subregular nilpotent representations of lie algebras in prime ...

A.5. Let R 0 be a root subsystem of R. Then g 0 = h L 2R 0 g the Lie algebra
of a reductive closed subgroup of G. We can then apply A.2 also to g 0 working
with b + \ g 0 instead of b + .We shall write then
Z (f; g 0 )=U (g 0 ) U (b + \g 0 ) Kf (1)
for the analogue of Z (f). In case 2 R 0 we can also extend A.3/4 and get
Z (f; ; g 0 )=U (g 0 ) U (p \g 0 ) L ; (f) (2)
as an analogue of Z (f; ). We do not introduce extra notations for the analogues
of Z ; (f) and L ; (f) because the analogous U (p \ g 0 ){modules are just the
restrictions of the corresponding U (p ){modules.
A.6. In some cases we shall have to consider for a given f 2 h at the same time
all such that Z (f; ) is de ned. In that situation it will be convenient towork
with a slightly modi ed notation.
Set
X = f f 2 h j f(h ) 2 Fp g: (1)
This is a union of p a ne subspaces of h of codimension 1 unless h =0(inwhich
case X = h ). Set
=f 2 g j (p )=0g: (2)
Extend each (f) tog such that (f)(n + +n , ) = 0. Then Z (f)+ (f; ) is de ned
for all f 2 X and 2 . We write
Z(f; ; )=Z (f)+ (f; ): (3)
Note that then also all Z(f + ; ; )with 2 X are de ned because f + 2 X.
A.7. Proposition: Let 1; 2 2 X. For each integer m the set
f(f; ) 2 X j dim Homg(Z(f + 1; ; );Z(f + 2; ; )) m g (1)
is closed inX .
Proof : The point istoshow that the Hom space in (1) can be described as a space
of solutions to a system of linear equations where the matrix of the system has
size independent of(f; ) and entries that are polynomial functions of f and .If
we have, say, l unknowns, then the Hom space has dimension m if and only if
the rank of the matrix is l , m if and only if all (l , m +1) (l , m + 1) minors
of the matrix are 0. This condition clearly de nes a closed subset of X .
In order to show that we are in a situation as described above,letmeintroduce
some notation. Let R be the set of all R + {tuples of non-negative integers. To
each r =(r( )) 2 R we associate elements
x ,
r
= Y
>0
r(
x, )
and x + Y r(
r = x
>0
)
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