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

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.