In this talk I will describe an ongoing project with Toby Gee, whose goal, for any finite extension K of Q_p and any choice of dimension d, is to construct an equidimensional Artin stack over F_p whose F_p-bar points parameterize homomorphisms of G_K into GL_n(F_p-bar) . Our construction will give a qualitative explanation of certain phenomena observed in the theory of reduction of potentially semistable Galois representations. It will also have relations to the Breuil--Mezard conjecture, and other aspects of the p-adic Langlands program, as I hope to explain.