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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Argu<strong>in</strong>g as <strong>in</strong> H.4 one checks that L has a ltration with factors Z (w ; )<br />
' L and Z (ws<br />
have<br />
; ). We want to apply Lemma H.6 to the second factor. We<br />
(ws ) ,1 _ = s<br />
_ = _ ,h ;<br />
_ _ i :<br />
Now the claim <strong>in</strong> a) follows immediately from H.6.<br />
In most cases Lemma H.4 imp<strong>lie</strong>s that 0L has a ltration with factors<br />
Z (w ; ) ' L and Z (ws0 ; ) ' Z (ws 0 ; ). The only exception occurs<br />
when hw( (0) + ); _ i = h (0) + ; _ i is congruent to 0 modulo p. The choice <strong>of</strong><br />
(0) imp<strong>lie</strong>s that this can happen only if = 0, hence only if R has type A1. Set<br />
that case aside for the moment. We want to apply Lemma H.6 to Z (ws 0 ; );<br />
we have<br />
(ws 0) ,1 _ = s 0<br />
_ = _ ,h 0; _ i _<br />
0 :<br />
S<strong>in</strong>ce _<br />
0 is the largest root <strong>in</strong> R_ and s<strong>in</strong>ce R has not type A1, wehave h 0; _ i2<br />
f0; 1g. It follows that (ws 0) ,1