subregular nilpotent representations of lie algebras in prime ...

We keep the assumptions from Sections B and C. We assume in addition:
(D1) The prime p is good for R.
and
(D2) G is almost simple.
D
The rst assumption is crucial so that we can apply the Kac-Weisfeiler conjecture
proved by Premet. The second assumption is mainly meant to simplify the
statement of the results.
D.1. We call a linear form 2 g subregular if its orbit under the coadjoint
action has dimension equal to 2(N , 1) where N = jR + j. If so, then the Kac-
Weisfeiler conjecture as proved by Premet says: If M is a U (g){module, then
dim(M) is divisible by p N ,1 .
Recall that we setp = b + + g, for each simple root ;we denote (as in
B.14) by P the corresponding parabolic subgroup of G. The following result is a
translation of well known results on orbits in g:
Lemma: There exists a unique subregular nilpotent orbit O in g . If is a simple
root then O intersects the set of all 2 g with (p )=0in an open and dense
subset. This intersection is one orbit under P .
Proof :We can (under our assumptions on p) identify g and g as G{modules. The
classi cation of the nilpotent orbits in g is the same as for the Lie algebra over C
of the same type (since p is good). In particular, there is exactly one subregular
nilpotent orbit in g; this yields the rst claim.
The elements 2 g with (b + ) = 0 and (x, ) = 0 correspond (under
g ' g ) to the elements in the nilradical n = L >0; 6= g of the parabolic
subalgebra p . The theory of the Richardson orbits (cf. [3], 5.2.3) says that there
exists exactly one nilpotent orbit for G that intersects n in an open and dense
subset. That intersection is one orbit under P ; furthermore the dimension of the
orbit (under G) is equal to the codimension in g of a Levi factor of p . Since that
codimension is equal to 2(N , 1), we get the remaining claims.
Remark: For each simple root 6= the set of all with (p ) = 0 and (x, ) 6= 0
is an open and dense subset of the set of all with (p ) = 0. It follows: The set
of all subregular 2 g with (p )=0and (x, ) 6= 0for all simple roots 6=
is an open and dense subset of the set of all with (p )=0.
D.2. Each subset J of the set of all simple roots de nes a facet F (J) contained in
C 0 0 as follows: A weight 2 C 0 0 belongs to F (J) if and only if h + ; _i > 0 for all
2 J and h + ; _i = 0 for all simple roots =2 J. Write $ for the fundamental
weight corresponding to a simple root . Then 2 X belongs to F (J) if there
are integers m > 0 such that + = P 2J m $ and if h + ; _i