  In this paper we continue the study of character sheaves on a reductive group
G. To each subset of the set of simple reflections in the Weyl group we
associate an algebra of the same kind as an Iwahori-Hecke algebra with unequal
parameters in terms of parabolic character sheaves. We also prove a Mackey type
formula for character sheaves. We define a duality operation for character
sheaves. We also prove a quasi-rationality property for character sheaves.
