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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).

Hoping this may help you,

Best regards,

Philippe Gaillard

Miles-Receive-Header: reply

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