subregular nilpotent representations of lie algebras in prime ...

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The assumption on the facet implies that
np h + ; _ i (n +1)p:
Set d = h + ; _ i,np and d 0 = h + ; _ i,np. Then d is the same number as
considered in B.8. If d 0 > 0, then it the analogue of d working with instead of ;
if d 0 = 0, then that analogue is equal to p, however. Note that s ;np = , d
and s ;np = , d 0 .
We get from B.4(1) isomorphisms T Z ( ) ' Z ( )andT Z ( , d ) '
Z ( , d 0 ). We claim that, modulo these isomorphisms:
Lemma: If d 0 > 0, then T (' ) is a non-zero multiple of ' . If d 0 =0, then
T (' ) is a non-zero multiple of the identity.
Proof :Ifd = p, i.e., if h + ; _ i =(n +1)p, then the assumption on the facets
implies that also h + ; _ i =(n +1)p and d 0 = p. In this case both ' and '
are equal to (x, ) p times the identity. Since the functor T takes a multiple of
the identity to the corresponding multiple of the identity, the claim follows in this
case.
Assume from now on that d