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.