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

TL;DR

This paper computes higher Ext-groups between locally analytic generalized Steinberg representations of GL_n over p-adic fields, providing explicit bases, cup product descriptions, and generalizing Breuil--Schraen's $ ext{L}$-invariants to higher dimensions.

Contribution

It introduces a new combinatorial approach to compute Ext-groups, explicitly describes their bases and cup products, and generalizes existing $ ext{L}$-invariants to GL_n(K).

Findings

Computed Ext-groups using spectral sequences from Tits complex.

Provided explicit bases for graded pieces of Ext-groups.

Generalized Breuil and Schraen's $ ext{L}$-invariants to GL_n(K).

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 $\mathscr{L}$-invariants for $\mathrm{GL}_2(\mathbb{Q}_p)$ and Schraen's higher $\mathscr{L}$-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 $\mathrm{GL}_3(\mathbb{Q}_p)$.