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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28<br />
Pro<strong>of</strong> :a)Wehave clearly s 2 WI and could choose ws = w s .Sowehave<br />
M s = sbm(s w s ; ):<br />
On the other hand s = ,h ; _ i is a positive rootnot<strong>in</strong>ZI. Therefore also<br />
0 = w s is positive. Us<strong>in</strong>g s = w ,1 s w ,we get 0 = s w , hence<br />
s 0s w =(s w s w ,1 s )s w = s w s :<br />
S<strong>in</strong>ce (s w ) ,1 0 = >0, we get from Proposition C.8 that<br />
sbm(s 0s w ; ) sbm(s w ; );<br />
hence (1). Now (2) follows from C.12(3) and (s ) _ = _ ,h ; _ i _ .<br />
b) Now our assumptions imply that s = + = s . It follows that + 2 WI<br />
and that 0 = w s = w ( + ) 2 WI .Weobserved above that 0 = s w ;<br />
so we can apply Lemma C.13 with w = s w .Now x = w 0s w is the same x as<br />
<strong>in</strong> the pro<strong>of</strong> <strong>of</strong> Lemma C.13 and satis es = x .Wehave<br />
M = sbm(w; ) and M + =sbm(ws ; )<br />
s<strong>in</strong>ce w s is a possible ws = w + . Now Lemma C.13 says that we have a<br />
surjective homomorphism<br />
The left hand side has dimension equal to<br />
Z (x ; ) ,! M =M + : (4)<br />
hx( + ); _ ip N ,1 = h + ; _ N ,1<br />
ip<br />
as long as h + ; _ i > 0; otherwise its dimension is equal to p N . In the rst<br />
case, the surjection <strong>in</strong> (4) has to be an isomorphism by dimension comparison;<br />
this imp<strong>lie</strong>s (3).<br />
Remark: In the situation from b) one can deduce the <strong>in</strong>clusion <strong>in</strong> (1) directly from<br />
Lemma C.7.a (without the more complicated argument <strong>in</strong> C.8) us<strong>in</strong>g x ,1 = >0<br />
and the equalities<br />
M = sbm(x; ) and M + = sbm(s x; ):