Extensions of locally analytic generalized parabolic Steinberg representations

Let $L$ be a finite extension of ${\mathbf{Q}}_p$. In this paper, we study the locally ${\mathbf{Q}}_p$-analytic generalized parabolic Steinberg representations of ${\mathrm{G}\mathrm{L}}_n(L)$, and compute the $\mathrm{E}\mathrm{x}\mathrm{t}$-groups of locally ${\mathbf{Q}}_p$-analytic generalized parabolic Steinberg representations. They carry the Breuil’s simple $\mathcal{L}$-invariants, which arise in the automorphic side of a conjectured $p$-adic local Langlands correspondence.