subregular nilpotent representations of lie algebras in prime ...

2
positive roots . (We denote by _ the coroot corresponding to .) Then there
are up to isomorphism jej simple modules in C .We can denote them by L with
2 e suchthat
dim L = h + ; _ ip N ,1 ; if 2 ,
(p ,h + ; _ 0 i)p N ,1 ; if = , 0.
De ne integers m for all 2 e by m, 0 = 1 and _
0 = P 2 m _ . Let Q
denote the projective cover of L in C . Then
[Q : L ]=jW j m m for all ; 2 e . (2)
Here W is the Weyl group of R and we write [M : L] for the multiplicity ofa
simple module L as a composition factor of a module M.
If vanishes on the \standard" Borel subalgebra (corresponding to the positive
roots), then one can de ne \baby Verma modules" Z ( ). One gets then
[Z ( ):L ]=m for all 2 e . (3)
The extension group (in C )oftwo non-isomorphic simple modules is given by
Ext 1 (L ;L ) '
K; if ( ; ) < 0,
0; if ( ; )=0,
unless R is of type A1 where one has to replace K by K 2 . I do not know how big
the Ext group is in case = ; it will be non-zero in most cases.
The result on the number of simple modules in C as well as the formula in
(2) for the Cartan matrix con rm conjectures by Lusztig.
If we consider more generally 2 X such that 0 h + ; _ i