  We pioneer the development of a rigorous infinite-dimensional framework for
the Kempf-Ness theorem, addressing the significant challenge posed by the
absence of a complexification for the symmetry group in infinite dimensions,
e.g, the diffeomorphism group. We propose a novel approach, based on Cartan
bundles, to generalize Kempf-Ness theory to infinite dimensions, invoking the
fundamental role played by the Maurer-Cartan form. This approach allows us to
define and study objects essential for the Kempf-Ness theorem, such as the
complex model for orbits and the Kempf-Ness function, as well as establishing
its convexity properties and defining a generalized Futaki character. We show
how our framework can be applied to the study of various problems in K\"ahler
geometry, deformation quantization, and gauge theory.
