Cell decomposition of some unitary group Rapoport–Zink spaces

Abstract

In this paper we study the $$p$$ -adic analytic geometry of the basic unitary group Rapoport–Zink spaces $$\mathcal {M}_K$$ with signature $$(1,n-1)$$ . Using the theory of Harder–Narasimhan filtration of finite flat groups developed in Fargues (Journal für die reine und angewandte Mathematik 645:1–39, 2010), Fargues (Théorie de la réduction pour les groupes p-divisibles, prépublications. http://www.math.jussieu.fr/~fargues/Prepublications.html , 2010), and the Bruhat–Tits stratification of the reduced special fiber $$\mathcal {M}_{red}$$ defined in Vollaard and Wedhorn (Invent. Math. 184:591–627, 2011), we find some relatively compact fundamental domain $$\mathcal {D}_K$$ in $$\mathcal {M}_K$$ for the action of $$G(\mathbb {Q}_p)\times J_b(\mathbb {Q}_p)$$ , the product of the associated $$p$$ -adic reductive groups, and prove that $$\mathcal {M}_K$$ admits a locally finite cell decomposition. By considering the action of regular elliptic elements on these cells, we establish a Lefschetz trace formula for these spaces by applying Mieda’s main theorem in Mieda (Lefschetz trace formula for open adic spaces (Preprint). arXiv:1011.1720 , 2013).