On generalization of Breuil--Schraen's $\mathcal{L}$-invariants to ${\mathrm{GL}}_n$

Abstract

Let $p$ be prime number and $K$ be a $p$-adic field. We systematically compute the higher $\mathrm{Ext}$-groups between locally analytic generalized Steinberg representations (LAGS for short) of ${\mathrm{GL}}_n(K)$ via a new combinatorial treatment of some spectral sequences arising from the so-called Tits complex. Such spectral sequences degenerate at the second page and each $\mathrm{Ext}$-group admits a canonical filtration whose graded pieces are terms in the second page of the corresponding spectral sequence. For each pair of LAGS, we are particularly interested their $\mathrm{Ext}$-groups in the bottom two non-vanishing degrees. We write down an explicit basis for each graded piece (under the canonical filtration) of such an $\mathrm{Ext}$-group, and then describe the cup product maps between such $\mathrm{Ext}$-groups using these bases. As an application, we generalize Breuil's $\mathcal{L}$-invariants for ${\mathrm{GL}}_2({\mathbb{Q}}_p)$ and Schraen's higher $\mathcal{L}$-invariants for ${\mathrm{GL}}_3({\mathbb{Q}}_p)$ to ${\mathrm{GL}}_n(K)$. Along the way, we also establish a generalization of Bernstein--Zelevinsky geometric lemma to admissible locally analytic representations constructed by Orlik--Strauch, generalizing a result in Schraen's thesis for ${\mathrm{GL}}_3({\mathbb{Q}}_p)$.

This is replaced by arXiv:2512.24279