Finite loop spaces are a generalization of compact Lie groups.
However,
they do not enjoy all of the nice properties of
\clg s. For example having a maximal
torus is a quite distinguished property. Actually, an old
conjecture, due to Wilkerson, says said that every finite connected loop space
with a maximal torus is
equivalent to a connected compact Lie group. We give some more evidence
for this conjecture
by showing that the associated action of the Weyl group
on the maximal torus
always represents the Weyl group as a crystallographic group. We also
develop the notion of normalizers of maximal tori
for finite connected loop spaces and prove for a large class of
finite connected loop spaces that a finite connected loop space
with maximal tori is equivalent to
a connected compact Lie group if it has the right normalizer
of the maximal torus.
Actually, in the cases
under consideration the information about the Weyl group is sufficient
to give the answer. All this is done by
first studying the analoguous local or completed problems.
In particular, we prove homotopy uniqueness results for a large class
of compact connected
Lie groups considered as p-compact groups where the only input is given by
the Weyl group data.