Towards the wall-crossing of locally $\mathbb{Q}_p$-analytic representations of $\mathrm{GL}_n(K)$ for a $p$-adic field $K$

Abstract

Let $K$ be a finite extension of $\mathbb{Q}_p$. We study the locally $\mathbb{Q}_p$-analytic representations $π$ of $\mathrm{GL}_n(K)$ of integral weights that appear in spaces of $p$-adic automorphic representations. We conjecture that the translation of $π$ to the singular block has an internal structure which is compatible with certain algebraic representations of $\mathrm{GL}_n$, analogously to the mod $p$ local-global compatibility conjecture of Breuil-Herzig-Hu-Morra-Schraen. We next make some conjectures and speculations on the wall-crossings of $π$. In particular, when $π$ is associated to a two dimensional de Rham Galois representation, we make conjectures and speculations on the relation between the Hodge filtrations of $ρ$ and the wall-crossings of $π$, which have a flavour of the Breuil-Strauch conjecture. We collect some results towards the conjectures and speculations.