subregular nilpotent representations of lie algebras in prime ...

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B.5. Proposition: Suppose that and belong to the same facet with respect
to Wp. Then T L is simple for each simple module L in C . The functor T
induces a bijection from the isomorphism classes of simple modules in C to the
isomorphism classes of simple modules in C ; the inverse is induced byT .
Proof : Consider a composition series
Z ( )=Mr Mr,1 M1 M0 =0
of Z ( ). The exactness of T and B.4(1) yield a chain of submodules
Z ( ) ' T Mr T Mr,1 T M1 T M0 =0:
Each T Mi=T Mi,1 ' T (Mi=Mi,1) is non-zero by Lemma B.4. It follows that
the length of Z ( ) is greater than or equal to the length of Z ( ). By symmetry
we get also the reversed inequality. Therefore both modules have the same
length. This implies that all T (Mi=Mi+1) are simple, hence the rst claim of the
proposition.
We getby symmetry: If L 0 is simple in C , then also T L 0 is simple. It follows
that T T L is simple for all L simple in C .Wehave by adjointness
Homg(T T L; L) ' Homg(T L; T L) 6= 0;
hence T T L ' L (both modules being simple). We get by symmetry: T T L 0 '
L 0 for all L 0 simple in C . The second claim follows.
B.6. Let I be a subset of the set of simple roots. Let PI be the corresponding
parabolic subgroup of G and pI its Lie algebra. So pI is the direct sum of b + and
the g with 0
and =2 RI) annihilates each Z ;I( ). Considered as a module over gI ' pI=n I ,
each Z ;I( ) identi es with the corresponding baby Verma module for U (gI).
The Weyl group WI of RI identi es with the subgroup of W generated by
the s with 2 I; similarly for the corresponding a ne Weyl group WI;p. The
group GI again satis es (B1) and (B2); for (B2) note that ZRI = ZI is a direct
summand of ZR. We de ne categories C(I) of nite dimensional U (gI){modules