We correct the proof of the main result of the paper, Theorem 5.7. Our corrected proof relies on weaker versions of a number of intermediate results from the paper. The original, more general, versions of these statements are not known to be true.

In this article a general framework for studying analytic representations of a real Lie group G is introduced. Fundamental topological properties of the representations are analyzed. A notion of temperedness for analytic representations is introduced, which indicates the existence of an action of a certain natural algebra 𝒜(G) of analytic functions of rapid decay. For reductive groups every Harish-Chandra module V is shown to admit a unique tempered analytic globalization, which is generated by V and 𝒜(G) and which embeds as the space of analytic vectors in all Banach globalizations of V.

We define integral formulas which produce certain matrix coefficients of cohomologically induced representations of real reductive groups. They are analogous to Harish-Chandra's Eisenstein integrals for matrix coefficients of ordinary induced representations, and generalize Flensted-Jensen's fundamental functions for discrete series.

We construct bases for the stable branching algebras for the symmetric pairs $(\mathrm{GL}_{2n},\mathrm{Sp}_{2n}),\ (\mathrm{Sp}_{2(n+m)}, \mathrm{Sp}_{2n}\times\mathrm{Sp}_{2m})$ and $(\mathrm{O}_{2n},\mathrm{GL}_{n})$. Each basis element is expressed as a sum of products of pfaffians.