  Our main theorem describes the degree 0 cohomology of Igusa varieties in
terms of one-dimensional automorphic representations in the setup of mod p
Hodge-type Shimura varieties with hyperspecial level at p, mirroring the well
known analogue for complex Shimura varieties. As an application, we obtain a
completely new approach to two geometric questions. (See Sect. 1.5 for a
comparison with independent results by van Hoften and Xiao via a different
approach.) Firstly, we verify the discrete part of the Hecke orbit conjecture,
which amounts to irreducibility of central leaves, generalizing preceding works
by Chai, Oort, Yu, et al. Secondly, 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.
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.
