Equidimensional adic eigenvarieties for groups with discrete series

Abstract

We extend Urban’s construction of eigenvarieties for reductive groups $G$ such that $G\left(ℝ\right)$ has discrete series to include characteristic $p$ points at the boundary of weight space. In order to perform this construction, we define a notion of “locally analytic” functions and distributions on a locally ${\mathbb{Q}}_p$-analytic manifold taking values in a complete Tate ${\mathbb{Z}}_p$-algebra in which $p$ is not necessarily invertible. Our definition agrees with the definition of locally analytic distributions on $p$-adic Lie groups given by Johansson and Newton.