> < ^ Date: Thu, 05 Apr 2001 16:33:35 +0200 (MET DST)
> < ^ From: Jan Draisma <Jan.Draisma@unibas.ch >
> < ^ Subject: Re: Chevalley module for SO(4,C)

Dear forum, and Philippe,

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.

(Rem: the diagonal subalgebra of g does leave a 1-d subspace of
V_1 tensor V_1^* invariant)