Pseudo--K\"ahler induction on Lie groups with entire Grauert tubes

Given a closed subgroup $H\subseteq G$ and a Hamiltonian $H$-space $Y$, one can construct an induced Hamiltonian $G$-space. In this paper we investigate this construction in the setting where $Y$ is endowed with a compatible complex structure. Our aim is to develop symplectic induction in this framework and to establish the corresponding K"ahler analogues of the classical results. The main idea is to replace the cotangent bundle $T^{\ast }G$, appearing in the standard construction of $\mathrm{Ind}$, with the Grauert tube of $G$. We focus on Lie groups admitting a globally defined Grauert tube and study the resulting pseudo–K"ahler induction procedure.