subregular nilpotent representations of lie algebras in prime ...

H.3. So far we havestudied only the categories C with 2 C 0 0. In order to get
all possible C we should also look at the 2 C0 with h + ; _ 0 i = p. Weknow
0 0 from H.1 that there exist in many cases 2 C0 with C = C 0, but there are also
cases where this does not hold.
When we try to extend the theory for weights in C 0 0 to the remaining weights
in C0, then we encounter two major problems: The proof in D.3 that certain
modules are simple will not work for all 2 C0, and there is not an easy de nition
of a invariant as in D.5. The rst problem occurs only for relatively few and
we are going to ignore them here.
De nition: We say that a weight 2 C0 is nice if for all simple roots , all
subregular 2 g with (p ) = 0, and all w 2 W with w ,1 simple and hw( +
); _ i > 0 the module Z (w ; ) is simple.
Note that Lemma D.3.a says that each 2 C 0
0 is nice. I hope that all 2 C0
are nice in good characteristic; what I can prove is this:
Lemma: Let 2 C0 with h + ; _ 0 i = p.
a) In case R is of type E8, F4, orG2, assume that p>h+1. If there exists a
simple root with $ minuscule such that y + p$ 2 C 0 0 , then is nice.
b) If there exists for each simple root with h + ; _ i > 0 a weight in the
same facet as with h + ; _ i =1,then is nice.
c) Suppose that R is not of type G2. Ifh + ; _ i > 0 for all simple roots , then
is nice.
Proof : Consider , , and w as in the de nition above.
a) Set 0 = y + p$ .Wehave
Z (w ; ) ' Z (wy ,1 0 ; ): (1)
Recall from H.1 that y permutes [f, 0g. Wehave therefore
(wy ,1 ) ,1 = y w ,1 2 [f, 0g:
So the right hand side in (1) is simple by Lemma D.4.a (in case y w ,1 = , 0)
or by Lemma D.3.a (in case y w ,1 simple) provided we know in the second case
that hwy ,1 ( 0 + ); _ i > 0. However, one checks easily that
hwy ,1 ( 0 + ); _ i = hw( + ); _ i + ph$ ;y w ,1 _ i > 0
using wy ,1 ( 0 + )=w( + )+pwy ,1 $ .
b) Under these assumptions Lemma D.2 holds with F (J) replaced by the facet of
. Therefore the proof of Lemma D.3.a works in this case.
c) Note that p = h + ; _ 0 i and 2 C0 imply p h ; _ 0 i = h , 1.
If R is of type A1, then = 0 and w =1. Then Z ( ; )= Z ( )isa
Steinberg module, hence irreducible.
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