subregular nilpotent representations of lie algebras in prime ...
subregular nilpotent representations of lie algebras in prime ...
subregular nilpotent representations of lie algebras in prime ...
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
46<br />
for all 2 R1 and m 2 M.<br />
Pro<strong>of</strong> : Let me abbreviate x = Q 2R +<br />
2<br />
x p,1<br />
, . We have toshowthat x x , xx<br />
annihilates M V .<br />
S<strong>in</strong>ce u is the nilradical <strong>of</strong> l u, wehave [l; u] u. Sox 2 l imp<strong>lie</strong>s that<br />
x x , xx 2 U(u):<br />
We can express this commutator <strong>in</strong> terms <strong>of</strong> a PBW basis <strong>of</strong> U(u). So x x , xx<br />
is a l<strong>in</strong>ear comb<strong>in</strong>ation <strong>of</strong> monomials <strong>of</strong> the form<br />
Y<br />
2R +<br />
2<br />
a( )<br />
x, Y<br />
2R +<br />
3<br />
b( )<br />
x<br />
with non-negative <strong>in</strong>tegers a( );b( ).<br />
Now x x , xx and all monomials <strong>in</strong> (3) are eigenvectors for the adjo<strong>in</strong>t<br />
action <strong>of</strong> T . Wecan therefore assume that only monomials occur that have the<br />
same weight asx x , xx , i.e., with<br />
X<br />
b( ) , X<br />
a( ) = , (p , 1) X<br />
: (4)<br />
2R +<br />
3<br />
2R +<br />
2<br />
If b( ) > 0 for some 2 R +<br />
3 , then the term <strong>in</strong> (3) annihilates M s<strong>in</strong>ce x belongs<br />
to the nilradical <strong>of</strong> pI, hence annihilates M. In order to prove our claim it therefore<br />
su ces to look at the terms <strong>in</strong> (3) with b( ) = 0 for all 2 R +<br />
3 . Then (4) reduces<br />
to X<br />
(p , 1 , a( )) = : (5)<br />
2R +<br />
2<br />
, then xa( )<br />
If a( ) p for some 2 R +<br />
2 , acts as 0 on V s<strong>in</strong>ce (u) = 0; hence so<br />
does the monomial <strong>in</strong> (3). So we mayassume that a( ) p , 1 for all 2 R +<br />
2 .<br />
Note that then a( )