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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We keep the assumptions from Sections B and C. We assume <strong>in</strong> addition:<br />
(D1) The <strong>prime</strong> p is good for R.<br />
and<br />
(D2) G is almost simple.<br />
D<br />
The rst assumption is crucial so that we can apply the Kac-Weisfeiler conjecture<br />
proved by Premet. The second assumption is ma<strong>in</strong>ly meant to simplify the<br />
statement <strong>of</strong> the results.<br />
D.1. We call a l<strong>in</strong>ear form 2 g <strong>subregular</strong> if its orbit under the coadjo<strong>in</strong>t<br />
action has dimension equal to 2(N , 1) where N = jR + j. If so, then the Kac-<br />
Weisfeiler conjecture as proved by Premet says: If M is a U (g){module, then<br />
dim(M) is divisible by p N ,1 .<br />
Recall that we setp = b + + g, for each simple root ;we denote (as <strong>in</strong><br />
B.14) by P the correspond<strong>in</strong>g parabolic subgroup <strong>of</strong> G. The follow<strong>in</strong>g result is a<br />
translation <strong>of</strong> well known results on orbits <strong>in</strong> g:<br />
Lemma: There exists a unique <strong>subregular</strong> <strong>nilpotent</strong> orbit O <strong>in</strong> g . If is a simple<br />
root then O <strong>in</strong>tersects the set <strong>of</strong> all 2 g with (p )=0<strong>in</strong> an open and dense<br />
subset. This <strong>in</strong>tersection is one orbit under P .<br />
Pro<strong>of</strong> :We can (under our assumptions on p) identify g and g as G{modules. The<br />
classi cation <strong>of</strong> the <strong>nilpotent</strong> orbits <strong>in</strong> g is the same as for the Lie algebra over C<br />
<strong>of</strong> the same type (s<strong>in</strong>ce p is good). In particular, there is exactly one <strong>subregular</strong><br />
<strong>nilpotent</strong> orbit <strong>in</strong> g; this yields the rst claim.<br />
The elements 2 g with (b + ) = 0 and (x, ) = 0 correspond (under<br />
g ' g ) to the elements <strong>in</strong> the nilradical n = L >0; 6= g <strong>of</strong> the parabolic<br />
subalgebra p . The theory <strong>of</strong> the Richardson orbits (cf. [3], 5.2.3) says that there<br />
exists exactly one <strong>nilpotent</strong> orbit for G that <strong>in</strong>tersects n <strong>in</strong> an open and dense<br />
subset. That <strong>in</strong>tersection is one orbit under P ; furthermore the dimension <strong>of</strong> the<br />
orbit (under G) is equal to the codimension <strong>in</strong> g <strong>of</strong> a Levi factor <strong>of</strong> p . S<strong>in</strong>ce that<br />
codimension is equal to 2(N , 1), we get the rema<strong>in</strong><strong>in</strong>g claims.<br />
Remark: For each simple root 6= the set <strong>of</strong> all with (p ) = 0 and (x, ) 6= 0<br />
is an open and dense subset <strong>of</strong> the set <strong>of</strong> all with (p ) = 0. It follows: The set<br />
<strong>of</strong> all <strong>subregular</strong> 2 g with (p )=0and (x, ) 6= 0for all simple roots 6=<br />
is an open and dense subset <strong>of</strong> the set <strong>of</strong> all with (p )=0.<br />
D.2. Each subset J <strong>of</strong> the set <strong>of</strong> all simple roots de nes a facet F (J) conta<strong>in</strong>ed <strong>in</strong><br />
C 0 0 as follows: A weight 2 C 0 0 belongs to F (J) if and only if h + ; _i > 0 for all<br />
2 J and h + ; _i = 0 for all simple roots =2 J. Write $ for the fundamental<br />
weight correspond<strong>in</strong>g to a simple root . Then 2 X belongs to F (J) if there<br />
are <strong>in</strong>tegers m > 0 such that + = P 2J m $ and if h + ; _i