1995, 553–572) gave a necessary and sufficient condition for Iwahori–Hecke algebras to be semisimple, using the modular representation theory. The author (J. Algebra183, 1996, 514–544) studied the semisimplicity of parabolic Hecke algebras when they have only one parameterq. In this paper we completely determine the cases when parabolic Hecke algebras are semisimple complementing our previous work applying the method of Gyoja.