subregular nilpotent representations of lie algebras in prime ...

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