  This article discusses the modular representation theory of finite groups of
Lie type from the viewpoint of Broue's abelian defect group conjecture.
  We discuss both the defining characteristic case, the inspiration for
Alperin's weight conjecture, and the non-defining case, the inspiration for
Broue's conjecture. The modular representation theory of general finite groups
is conjectured to behave both like that of finite groups of Lie type in
defining characteristic, and in non-defining characteristic, to a large extent.
  The expected behaviour of modular representation theory of finite groups of
Lie type in defining characteristic is particularly difficult to grasp along
the lines of Broue's conjecture and we raise a new question related to the
change of central character.
  We introduce a degeneration method in the modular representation theory of
finite groups of Lie type in non-defining characteristic. Combined with the
rigidity property of perverse equivalences, this provides a setting for two
variable decomposition matrices, for large characteristic. This should help
make progress towards finding decomposition matrices, an outstanding problem
with few general results beyond the case of general linear groups. This last
part is based on joint work with David Craven and Olivier Dudas.
