subregular nilpotent representations of lie algebras in prime ...

Subregular nilpotent representations of Lie algebras
in prime characteristic
Jens Carsten Jantzen
Introduction
Let G be a reductive algebraic group over an algebraically closed eld K of
prime characteristic p>0. This paper deals with certain representations of the Lie
algebra g of G. For the purpose of this introduction assume that G is semi-simple
and simply connected, that the root system R of G is irreducible, and that p is
larger than the Coxeter number h of R. (If R is of type E8 or F4, assume that
p>h+ 1; this restriction should be unnecessary, but my proofs require it.)
Each simple g{module has a p{character; that is a linear form on g such
that all x p , x [p] , (x) p 1withx 2 g annihilate the module. Here x p is the p{th
power of x in the universal enveloping algebra U(g) and x 7! x [p] is the p{th power
map on the Lie p{algebra g. A general result due to Kac and Weisfeiler reduces
the problem of describing all simple modules basically to the case where the p{
character is nilpotent; this means that vanishes on some Borel subalgebra
of g. Due to work by Curtis, Friedlander, Parshall, and Panov one then has a
classi cation of the simple modules in case has a certain special form (\standard
Levi form"). For not of this form so far no classi cation of the corresponding
simple modules has been known.
We look in this paper at the case where is subregular nilpotent. Here \subregular"
means that the orbit of under the coadjoint actionofG has dimension
2(N , 1) where 2N = jRj. A subregular nilpotent has standard Levi form if and
only if R has type An or Bn. Inthosetwo cases I have given a detailed description
of the corresponding simple modules in [10]. For the other types the results in this
paper are new. In order to describe them I rst need some notation.
Let T be a maximal torus in G, letX be the character group of T and h the
Lie algebra of T . Choose a basis for the root system R X. Set equal to half
the sum of the positive roots, and let 0 denote the unique short root that is a
dominant weight. (If all roots have the same length, then all roots are short, and
none is long.) Set e = [f, 0g.
Consider the algebra U(g) G of G{invariants in U(g). Each 2 X de nes a
\central character" cen : U(g) G ! K such that U(g) G acts via cen on a highest
weight module with highest weight . Fix a subregular nilpotent . Denote for
each 2 X the category of all nite dimensional g{modules M that are annihilated
by all x p , x [p] , (x) p 1 with x 2 g and such that U(g) G acts via cen on all
composition factors of M.
Assume rst that all roots in R have the same length. (This is the case where
our results are most complete.) Let 2 X such that 0 < h + ; _ i