The paper deals with constructing a complexification of a real analytic manifold M classically known as a “Grauert tube ” with additional structures. A Grauert tube is a Stein manifold X together with a real analytic totally real embedding i: M ↩ → X and with an antiholomorphic involution σ: X → X having M (identified with i(M)) as its fixed point set [H. Grauert, Ann. of Math. (2) 68 (1958), 460–472; MR0098847 (20 #5299)]. In the reviewer’s thesis, as an additional structure a proper action of a real Lie group G on M was considered and it was proved that a Stein G-tube can be constructed. That means that in addition X can be chosen to admit a proper G action (by holomorphic transformations) such that the embedding i and the involution σ are G-equivariant. In addition, M is a strong G-equivariant deformation retract of X. In fact any G-stable neighborhood of M in X can be shrunk to (contains) a G-stable and G-retractable Stein neighborhood. The tube is unique as a germ around M. For these results see also [P. Heinzner, A. T. Huckleberry and F. Kutzschebauch, in Complex analysis and geometry (Trento, 1993), 229–273, Dekker, New York, 1996; MR1365977 (96j:57047)]. In the present paper, as an additional structure a closed G-invariant 2-form τ on M is extended to a G-invariant Kähler form ω on a Stein G-tube X; i.e., i ∗ (ω) = τ. It is also proved that if τ admits a moment map, than there is a moment map for ω on X extending it. The Kähler form ω and the moment map are unique up to diffeomorphisms around M fixing M pointwise. With a compact group action this additional structure on the “Grauert tube ” was already considered in [P. Heinzner, A. T. Huckleberry and F. Loose, J. Reine Angew. Math. 455 (1994), 123–140
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