subregular nilpotent representations of lie algebras in prime ...

44
E
We return to the more general set-up from Section A. We assume that G satis es
(B1) and (D1).
E.1. Let 2 g have Jordan decomposition = s + n. Assume that we have
s(n , + n + )=0 and n(b + )=0: (1)
Then the assumption that = s + n is a Jordan decomposition means that
s(h )=0 for all 2 R with n(x ) 6= 0. (2)
Set R1 equal to the set of all roots with s(h ) = 0. This is a root subsystem
of R; it satis es R1 = R \ QR1 because we assume the characteristic to be good.
The centraliser l of s is given
l = h
M
2R1
g : (3)
This is a Levi subalgebra of some parabolic subalgebra of g. This parabolic subalgebra
is not uniquely determined. We choose it as follows: We rstchoose a basis
of the root system R1 such that R1 \ R + is the set of positive roots with respect
to this basis. Then we extend this basis of R1 to a basis of R. (This is possible
since R1 = R \ QR1, see [1], Ch. VI, x1, prop. 24.) Set u equal to the direct sum
of all g with positive with respect to this new basis and not in R1. Then l u
is a parabolic subalgebra of g with nilradical u.
Now Kac & Weisfeiler (or rather Friedlander & Parshall, [6], Thm. 3.2) tell
us: The functor V 7! V u is an equivalence of categories from U (g){modules to
U (l){modules. We have
dim(V )=p dim u dim(V u ) (4)
for all these V . (It is here that we need (B1) and (D1). In [6] one assumes also
that G is semi-simple, but that is not necessary, cf. [11], 7.4.)
E.2. Keep the assumptions from E.1. We want toevaluate the functor V 7! V u
on certain induced modules.
Let I be a subset of the set of simple roots and let pI b + be the corresponding
standard parabolic subalgebra. Assume that satis es in addition
s(h )=0 for all 2 I and n(pI) =0: (1)
Let gI be the standard Levi factor of pI (i.e., the direct sum of h and all g with
2 R \ ZI).
Let M be a nite dimensional U (gI){module. Extend M to a pI{module such
that the nilradical of pI acts trivially. We get thus a U (pI){module because
vanishes on that nilradical. This leads then to the induced module U (g) U (pI)M.
We want to describe (U (g) U (pI) M) u .We shall identify each m 2 M with the
element 1 m in the induced module.
We shall need some additional notation: Set R +
2
(resp. R+ 3 ) equal to the set
of all positive rootswith g, u (resp. g u). So u is the direct sum of all
g with 2 (,R +
2 ) [ R+ 3 .