subregular nilpotent representations of lie algebras in prime ...

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Consider nally R of type F4. Wemay assume that (p 1) = 0. The set of
roots orthogonal to 1 is a root system of type C3. (Note that 1 is conjugate
to the largest root, i.e., to $1.) A basis of this root system is 4, 3, "2 + "3 =
1 +2 2 +2 3. The stabiliser W1 of 1 is the Weyl group of this subsystem.
The short roots orthogonal to 1 formarootsystemoftype D3 = A3. The
corresponding Weyl group W2 is a subgroup of index 2 in W1; wehave
W1 = W2 [ W2s"2+"3: (3)
We have 2f 1; 2g. The simple modules in C with invariant f g are
(by D.3.a, D.5.c and D.14) the Z (x ; 1) withx 2 W and x = 1. The x with
this property are a coset for W1, in fact equal to W1 in case = 1, equal to
W1s2s1 in case = 2. Proposition F.1 implies for all w 2 W2 (and x as above)
that Z (wx ; 1) ' Z (x ; 1). This yields for all x 2 W1 in case = 1
and in case = 2
Z (x ; 1) ' Z ( ; 1); if x 2 W2,
Z (s"2+"3 ; 1); if x=2 W2,
Z (xs2s1 ; 1) ' Z (s2s1 ; 1); if x 2 W2,
Z (s"2+"3s2s1 ; 1); if x=2 W2.
Remark: Lusztig's conjectures predict that we should have two classes also for
types Cn and F4.
F.7. Given 2 g each g 2 CG( ) (the stabiliser of under the coadjoint action)
permutes the isomorphism classes of simple U (g){modules via L 7! g L. Since
there are only nitely many classes, one sees easily that CG( ) 0 , the connected
component of the identity inCG( ), acts trivially. So does the centre Z(G).
This means that we are really looking at an action of the \component group"
A( )=CG( )=(Z(G)CG( ) 0 ). (It is really the component group for the adjoint
group.)
Note that this action is \the same" for all ina xedG{orbit: Given h 2 G we
have CG(h )=hCG( )h ,1 ; therefore conjugation with h induces an isomorphism
A( ) ,! A(h ). Furthermore, the map L 7! h L induces a bijection from the
set of isomorphism classes of simple U (g){modules to the corresponding set for
Uh (g). This bijection is compatible with the actions of A( ) and A(h )identi ed
as above.
We have used before that U(g) G acts on g L as it does on L. Therefore the
action of A( ) permutes (for nilpotent) the simple modules in each C . (For
general one would have tointroduce a new notation.) By D.5(3) the action of
A( ) preserves the invariant (when de ned).
Suppose now that is subregular and nilpotent. If all roots have the same
length, then F.5 and the remarks above show that A( ) acts trivially. Actually,
we have in these cases A( ) = 1 at least for large p, see [20], 7.5.
In the next subsections we are going to prove:
(4)
(5)