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