Equidimensional adic eigenvarieties for groups with discrete series

TL;DR

This paper extends the construction of eigenvarieties for certain reductive groups to include characteristic p points at the boundary of weight space, introducing a new notion of locally analytic functions and distributions.

Contribution

It introduces a new framework for defining locally analytic functions and distributions on locally $bQ_p$-analytic manifolds with values in complete Tate $bZ_p$-algebras, extending Urban's eigenvariety construction.

Findings

Extended eigenvariety construction to include boundary points in weight space.

Defined a new notion of locally analytic functions and distributions.

Aligned the new definition with Johansson and Newton's distributions.

Abstract

We extend Urban's construction of eigenvarieties for reductive groups $G$ such that $G(\mathbb{R})$ 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.