Indecomposable representations with invariant inner product. A theory of the Gupta-Bleuler triplet (Q1068965)

The representations \(\pi\) of a Lie group G on a space \({\mathfrak H}\) (with an invariant indefinite inner product of the following form \[ (1.1)\quad \pi_ n\to \pi_{n-1}\to...\to \pi_ 1,\quad {\mathfrak H}={\mathfrak H}_ n\supset {\mathfrak H}_{n-1}\supset...\supset {\mathfrak H}_ 1 \] are studied. Theorem 1. Let \(\pi\) be a representation of a group G on a complex vector space \({\mathfrak H}\) with a G-invariant non-degenerate Hermitian form. Let \({\mathfrak H}_ 1\) be a G-invariant closed subspace of \({\mathfrak H}\) such that the restriction \(\pi_ 1\) of \(\pi\) to \({\mathfrak H}_ 1\) is irreducible and there are no G-invariant, closed topological complements of \({\mathfrak H}_ 1\) in \({\mathfrak H}\). Then (1) \({\mathfrak H}_ 1\) is a null space, (2) \(\pi\) is of the form (1.1) with \(n=2\) or 3 such that \(\pi_ n\) on \({\mathfrak H}_ n/{\mathfrak H}_{n-1}\) is conjugate to \(\pi_ 1\) on \({\mathfrak H}_ 1\), and (3) \({\mathfrak H}_ 2/{\mathfrak H}_ 1\) has a \(\pi_ 2(G)\)-invariant non-degenerate Hermitian form. A necessary and sufficient condition on representations \(\pi_ j\) to have G-invariant non-degenerate inner product, in terms of cohomologies, is given.