
--- Bringing it all together -------------------------------------------------

10-13-04

Things are starting to look up, and both approaches seem to converge nicely.

The basic parameter space is the set of all elements of G which conjugate
one of the representatives of standard real Cartans into our prototype Cartan,
up to the action of the normaliser of H_i in G(R). For each fixed i, the
set of conjugations obviously has a transitive action of the normalizer of
T in G. 

... something is missing here. I wonder if the full normalizer should act ?
should we restrict conjugations somehow ? ...

Anyway we should make it so that a parameter is really the choice of an
involution on \frak(t), together with a grading of the imaginary roots.
These data allow to completely reconstruct the involution on \frak(g),
hence on G, and therefore should lead to a conjugacy (isomorphism?) with
our original G(R).

Then things come together beautifully : classification of gradings turns
out to be very easy. Underlying to our parameter there is the involution
of the Cartan, which amounts to an involution in the Weyl group, modulo
perhaps an involution of the graph. The real Weyl group W(G,H_i) should be
the stabilizer of a parameter.

Moreover, fitting all real forms in a given inner class together works
perfectly : they all share a fundamental Cartan. For this Cartan, all
gradings are acceptable, their classes correspond precisely to the
classifiacation of the inner forms in this inner class. Each grading
class "propagates" partially downwards, picking out disjoint grading
classes on the various M-groups (which are the same for all the groups
in the inner class, but with varying real forms; actually no two M-groups
are ever isomorphic as real groups between different real forms in the
inner class.) Furthermore there is a wonderful descent algorithm
explained to me by David Vogan, which will tell how the grading must
behave under Cayley descent --- this is how we may describe the full
parameter space easily, using descent and saturation (perhaps a little
bit more easily than using ascents).

Actually this is just the "base" of the parameter space; there are fibers
over each parameter which are really component groups of the corresponding
tori.

What remains to be clarified is the influence of the covering group. It should
be the case that not all real forms are acceptable for all covering groups.
To what extent this is bizarre or not I need to discuss with David.
