2026/07/20 by Alexandre Pyvovarov
#math.RT #math.GN #math.NT
Let F and E be finite extensions of \mathbb Qp, let \mathbb G be a reductive group over F, and put G=\mathbb G(F). Let V be a locally algebraic representation of the form V=πsm⊗Eσalg, where πsm is smooth admissible and σalg is finite-dimensional algebraic. We study the extended Goldman--Iwahori distance on the set of non-Archimedean norms on V. After fixing a reference norm α0, its finite-distance component \mathscr Nα0(V) is the bounded projective limit of the extended Bruhat--Tits buildings attached to VK=πsmK⊗Eσalg. It is complete for the resulting uniform sup metric; this metric is of ℓ^∞ type and is generally not CAT(0). We prove directly that a G-orbit in \mathscr Nα0(V) is bounded if and only if this component contains a G-invariant norm. The invariant norm is the pointwise supremum of the orbit. We formulate an integral group-algebra and type-Hecke condition necessary for an invariant norm. For G=GLn(F) we specialise to V=BS(r)=πgen(r)⊗E πalg(r).