subregular nilpotent representations of lie algebras in prime ...

Remark: Suppose that all roots in R have the same length. Then the theorem
says that C has 1 + jJ( )j isomorphism classes of simple modules. For p{regular
, i.e., if J( ) consists of all simple roots, this con rms a conjecture of Lusztig,
see [15], 14.5, [17], 2.4, and [18], 17.2.
F.6. Let 2 C 0 0 . It is clear that modules in C with distinct invariants are not
isomorphic to each other. Given Theorem F.5 the main open problem (besides
type G2 and the restriction on p in the types E8 and F4) is the classi cation of
the simple modules with invariant a long simple root. Unfortunately our results
are not as complete as in the case treated in F.5.
Let again be subregular. Assume that R has type Bn, Cn, orF4; ifRis of
type F4, assume that p>h+1.
Proposition: Let 2 C 0
0 . Let be a long root with 2 J( ). IfRhas type Bn,
then there aretwo isomorphism classes of simple modules with invariant f g.
In the other cases there areat most two such isomorphism classes.
Proof : As in F.5 we get: If this theorem holds for one subregular , then it holds
for all subregular .
If R is of type Bn then we assume that has the form as in [10], Section
3. In this case the claim follows from the results in [10]: If = i, thenthetwo
isomorphism classes are represented by L ( i) and L ( 2n,i) in the notation from
[10].
Assume now that R has type Cn. Then has to be equal to n. Wemay
assume that (p n) = 0. Set
W1 = fw 2 W j w n = ng:
Recall that all Z (x ; n) with x 2 W1 are simple with invariant f ng, see D.3.a
and D.5.c. On the other hand, each simple module L in C with (L) =f ng is
by D.13 isomorphic to some Z (x ; n) withx 2 W1
Since W1 is generated by alls with orthogonal to n =2"n, it is clear
that W1 is the Weyl group of the root system R1 = R \ P i