H⁰ of Igusa varieties via automorphic forms

Our main theorem describes the degree 0 cohomology of non-basic Igusa varieties in terms of one-dimensional automorphic representations in the setup of mod p Hodge-type Shimura varieties with hyperspecial level at p. As an application we obtain a completely new approach to two geometric questions. Firstly, we deduce irreducibility of Igusa towers and its generalization to non-basic Igusa varieties in the same generality, extending previous results by Igusa, Ribet, Faltings–Chai, Hida, and others. Secondly, we verify the discrete part of the Hecke orbit conjecture, which amounts to the assertion that the irreducible components of a non-basic central leaf belong to a single prime-to-p Hecke orbit, generalizing preceding works by Chai, Oort, Yu, et al. We also show purely local criteria for irreducibility of central leaves. Our proof is based on a Langlands–Kottwitz type formula for Igusa varieties due to Mack-Crane, an asymptotic study of the trace formula, and an estimate for unitary representations and their Jacquet modules in representation theory of p-adic groups due to Howe–Moore and Casselman.