Infinitesimal characters for the completed cohomology of $\mathrm{GL}_n$ over CM fields

TL;DR

This paper proves that the infinitesimal characters of locally analytic vectors in completed cohomology of GL_n over certain CM fields are determined by Sen operators, confirming a conjecture in this setting.

Contribution

It establishes a link between infinitesimal characters and Sen operators for completed cohomology of GL_n over CM fields, confirming a specific conjecture.

Findings

Infinitesimal characters are determined by Sen operators.

The result applies to non-Eisenstein, generic maximal ideals.

Confirms a conjecture of Dospinescu-Paškūnas-Schraen.

Abstract

Let $p$ be a prime, and let $F$ be a CM field containing an imaginary quadratic field in which $p$ splits. We show that the locally analytic vectors of Hecke eigenspaces in the ($p$-adic) completed cohomology of $\mathrm{GL}_n/F$, localized at a non-Eisenstein decomposed generic maximal ideal, admit infinitesimal characters determined by the Sen operators of the corresponding Galois representations, thus confirming a conjecture of Dospinescu-Pa\v{s}k\={u}nas-Schraen in this case.