Dual Shellability of Admissible Set and Cohen-Macaulayness of Local Models

Abstract: We prove G\"ortz's combinatorial conjecture \cite{Go01} on dual shellability of admissible sets in Iwahori-Weyl groups, proving that the augmented admissible set $\widehat{\mathrm{Adm}}(\mu)$ is dual shellable for any dominant coweight $\mu$. This provides a uniform, elementary approach to establishing Cohen-Macaulayness of the special fibers of the local models with Iwahori level structure for all reductive groups-including residue characteristic $2$ and non-reduced root systems-circumventing geometric methods. Local models, which encode singularities of Shimura varieties and moduli of shtukas, have seen extensive study since their introduction by Rapoport-Zink, with Cohen-Macaulayness remaining a central open problem. While previous work relied on case-specific geometric analyses (e.g., Frobenius splittings \cite{HR23} or compactifications \cite{He13}), our combinatorial proof yields an explicit labeling that constructs the special fiber by sequentially adding irreducible components while preserving Cohen-Macaulayness at each step, a new result even for split groups.