subregular nilpotent representations of lie algebras in prime ...
subregular nilpotent representations of lie algebras in prime ...
subregular nilpotent representations of lie algebras in prime ...
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simple module <strong>in</strong> C <strong>of</strong> the same dimension with (L 0 )=f ng by B.13(2) and<br />
dim T L 0 = p N ,1 .<br />
S<strong>in</strong>ce L 0 is <strong>in</strong> C it has to be a composition factor <strong>of</strong> Z ( ). All Mi=Mi+1 with<br />
i 6= n,1 and M2n,2 are simple with <strong>in</strong>variant di erent fromf ng. Therefore L 0<br />
has to be a composition factor <strong>of</strong> Mn,1=Mn. It follows that this factor module has<br />
length equal to 2 and that the second composition factor, say L 00 , has to have the<br />
same dimension. Furthermore, the exactness <strong>of</strong> the translation functors imp<strong>lie</strong>s<br />
that T (Mn,1=Mn) has a ltration with factors T L 0 and T L 00 . By comparison<br />
<strong>of</strong> dimensions we get that also T L 00 has dimension equal to p N ,1 . This shows<br />
that n 2 (L 00 ). On the other hand, s<strong>in</strong>ce L 00 is a composition factor <strong>of</strong> Mn,1=Mn<br />
we have<br />
(L 00 ) (Mn,1=Mn) =f ng:<br />
So we get equality: (L 00 )=f ng. This completes the pro<strong>of</strong> <strong>of</strong> the claim concern<strong>in</strong>g<br />
the composition factors <strong>of</strong> Mn,1=Mn, hence that <strong>of</strong> the theorem for type<br />
Cn.<br />
D.10. Consider R <strong>of</strong> type F4. Wechoose such that = 4 <strong>in</strong> the notations from<br />
[1]. So is short. We can choose the cha<strong>in</strong> <strong>in</strong> D.7(1) such that the correspond<strong>in</strong>g<br />
simple roots are (<strong>in</strong> this order)<br />
3; 2; 1; 3; 2; 3:<br />
We get r = 7 and 7 = 1 +2 2 +3 3 + 4, hence<br />
_<br />
7 =2 _<br />
1 +4 _<br />
2 +3 _<br />
3 + _<br />
4 = _<br />
0 , _<br />
4 :<br />
Set M8 = sbm(w0; ). We get from D.7(4){(6)<br />
dim(M0=M1) = dim(M7=M8) =h + ; _<br />
4 ipN ,1 ;<br />
dim(M1=M2) = dim(M4=M5) = dim(M6=M7) =h + ; _<br />
3 ip N ,1 ;<br />
dim(M2=M3) = dim(M5=M6) =2h + ; _<br />
2 ip N ,1 ;<br />
dim(M3=M4) =2h + ; _<br />
1 ip N ,1 :<br />
Lemma D.3.b imp<strong>lie</strong>s that M0=M1, M7=M8, M1=M2, M4=M5, andM6=M7 are<br />
simple or 0; if non-zero, then the rst two modules have <strong>in</strong>variant f 4g, and the<br />
rema<strong>in</strong><strong>in</strong>g three modules have <strong>in</strong>variant f 3g.<br />
Each factor module M2=M3, M5=M6, M3=M4 has length at most equal to 2<br />
by the Remark D.2. We have<br />
(M2=M3) = (M5=M6) =f 2g and (M3=M4) =f 1g:<br />
More precisely, wehave, if 2 2 J( ),<br />
dim T $2,<br />
(M2=M3) = dim T $2, N ,1<br />
(M5=M6) =2p<br />
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