> < ^ Date: Tue, 10 Apr 2001 10:45:04 +0200 (MET DST)
> < ^ From: Jan Draisma <Jan.Draisma@unibas.ch >
< ^ Subject: Re: Chevalley module for SO(4,C)

Dear Gap-forum, and Philippe,

I was mistaken: it is handy to view so(4,C) as sl(2,C)+sl(2,C). Then,
the 8-dim. module
(V_1 x V_1)+(V_0 x V_1)+(V_1 x V_0)
where x denotes the tensor product, and V_i the sl(2,C)-module with
highest weight i, should work. (Hence, V_1 is the standard module, and
V_0 the trivial one; in fact it turns out to be easier to take V_1 x
V_1^* (2 by 2 matrices) for the former direct sum.) Indeed, the maximal
subalgebras are the diagonal one, which leaves a line in V_1 x V_1
invariant, and sl(2,C)+B, and B+sl(2,C), which lines in V_0 x V_1 and
V_1 x V_0 invariant, respectively.

Note that here is a subtlety which seems to have be overlooked in the
paper you quote: there is an automorphism that interchanges the latter
two subalgebras, but this automorphism takes one module to one which
is not isomorphic to it; therefore we need two modules. Of course, as
there is only a finite number of irreducible modules of a given
dimension, their proof remains correct.

In GAP, you can construct the above modules using the function
HighestWeightModule.

Hope this helps,

Jan

Dear gap-forum, and Jan,

> I'm far to be an expert in representation theory and in GAP, but I'm
>looking for a Chevalley module V for G=SO(4,C), taht is to say a faithful
>finite dimensional SO(4,C)-module such that:
> 1. V contains no one-dimensional G-modules
> 2. any proper connected closed subgroup H $\in$ G leaves a one-dimensional
> subspace W $\in$ V invariant.

I don't understand your question. Looking at the Lie algebra level,
you need a faithful module V for g=sl_2+sl_2 such that any proper subalgebra
leaves a 1-dim subspace invariant, contrary to g itself. Any g-module is
a direct sum of submodules V_i tensor V_j, where i,j>0 denote the highest
weights; i.e. the first sl_2 acts trivially on V_j and the second acts
trivially on V_i. Now h=the first sl_2-factor is a proper subalgebra
of g. Suppose that there exists a 1-d W<=V such that hW<=W. W has a
non-zero projection to some V_i tensor V_j, and we find a 1-d. subspace
W' of the latter with the same property. However, the sl_2 in h will
not leave W' invariant. So, no such module exists.

I look actually for a Chevalley module in order to use a theorem of C.Mitschi
and M.F.Singer to find a realization over C(x) of SO(4,C) by a system Y'=AY.
In their article, they prove that every connected semisimple linear group
defined over C has a Chevalley module. The references of their article are:
Connected Linear Groups as Differential Galois Groups; J.of Algebra,184(1996),
p.333-361 (p.344 for the lemma which interests us).