subregular nilpotent representations of lie algebras in prime ...

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