subregular nilpotent representations of lie algebras in prime ...

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E.6. We now drop the assumptions xed throughout E.1{5 and consider a more
general situation.
Let = 0 + 1 2 g with
0(n , + n + )=0 and 1(b + )=0: (1)
So this looks like E.1(1); but we no longer assume that = 0 + 1 is a Jordan
decomposition, i.e., the analogue of E.1(2) will not hold in general.
On the other hand, we x again a set I as in E.2 and assume the analogue of
E.2(1):
0(h )=0 for all 2 I and 1(pI) =0: (2)
Let PI be standard parabolic subgroup of G with Lie algebra pI. Denote its
unipotent radical by Ru(PI). For each g 2 Ru(PI) and all x 2 pI the di erence
Ad(g ,1 )(x),x belongs to the nilradical of pI. Since vanishes on that nilradical,
we get
(Recall that (g )(x) = (Ad(g ,1 )(x)).
(g ) jpI = jpI : (3)
Lemma: There exist g 2 Ru(PI) and 0 1 2 g with 0 1(pI) =0such that g has
Jordan decomposition g = 0 + 0 1 .
Proof : This follows from the proof in Subsection 3.8 in [13]. If we apply the
construction there to our (as their l), then we get an element g 2 G such that
g has Jordan decomposition g = 0 + 0 1 where 0 1 = g , 0. Wejusthave
to check that the g used there actually is in Ru(PI). (If so, then we get 0 1(pI) =0
from 1(pI) = 0, since 0 1 coincides by (3) with 1 on pI.)
Well, in [13] one considers the set (denoted by ) of all positive roots with
(h ) 6= 0. Then g is constructed as a product of elements from the root subgroups
U with 2 . Now our assumption (2) says that R + n ZI. This implies that
g 2 Ru(PI) as claimed.
E.7. Keep the assumptions of E.6 and choose g as in Lemma E.6. We can
apply E.1{5 with s = 0, n = 0 1 and with replaced by g . Set in particular
R1 = f 2 R j 0(h )=0g and l = h L 2R1 g .Weget also 1 as in E.5(1).
Proposition: There exists an equivalence ofcategories
such that
F : U (g){modules ! Ug (l){modules
F(U (g) U (pI) M) ' Ug (l) U (l\pI) (M 1) (1)
for all nite dimensional U (gI){modules M extended trivially to a pI{module.
Proof :We construct F as a composition of two equivalences. The rst one is
U (g){modules ! Ug (g){modules; V 7! g V (2)