# Geometric applications of irreducible representations of Lie groups

### Antonio J. Di Scala

Politecnico di Torino, Italy### Thomas Leistner

The University of Adelaide, Australia### Thomas Neukirchner

Humboldt-Universität zu Berlin, Germany

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## Abstract

In this note we give proofs of the following three algebraic facts which have applications in the theory of holonomy groups and homogeneous spaces: Any irreducibly acting connected subgroup G ⊂ GL(n,ℝ) 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. 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. Finally, we give some applications and present some open problems.