Le lemme de Schur pour les représentations orthogonales.
Résumé
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)' is isomorphic to \R, \C, or \H. 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^\C is unitarily equivalent to π if σ(G)' is isomorphic to \R, to π \oplus π̄ if σ(G)' is isomorphic to \C, and to π \oplus π if σ(G)' is isomorphic to \H (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.