Cohomological Mackey formula for representations of reductive groups

I will describe the construction of induction and restriction morphisms on the critical cohomology associated with a function on a representation of a reductive group. The induction morphism plays a key role in obtaining a cohomological integrality decomposition, which is a decomposition into finite-dimensional pieces with enumerative significance. After discussing this decomposition and its geometric meaning, I will present a cohomological version of the Mackey formula that relates the induction and restriction operations.