Le lemme de Schur pour les représentations orthogonales

Let σ be an orthogonal representation of a group G on a real Hilbert space. We show that σ is irreducible if and only if its commutant σ(G)ʹ is isomorphic to , or . This result is an analogue of the classical Schur lemma for unitary representations. In both cases (orthogonal and unitary), a representation is irreducible if and only if its commutant is a field. If σ is irreducible, we show that there exists a unitary irreducible representation π of G such that the complexification σ is unitarily equivalent to π if σ(G)ʹ , to π π̄ if σ(G)ʹ , and to π π if σ(G)ʹ (here π̄ denotes the contragredient representation of π). These results are classical for a finite-dimensional σ, but seem to be new in the general case.