On induction of class functions

Let
G G be a connected reductive group defined over a finite field F q \mathbf {F}_q and let L L be a Levi subgroup (defined over F q \mathbf {F}_q ) of a parabolic subgroup P P of G G . We define a linear map from class functions on L ( F q ) L(\mathbf {F}_q) to class functions on G ( F q ) G(\mathbf {F}_q) . This map is independent of the choice of P P . We show that for large q q this map coincides with the known cohomological induction (whose definition involves a choice of P P ).