subregular nilpotent representations of lie algebras in prime ...

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Theorem: Assume that is subregular nilpotent with (b + )=0. Exclude the
case where R is of type G2; ifR is of type E8 or F4, assume that p>h+1. Let
m . We can denote the factors in a
2 C 0 0 . Then Z ( ) has length 1+P 2J( )
composition series of Z ( ) by L0 and L ;i with
that
2 J( ) and 1 i m such
dim L ;i = h + ; _ N ,1
ip and (L ;i) =f g (2)
for all and i, while
dim L0 =(p ,h + ; _ N ,1
0 i)p
and (L0) =J( ) [f0g: (3)
Remarks: 1) The restrictions on the type are hoped to be unnecessary. IfRis of
type E8 or F4 and p h + 1, then the theorem will hold for all 2 C 0
0 where the
socle of Z ( ) has dimension equal to (p ,h + ; _
0 i)pN ,1 . The same remark
applies to the results in D.12/13.
2) The theorem does not say whether di erent factors in the composition series are
isomorphic to each other. If L and L 0 are composition factors with (L) 6= (L 0 ),
then clearly L 6' L 0 . So the question is whether for xed 2 J( )them factors
L ;i are isomorphic to each other. We shall see in Section F that such isomorphisms
exist in certain cases.
3) The theorem con rms in part my speculations in [11], 11.15 (where I look only
at those 2 C 0 0 that are p{regular, i.e., satisfy h + ; _ i > 0 for all 2 R + ).
The part of those speculations not con rmed is that the factors in a composition
series should be pairwise non-isomorphic. As mentioned in the preceding remark,
that turns out to be wrong.
D.7. We now begin to prove the theorem. Lemma B.13 and D.5(3) show: If
Theorem D.6 holds for one subregular , then it holds for all subregular . We
assume from now on that is a simple root with (p )=0and (x, ) 6= 0 for
all simple roots 6= . Remark D.1 shows that we can nd with this property
for each . Later on we shall make speci c choices for . [Usually this is done
such that the right hand side in (6) below is as small as possible.]
Set L0 equal to the simple socle of Z ( ). It satis es D.6(3) by Proposition
D.4.b and D.5(2).
By our choice of the set I as in C.11(1) consists of all simple roots 6= .We
shall use the construction from C.12{14 in order to nd the remaining composition
factors of Z ( ).
We can construct inductively a chain
1 = ; 2;:::; r (1)
of roots in WI such that h r; _ i 0 for all 2 I and such that there exists for
each i