Indecomposable representations with invariant inner product

Consequences of the existence of an invariant (necessarily indefinite) non-degenerate inner product for an indecomposable representation π of a groupG on a space (mathfrak{H}) are studied. If π has an irreducible subrepresentation π₁ on a subspace (mathfrak{H}_1 ) , it is shown that there exists an invariant subspace (mathfrak{H}_2 ) of (mathfrak{H}) containing (mathfrak{H}_1 ) and satisfying the following conditions: (1) the representation π ₁ ^# =π mod (mathfrak{H}_2 ) on (mathfrak{H}) mod (mathfrak{H}_2 ) is conjugate to the representation (π₁, (mathfrak{H}_1 ) ), (2) (mathfrak{H}_1 ) is a null space for the inner product, and (3) the induced inner product on (mathfrak{H}_2 ) mod (mathfrak{H}_1 ) is non-degenerate and invariant for the representation $$pi _2 = (pi _2 |_{mathfrak{H}_2 } )bmod mathfrak{H}_1 ,$$ a special example being the Gupta-Bleuler triplet for the one-particle space of the free classical electromagnetic field with (mathfrak{H}_1 ) =space of longitudinal photons and (mathfrak{H}_2 ) =the space defined by the subsidiary condition.