Infinitesimal action on completed cohomology for GL_n over a CM field

I will talk about joint work with Vaughan McDonald which confirms a conjecture of Dospinescu-Pa\v{s}k\={u}nas-Schraen (localised suitably) for reductive group $GL_n/F$ for $F$ a CM field containing an imaginary quadratic field where a fixed prime $p$ splits.

In the first part of the talk, I will recall some facts about completed cohomology of a reductive group $G$ and give a motivation (at least for me) for one to think there might be a relationship between:
1.  action of the centre of universal algebra $Z(g)$ of $G$ on locally analytic vectors of completed cohomology, and
2. Hodge--Tate--Sen weights of Galois representations attached to Hecke eigenspaces

Then I will introduce the relevant constructions from Dospinescu-Pa\v{s}k\={u}nas-Schraen in some detail, state their conjecture on the relationship between 1) and 2), and state our theorem.

In the second part of the talk, I will show the proof strategy and highlight the most important ingredients.