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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)

Perhaps I'm mistaken, or I misunderstood your question. Please let me

know.

Regards,

Jan

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