subregular nilpotent representations of lie algebras in prime ...

64
Remarks: 1) This result con rms for p{regular (i.e., for 2 C 0 0 with jW j = W )
the revised conjecture by Lusztig, as in [17], 2.6. (The formulation there looks
somewhat di erent, but can be checked to yield the same numbers.)
2) Suppose that R is of type Bn, Cn, orF4. Thenwecan apply G.3(5) for L = L0 and for L = L with short. Therefore (1) holds, (2) holds if is short, and
(3) holds if both and are short. If is a long root in J( ) and if there (as
expected) exist two isomorphism classes of simple modules with invariant f g
in C ,thenweget [Q0 : L ;i] =jW j m
2
(5)
and
[Q : L ;i] =jW
m m
j
2
(6)
for all short 2 J( ) [and for i =1; 2] in both cases. By the symmetry of the
Cartan matrix we get also [Q ;i : L0] and [Q ;i : L ].
3) For R of type Bn we can choose to have standard Levi form. Then QL has a
Z{ ltration for each simple L in C, cf. [11], 10.11. This implies that G.3(5) holds
for all L even though QL =2 P 0 in general (see the nal remark in G.4). We getin
this case that
[Q ;i : L ;j ]=jW j
m m
4
for all long ; 2 J( ). One may speculate whether (8) also holds in types Cn
and F4.
We keep the assumptions from Section D.
H
H.1. Recall from D.6(1) that we write _ 0 = P 2 m _ where is the set of
simple roots.
The fundamental weight $ corresponding to a simple root is minuscule
(in the sense of [1], Ch. VI, x1, exerc. 24) if and only if m =1. For all with
this property sety = y 0 w0 2 W where w0 is the longest element inW and where
y 0 is the longest element in the subgroup of W generated by all s with 6= .
Now Prop. 6 in [1], Ch. VI, x2, (applied to R _ instead of R) shows that
y C0 + p$ = C0. More precisely, the map x 7! y x + p$ maps the `real'
alcove ofallx 2 X Z R with 0 hx + ; _ i p for all 2 R + to itself. It
therefore permutes the \walls" of this alcove, i.e., the hyperplanes with equations
hx + ; _ i = 0 with 2 and the hyperplane with equation hx + ; _ 0 i = p. So
y permutes [f, 0g. In fact, one checks easily that y (, 0) = and that
hx + ; _
0 i = p () hy (x + )+p$ ; _ i =0: (1)
The simple root ,w0 satis es y (,w0 )=,y 0 ( ) < 0 since y 0 ( ) > 0. It
follows that y (,w0 )=, 0, hence that
hx + ; (,w0 ) _ i =0 () hy (x + )+p$ ; _
0 i = p: (2)
(8)