Irreducibly acting subgroups of \(Gl(n,\rr)\)

In this note we prove the following three algebraic facts which have applications in the theory of holonomy groups and homogeneous spaces: Any irreducibly acting connected subgroup \(G \subset Gl(n,\rr)\) is closed. Moreover, if \(G\) admits an invariant bilinear form of Lorentzian signature, \(G\) is maximal, i.e. it is conjugated to \(SO(1,n-1)_0\). Finally we calculate the vector space of \(G\)-invariant symmetric bilinear forms, show that it is at most 3-dimensional, and determine the maximal stabilizers for each dimension.