On $\mathrm{Ext}^{\bullet}$ between locally analytic generalized Steinberg with applications
Zicheng Qian
TL;DR
This work advances the p-adic representation-theoretic study of GL _n(K) by computing first non-vanishing Ext groups between locally analytic generalized Steinberg representations and connecting these invariants to higher L-invariants. It develops a robust framework around the Tits double complex and a Coxeter filtration to control Ext spaces $\mathbf{E}_{J}=\mathrm{Ext}_G^{\#J}(1_G,V_{\Delta\setminus J}^{\mathrm{an}})$, including a precise dimension formula for graded pieces and a canonical injective cup-product structure $\kappa_{J,J'}$. A central achievement is identifying a Drinfeld-space–driven realization: the universal filtered $(\varphi,N)$-module arising from $\mathrm{St}_G^{\mathrm{an}}$ corresponds to $\mathcal{D}_{n-1}\otimes_{\mathcal{R}}\mathcal{R}(\theta^{1-n})$, providing a $p$-adic local–global bridge via the de Rham cohomology of the Drinfeld upper-half space. The paper then defines Breuil–Schraen L-invariants as codimension-1 subspaces of $\mathbf{E}_{n-1}$ compatible with cup-product, and outlines their conjectural link to Fontaine–Mazur L-invariants through regulator maps, building a program to relate automorphic and Galois invariants. Overall, the results give a comprehensive, combinatorially explicit picture of Ext-groups for generalized Steinberg representations and open a path to higher-rank $p$-adic L-invariants in the GL extsubscript{n} setting.
Abstract
Let $n\geq 2$ be an integer, $p$ be a prime number and $K$ be a finite extension of $\mathbb{Q}_p$. Motivated by Schraen's thesis and Gehrmann's definition of automorphic simple $\mathscr{L}$-invariants, we study the first non-vanishing extension groups between a pair of locally $K$-analytic generalized Steinberg representations of $\mathrm{GL}_n(K)$. We study subspaces of these extension groups defined by using either relative conditions with respect to Lie subalgebras of $\mathfrak{s}\mathfrak{l}_{n}$ (isomorphic to $\mathfrak{s}\mathfrak{l}_{m}$ for some $2\leq m<n$) or maps between locally $K$-analytic generalized Steinberg representations of $\mathrm{GL}_n(K)$ with different highest weights. The applications of these computations are two-fold. On one hand, we prove that a certain universal successive extension of filtered $(\varphi,N)$-modules can be realized as the space of homomorphisms from a suitable shift of the dual of locally $K$-analytic Steinberg representation into the de Rham complex of the Drinfeld upper-half space, generalizing one main result of Schraen's thesis from $\mathrm{GL}_{3}(\mathbb{Q}_p)$ to $\mathrm{GL}_{n}(K)$. On the other hand, we give a definition of higher $\mathscr{L}$-invariants for $\mathrm{GL}_n(K)$ (which we call Breuil-Schraen $\mathscr{L}$-invariants) and discuss its possible explicit relation to Fontaine-Mazur $\mathscr{L}$-invariants, using ideas from Breuil-Ding's higher $\mathscr{L}$-invariants for $\mathrm{GL}_{3}(\mathbb{Q}_p)$.