If G is a (split) Kac–Moody group over a field K endowed with a real valuation $\omega$, we build an action of G on a geometric object $\mathcal I$. This object is called a building, as it is an union of apartments, with the classical properties of systems of apartments. However, these apartments are more exotic: that associated to a torus T may be seen as the gluing of all Satake compactifications of affine apartments of T with respect to spherical parabolic subgroups of G containing T. Another geometric realization of these apartments makes them look more like the apartments of $\Lambda$-buildings; then the translations of the Weyl group act only on infinitely small elements of the apartment, so we call these buildings microaffine.

We show that every countable direct system of finite-dimensional real or complex Lie groups has a direct limit in the category of Lie groups modelled on locally convex spaces. This enables us to push all basic constructions of finite-dimensional Lie theory to the case of direct limit groups. In particular, we obtain an analogue of Lie's third theorem: every countable-dimensional locally finite real or complex Lie algebra arises as the Lie algebra of some regular Lie group (a suitable direct limit group).

We combine the geometric realization of principal series representations on partially holomorphic cohomology spaces, with the Bott–Borel–Weil theorem for direct limits of compact Lie groups, obtaining limits of principal series representations for direct limits of real reductive Lie groups. We introduce the notion of weakly parabolic direct limits and relate it to the conditions that the limit representations are norm-preserving representations on a Banach space or unitary representations on a Hilbert space. We specialize the results to diagonal embedding direct limit groups. Finally we discuss the possibilities of extending the results to limits of tempered series other than the principal series.