Types and collapse for cuspidal representations of groups acting on trees
arXiv:2607.27590
The paper refines a geometric construction of supercuspidal representations of rank‑one p‑adic groups by describing the inducing data through minimal K‑types and showing that the associated equivariant sheaves on trees collapse to injective sheaves, with ideas that extend to higher rank groups.
Abstract
In a previous paper, the second author proved that every supercuspidal representation of a rank-one p-adic group is induced from a compact-mod-center open subgroup. The method was geometric, localizing representations to obtain equivariant sheaves on trees. Here we provide two refinements. The first is a geometric description of the inducing data, via a geometrically minimal K-type. Second is a proof that the equivariant sheaves collapse onto injective sheaves. The two notions of geometrically minimal K-types and collapsibility generalize to higher rank groups, suggesting a pair of conjectures.
26 pages