subregular nilpotent representations of lie algebras in prime ...

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Remark: Our assumptions (B1) and (B2) are not needed to prove (1). For (2) we
just need (B1). Similarly, one can check that the rst seven paragraphs in B.13
below do not require the assumptions while (B1) su ces for the remainder of B.13
and for B.14.
B.13. In this and the next subsection we drop our assumption that (b + )=0
and consider arbitrary 2 g .(Butwe shall soon assume to be nilpotent.) Let
C( ) denote the category of all nite dimensional U (g){modules; as before we just
write C when it is clear which we consider.
Let g 2 G. We write Ad(g) for the adjointactionofg on g and for the induced
action on U(g). If M is a g{module, then we write g M for M \twisted by g", i.e.,
we take g M = M as a vector space and let any x 2 g (or in U(g)) act on g M as
Ad(g ,1 )(x) doesonM. (See also the more general discussion at the beginning of
[10], 1.13.)
Clearly M 7! g M is an exact functor that takes simple modules to simple
modules. If M is a G{module considered as a g{module via the derived action, then
we have an isomorphism g M ,! M given by m 7! gm. IfM is a U (g){module,
then g M is a Ug (g){module where g is the image of under the coadjoint action
of g given by (g )(x) = (Ad(g ,1 )x). The functor M 7! g M is an equivalence of
categories from C( )toC(g ). It takes simple modules to simple modules.
Assume now that is nilpotent. This means that vanishes on a Borel
subalgebra of g, or, equivalently, that there exists g 2 G with (g )(b + ) = 0, see
[13].
If M is a U (g){module, then each u 2 U(g) G acts on each g M as it does on
M. Suppose for the moment that (g )(b + )=0;ifM is simple, then U(g) G acts
on g M (and hence also on M) as on some Zg ( )with 2 X. Set C( ) equal to
the full subcategory of C( ) consisting of those N in C( ) such thatU(g) G acts
on each composition factor of N as on Zg ( ). Then C( ) is the direct sum of all
C( ) with in a suitable set of representatives.
Note: If we have already (b + ) = 0, then the de nition of C( ) given above
coincides with that one that we get by applying B.1 directly, because each u 2
U(g) G acts on Z ( ) and on Zg ( )by the same scalar (for each 2 X), cf.
[10], 1.7. This observation implies for arbitrary that the de nition of C( ) is
independent of the choice of g with (g )(b + )=0.
We have (foreach nilpotent ) projection functors pr : C( ) !C( ) and
translation functors T : C( ) !C( ) de ned as in B.1/2.
Let g 2 G. IfM is a U (g){module in C( ) ,then g M belongs to C(g ) . The
functor M 7! g M restricts to an equivalence of categories from C( ) to C(g ) .
We get for arbitrary M in C( ) that
This implies for M in C( ) that
pr ( g M)= g (pr M): (1)
g (T M) ' T ( g M): (2)