subregular nilpotent representations of lie algebras in prime ...

24
C.8. We nowwant to generalise Lemma C.7.a to positive roots that are not
necessarily simple:
Proposition: Let w 2 W and 2 R + with w ,1 >0. Then sbm(s w; )
sbm(w; ).
Proof : This will follow in the same way as in C.7 if we can nd a homomorphism
' w;s w : Z (s w ) ! Z (w ) with ' s w = ' w ' w;s w: (1)
Suppose rst that G is semi-simple and simply connected; drop the assumption
that (B2) should hold. In order to construct the map in (1) we make a detour
to characteristic 0. Let gC be a complex semisimple Lie algebra of the same type
and a Chevalley basis
of gC. Denote by gZ the span over Z of our Chevalley basis. This is a Lie algebra
induced by thatof
as g. Fix a triangular decomposition gC = n ,
C hC n +
C
over Z with a triangular decomposition gZ = n ,
Z
hZ n +
Z
gC. Wecan and shall assume that g = gZ Z K, similarly for n and h. Wehave
then also U(g) =U(gZ) Z K and similarly for n and h. Wedenote (by abuse
of notation) the vectors in our Chevalley basis of gC by x and h ( ; 2 R,
simple). We assume that we have chosen the x
we usually denote by x ).
1 as our root vectors in g (which
The group X can be identi ed with the lattice of integral weights of hC. We
have for each 2 X aVerma module M( )C for gC with highest weight ;we
denote its standard generator by z .Wede ne for each w 2 W a homomorphism
fw : M(w )C ! M( )C in the same way aswede ned in C.6(2) the 'w . There
is a unique element
u w 2 U(n ,
C ) with f w(zw )=u wz : (2)
The construction shows that uw is a product of powers of the x, with simple,
hence contained in U(n ,
Z ), and that
' w(vw )=(u w 1)v : (3)
Let 1, 2;:::; n denote the simple roots. If we writeu(w; ) as a linear combination
of the usual PBW basis of U(n ,
C ), then
u w =
nY
i=1
x ri
, +lower order terms if , w =
i
nX
i=1
ri i (4)
where \lower order terms" refers to the canonical ltration of an enveloping algebra
as in [4], 2.3.1.
Now consider w and as in the proposition. The theory of Verma modules
shows that there exists a unique homomorphism
f w;s w : M(s w )C ! M(w )C with f w f w;s w = f s w; (5)