subregular nilpotent representations of lie algebras in prime ...

D.13. We want tohave an analogue to Lemma D.12 for the simple modules
whose invariant is a long simple root. Suppose that R is of type Bn, Cn, orF4;
if R is of type F4 assume that p>h+1. Set = 1 if R is of type Bn or F4, set
= n if R is of type Cn. So is a long simple root.
Let be subregular with (p )=0. We claim under these assumptions:
Lemma: Let 2 C 0 0 . Let be a long simple root with 2 J( ). IfLis a simple
module in C with (L) =f g, then there exists an element x 2 W with x =
and L ' Z (x ; ).
Proof : As in D.12 it su ces to prove the claim for one special subregular with
(p )=0.
Consider rst R of type Bn. Then we may assume that has the standard
Levi form considered in [10], Section 3. Now the claim follows from [10], 3.13.
Consider next R of type Cn. Wemay assume that (x, i) 6= 0 for all i