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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where E is the simple G{module with highest weight <strong>in</strong>W ( , ). (This makes<br />
sense: see the argument <strong>in</strong> [12], 4.7 where there is a more restrictive assumption<br />
on , which however does not a ect the argument here.) We haveclearly T =<br />
w +p<br />
Tw +p for all w 2 W and 2 X. Note that we get (<strong>in</strong> general) more than one<br />
functor C !C for xed and :if and 0 are two dist<strong>in</strong>ct weights <strong>in</strong> C with<br />
0 2 W + pX, thenT<br />
0<br />
and T will be two (<strong>in</strong> general:) dist<strong>in</strong>ct functors from<br />
C to C = C 0.<br />
We haveas usual that each T is exact and that T and T are adjo<strong>in</strong>t to<br />
each other: The natural isomorphism Homg(E<br />
<strong>in</strong>duces an isomorphism<br />
M;N) ,! Homg(M;E N)<br />
cf. [12], 4.7.<br />
Homg(T (M);N) ,! Homg(M;T (N)); (2)<br />
B.3. For the next result we have to use the assumption (B2) made at the beg<strong>in</strong>n<strong>in</strong>g<br />
<strong>of</strong> this section.<br />
Proposition: Let ; 2 X and w 2 Wp. Suppose that there exists a closed<br />
alcove with respect to Wp conta<strong>in</strong><strong>in</strong>g and . Then T Z (w ) has a ltration<br />
with factors Z (ww1 ) with w1 2 Wp, w1 = . There is one factor for each<br />
weight <strong>of</strong> the form ww1 .<br />
Pro<strong>of</strong> : Consider E as <strong>in</strong> the de nition <strong>of</strong> T . The tensor identity (see [12], 1.12(1))<br />
imp<strong>lie</strong>s that E Z (w ) has a ltration with factors Z (w + )with runn<strong>in</strong>g<br />
over the weights <strong>of</strong> E (counted with multiplicities). Then T Z (w ) has<br />
a ltration with factors Z (w + ) where runs over all weights <strong>of</strong> E with<br />
w +p 2 W +pX. Suppose that has this property; so there are w1 2 W and<br />
1 2 X with w + = w1 + p 1. Wehave w 2 + ZR and 2 ( , )+ZR<br />
and w1 2 + ZR, hence p 1 2 ZR. In other words, we have p( 1 + ZR) =0<br />
<strong>in</strong> X=ZR. Assumption (B2) yields now 1 2 ZR, hence w1 + p 1 2 Wp .<br />
So T Z (w ) has a ltration with factors Z (w + ) where runs over those<br />
weights <strong>of</strong> E with w + p 2 Wp + pX. Now the claim follows from standard<br />
results, cf. [9], II.7.13.<br />
B.4. Note that Proposition B.3 imp<strong>lie</strong>s <strong>in</strong> particular: If is <strong>in</strong> the closure <strong>of</strong> the<br />
facet <strong>of</strong> with respect to Wp, then we have<br />
for all w 2 Wp.<br />
11<br />
T Z (w ) ' Z (w ) (1)<br />
Lemma: If is <strong>in</strong> the closure <strong>of</strong> the facet <strong>of</strong> with respect to Wp, then we have<br />
T L 6= 0for all simple modules L <strong>in</strong> C .<br />
Pro<strong>of</strong> : There exists w 2 W with Homg(Z (w );L) 6= 0. Now (1) and the<br />
adjo<strong>in</strong>tness property B.2(2) imply that<br />
hence the claim.<br />
Homg(Z (w );T L) ' Homg(Z (w );L) 6= 0;<br />
Remark: The claim extends <strong>of</strong> course to all non-zero modules <strong>in</strong> C .