Orbital characters and their applications

In this paper, we provide some formulae to calculate orbital characters. Especially we give a lifting theorem by which one can read off the value of orbital characters from local subgroups. With orbital characters, we can show that Reynolds ideal is dual to the vector space of simple modules and this leads to a new method to count simple modules. We also give a natural morphism from local to general vector space of simple modules and determine it completely. Some other applications to modular representation are also included.