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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68<br />
Assume that R is not <strong>of</strong> type A1. Iwant toshow that the assumption <strong>in</strong> b)<br />
is satis ed. Let be a simple root. If $ is a m<strong>in</strong>uscule fundamental weight with<br />
6= ,thenwe can choose =(p ,h ; _ 0 i)$ .<br />
We are left with the cases where there are no m<strong>in</strong>uscule fundamental weights<br />
at all, or where $ is the only m<strong>in</strong>uscule fundamental weight. (So we are not <strong>in</strong><br />
type An.) In these cases one can nd a fundamental weight $ with h$ ; _ 0 i =2<br />
and 6= . (Recall that we exclude G2.) Furthermore both h ; _ 0 i and p are<br />
odd (s<strong>in</strong>ce we are done with type An). So we can choose = r$ with r =<br />
(p ,h ; _<br />
0 i)=2.<br />
Remark: If R is <strong>of</strong> exceptional type, then there exist simple roots and with<br />
h$ ; _ 0 i = 2 and h$ ; _ 0 i = 3. Consider positive<strong>in</strong>tegers a and b with 2a+3b = p.<br />
(They exist for each p 5.) Then = a$ + b$ , satis es h + ; _ 0 i = p.<br />
The weights <strong>in</strong> the facet <strong>of</strong> are all = a 0 $ + b 0 $ , with a 0 ;b 0 > 0and<br />
2a 0 +3b 0 = p. We have h + ; _ i = 1 if and only if a 0 = 1, if and only if<br />
3b 0 = p , 2. So we can nd with this property only <strong>in</strong> case p 2 (mod 3). So<br />
does not satisfy the condition <strong>in</strong> b). Us<strong>in</strong>g Remark 2 <strong>in</strong> H.1 one can see that<br />
also does not satisfy the condition <strong>in</strong> a).<br />
H.4. Let 2 C0 with h + ; _ 0 i = p. Suppose that is a weight with 2 C 0 0<br />
such that is <strong>in</strong> the closure <strong>of</strong> the facet <strong>of</strong> . (Given we can nd with this<br />
property as long as there exists a simple root with h + ; _ = , $ works. If there is no such , then<br />
i > 1: In that case<br />
+ = P 2J( ) $ and then no<br />
as above can exist. This can happen only <strong>in</strong> case p h , 1.)<br />
Lemma: Let and be asabove. Let be a simple root and let 2 g with<br />
(p )=0. Let w 2 W .<br />
a) We have T Z (w ; ) = 0 if w ,1 < 0 and hw( + ); _ i = ,p or if<br />
w ,1 >0 and hw( + ); _ i =0< hw( + ); _ i. In all other cases we have<br />
T Z (w ; ) ' Z (w ; ).<br />
b) Suppose that hw( + ); _ i6 0 (mod p). Then T Z (w ; ) has a ltration<br />
with factors Z (w 0 ; ) with 0 2 (StabWp ) .<br />
Pro<strong>of</strong> : a) This is more or less a special case <strong>of</strong> Corollary B.11. One uses that<br />
hw( + ); _ i > ,p for all w 2 W .<br />
b) We apply Proposition B.7 with I = f g to (w ; w ) <strong>in</strong>stead <strong>of</strong> ( ; ). S<strong>in</strong>ce<br />
w is <strong>in</strong> the closure <strong>of</strong> the facet <strong>of</strong> w , the assumption hw( + ); _ i 6 0<br />
(mod p) imp<strong>lie</strong>s that also hw( + ); _ i6 0 (mod p). So both w and w<br />
have trivial stabiliser <strong>in</strong> WI;p. Proposition B.7 says now that<br />
has a ltration with factors<br />
T Z (w ; )=T w<br />
w <strong>in</strong>dI L ; (w )<br />
<strong>in</strong>dI T (I) wiw<br />
w L ; (w ) (1)<br />
where the wi belong to StabWpw and are a system <strong>of</strong> representatives for the<br />
cosets modulo StabWpw .