Infinitesimal characters and Lafforgue's pseudocharacters

TL;DR

This paper extends the construction of infinitesimal characters to p-adic families of Lafforgue's pseudocharacters, enabling analysis of the center's action on locally analytic vectors in completed cohomology.

Contribution

It introduces a novel method to associate infinitesimal characters to p-adic families of pseudocharacters, building on previous work by Dospinescu, Schraen, and the first author.

Findings

Infinitesimal characters are associated to p-adic families of pseudocharacters.

The action of the center of the universal enveloping algebra is studied on locally analytic vectors.

The construction provides new insights into the structure of completed cohomology.

Abstract

We associate infinitesimal characters to $p$-adic families of Lafforgue's pseudocharacters of the absolute Galois group of a $p$-adic local field by extending a construction of Dospinescu, Schraen and the first author. We use this construction to study the action of the centre of the universal enveloping algebra on the locally analytic vectors in the Hecke eigenspaces of the completed cohomology.