subregular nilpotent representations of lie algebras in prime ...

46
for all 2 R1 and m 2 M.
Proof : Let me abbreviate x = Q 2R +
2
x p,1
, . We have toshowthat x x , xx
annihilates M V .
Since u is the nilradical of l u, wehave [l; u] u. Sox 2 l implies that
x x , xx 2 U(u):
We can express this commutator in terms of a PBW basis of U(u). So x x , xx
is a linear combination of monomials of the form
Y
2R +
2
a( )
x, Y
2R +
3
b( )
x
with non-negative integers a( );b( ).
Now x x , xx and all monomials in (3) are eigenvectors for the adjoint
action of T . Wecan therefore assume that only monomials occur that have the
same weight asx x , xx , i.e., with
X
b( ) , X
a( ) = , (p , 1) X
: (4)
2R +
3
2R +
2
If b( ) > 0 for some 2 R +
3 , then the term in (3) annihilates M since x belongs
to the nilradical of pI, hence annihilates M. In order to prove our claim it therefore
su ces to look at the terms in (3) with b( ) = 0 for all 2 R +
3 . Then (4) reduces
to X
(p , 1 , a( )) = : (5)
2R +
2
, then xa( )
If a( ) p for some 2 R +
2 , acts as 0 on V since (u) = 0; hence so
does the monomial in (3). So we mayassume that a( ) p , 1 for all 2 R +
2 .
Note that then a( )