subregular nilpotent representations of lie algebras in prime ...

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Remark: Let denote the usual Chevalley-Bruhat order on W (with smallest
element 1). The proposition implies for all w1;w2 2 W :
w1 w2 =) sbm(w1; ) sbm(w2; ): (11)
C.9. Lemma: Let be a weight in the closure of the facet of with 2 C 0
0 .
For all w 2 W and 2 R simple and w ,1 >0 we can identify T (' w;s w) with
a non-zero multiple of ' w;s w . We can identify T (' w ) with a non-zero multiple
of ' w and have
T sbm(w; ) ' sbm(w; ) (1)
for all w 2 W .
Proof : In order to prove the claim for the ' w;s w we distinguish two cases:
If hw( + ); _ i = 0, then the assumption on implies that also hw( +
); _ i = 0. Then both ' w;s w and ' w;s w are the identity. SoisT (' w;s w); the
claim follows in this case.
If hw( + ); _ i > 0, then the claim follows easily from Proposition B.10
because ' w;s w is a non-zero multiple of the map ' w considered there.
The rest of the lemma follows now from the de nition of ' w, the functor
property ofT and its exactness.
C.10. We can apply Lemma C.9 to = , . Since all ' , non-zero, we get that
w are the identity, hence
'w 6= 0 for all w 2 W . (1)
This holds in particular for w = w0. Proposition C.4.b implies therefore that
C.11. Set
sbm(w0; )=socZ ( ): (2)
I = f 2 R j simple; (x, ) 6= 0g: (1)
Let WI denote the subgroup of W generated by alls with 2 I.
Lemma: a) We have sbm(w1; ) = sbm(w2; ) for all w1;w2 2 W with WIw1 =
WIw2.
b) Let w 2 WI and be a simple root with =2 I. Then
and
where N = jR + j.
sbm(s w; ) ' Z (s w ; ) (2)
dim sbm(s w; )=(p ,hw( + ); _ N ,1
i)p
Proof : The claim in a) follows easily from Lemma C.7.b. In b) we have w ,1 >0
since w 2 WI and =2 I. It follows that ' s w = ' w ' w;s w. Furthermore ' w is a
composition of certain ' w 0 ;s w 0 with 2 I, hence an isomorphism. Therefore the
image sbm(s w; )of' s w is isomorphic to the image of ' w;s w .Now the claims
in b) follow from Lemma C.5.
(3)