A note on extensions of $p$-adic representations of $\mathrm{GL}_2(\mathbb{Q}_p)$

Abstract

We compute extension groups in the category of duals of $p$-adic Banach space representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. Focusing on representations arising from the $p$-adic local Langlands correspondence for generic Galois representations, we classify these extensions completely. These results are then applied to prove the vanishing of extensions between the duals of reducible representations and supercuspidal isotypic components of the ètale cohomology of the finite level Drinfeld spaces.