Ideal triangles in Euclidean buildings and branching to Levi subgroups

Let G denote a connected reductive group, defined and split over Z, and let M ⊂ G denote a Levi subgroup. In this paper we study varieties of geodesic triangles with fixed vector-valued side-lengths α, β, γ in the Bruhat-Tits buildings associated to G, along with varieties of ideal triangles associated to the pair M ⊂ G. The ideal triangles have a fixed side containing a fixed base vertex and a fixed infinite vertex ξ such that other infinite side containing ξ has fixed “ideal length” λ and the remaining finite side has fixed length μ. We establish an isomorphism between varieties in the second family and certain varieties in the first family (the pair (μ, λ) and the triple (α, β, γ) satisfy a certain relation). We apply these results to the study of the Hecke ring of G and the restriction homomorphism R(Ĝ) → R(M̂) between representation rings. We deduce some new saturation theorems for constant term coefficients and for the structure constants of the restriction homomorphism.