
        Parametrization of representations

This is clearly one of the most fundamental issues of the program, at least
for everything that revolves around k-l computations. We wish to look at
all irreducible (g,K)-modules for a certain fixed infinitesimal character,
perhaps for all the real forms in a given inner class of our complex group G.

There are at least three descriptions : the one in terms of conjugacy classes
of real parabolics, and Langlands parameters; the one in terms of local systems
on K-orbits in G/B; the one in terms of L-groups. Of course the three should
be reconciled.

Currently it seems that the most accessible computationally is the first one;
the other two should rather be used for sanity checking. The idea is to fix
a complex torus T, and to bring everything back to T. Let us fix a positive
Weyl chamber in the Lie algebra t once and for all, and let's assume that
we are dealing with a regular infinitesimal character. Then for each
Cartan involution theta on G, and each theta-stable Cartan subgroup H, there
is a definite Weyl regular Weyl group orbit in h* attached to our given
infinitesimal character. For each lambda in that orbit, there is a conjugation
in G which takes T to H and our chosen positive chamber to the one that
contains lambda; this conjugation is unique up to right multiplication by
an element of T. It is also unique up to left multiplication by an element
of H.

Now we wish to keep track of our involution theta. Notice that theta defines
a grading of the imaginary roots in H (the set of imaginary roots depends only
on the restriction of theta to H, but the grading depends --- in fact, defines
--- the global theta.) Then two involutions stabilizing H are conjugate
under H_ad iff they are conjugate under H^theta iff they define the same 
grading of the imaginary roots. Therefore G_ad-conjugacy classes of triples 
(H,theta,lambda) correspond bijectively to pairs (theta,gr) in T, where
theta is an involution of the root datum in T and gr is a grading of the
imaginary roots. So these pairs are our fundamental set of parameters for the
adjoint group.

Passing to a general group is just an issue of how much conjugacy under the
group (equivalently, under its image in the adjoint group) differs from
conjugacy under the adjoint group.
