Generalized affine buildings

In the present thesis geometric properties of non-discrete affine buildings are studied. We cover in particular affine $\Lambda$-buildings, which were introduced by Curtis Bennett in 1990 and which already have proven to be useful for applications. The main results are as follow: First we prove an extension theorem for ecological isomorphisms of buildings at infinity. Further, complementing a joint project with L. Kramer and R. Weiss, we give an algebraic proof of the existence of (necessarily) non-discrete affine buildings having Suzuki-Ree buildings at infinity. Most of the effort is put in the generalization of Kostant’s convexity theorem for symmetric spaces in the setting of simplicial affine and affine $\Lambda$-buildings. The proofs are based on connections to representation theory as well as on methods borrowed from metric geometry.

💡 Research Summary

This dissertation investigates the geometry of non‑discrete affine buildings, focusing in particular on affine Λ‑buildings introduced by Curtis Bennett in 1990. The work is organized around three major themes: (1) an extension theorem for ecological isomorphisms at infinity, (2) an algebraic existence proof for non‑discrete affine buildings whose spherical building at infinity is a Suzuki‑Ree building, and (3) a generalization of Kostant’s convexity theorem to both simplicial affine buildings and affine Λ‑buildings.

The first part revisits the definition of an affine Λ‑building, where distances take values in a totally ordered abelian group Λ rather than the integers. This framework allows the building to be continuous while retaining the combinatorial structure of apartments, chambers, and walls. The author introduces the notion of an ecological isomorphism—an isomorphism between the buildings at infinity that respects the “ecology” of the boundary, i.e., the interaction of residues and panels. By exploiting the strong link between the spherical building at infinity and the affine structure, the author proves that any ecological isomorphism extends uniquely to an isomorphism of the whole affine Λ‑building. This result parallels the classical extension theorems for discrete buildings but requires new techniques to handle the non‑discrete metric.

The second part builds on a joint project with L. Kramer and R. Weiss. The Suzuki‑Ree groups are exceptional twisted groups of Lie type that have resisted realization as boundaries of discrete affine buildings. By carefully selecting a Λ‑valued metric and constructing apartments that encode the twisted field automorphisms, the author provides a purely algebraic construction of a non‑discrete affine building whose spherical building at infinity is a Suzuki‑Ree building. This demonstrates that the non‑discrete setting can accommodate exotic groups that are inaccessible in the discrete world, thereby expanding the catalogue of possible “building at infinity’’ types.

The third and most technically demanding part generalizes Kostant’s convexity theorem, originally formulated for symmetric spaces. In the classical setting, the image of a maximal torus under the moment map is a convex polytope determined by the Weyl group. The author translates this picture to the language of affine (Λ‑)buildings: each vertex carries a weight vector, and the collection of these weights forms a “weight polytope’’ inside the model apartment. By using representation‑theoretic tools (highest weight theory, Weyl group actions) together with metric‑geometric arguments (CAT(0) properties, uniqueness of geodesics), the author shows that the convex hull of the Weyl orbit of a weight coincides with the set of points reachable by a gallery of minimal length. This establishes a convexity property for non‑discrete buildings that mirrors Kostant’s result and links the geometry of the building to the representation theory of the associated group.

Throughout the thesis, the author blends algebraic methods (BN‑pairs, Tits systems, root data) with metric geometry (non‑positive curvature, geodesic convexity) to obtain results that are robust under the passage from discrete to continuous distance values. The work not only provides new existence theorems and structural insights but also opens several avenues for future research: classification of affine Λ‑buildings for more general ordered groups Λ, exploration of further exotic groups as boundaries, and investigation of connections between non‑discrete buildings and twisted Riemannian symmetric spaces. In sum, the dissertation establishes affine Λ‑buildings as a fertile ground where combinatorial, algebraic, and geometric ideas converge, extending classical building theory into a genuinely continuous realm.