On $\mathrm{Ext}^{\bullet}$ between locally analytic generalized Steinberg with applications

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 $\mathrm{GL}_{3}(\mathbb{Q}_p)$.