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 $\pi$ of ${\mathrm{GL}}_n(K)$ of integral weights that appear in spaces of $p$-adic automorphic representations. We conjecture that the translation of $\pi$ 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 $\pi$. In particular, when $\pi$ is associated to a two dimensional de Rham Galois representation, we make conjectures and speculations on the relation between the Hodge filtrations of $\rho$ and the wall-crossings of $\pi$, which have a flavour of the Breuil-Strauch conjecture. We collect some results towards the conjectures and speculations.

82 pages, with an appendix by Yiwen Ding, Yongquan Hu and Haoran Wang