Extensions for supersingular representations of $G{L}_2(Q_p)$

Abstract

Let $p>2$ be a prime number. Let $G:=G{L}_2(Q_p)$ and $\pi$, $\tau$ smooth irreducible representations of $G$ on ${\bar{F}}_p$-vector spaces with a central character. We show if $\pi$ is supersingular then $Ex{t}_G^1(\tau ,\pi )\ne 0$ implies $\tau \cong \pi$. This answers affirmatively for $p>2$ a question of Colmez. We also determine $Ex{t}_G^1(\tau ,\pi )$, when $\pi$ is the Steinberg representation. As a consequence of our results combined with those already in the literature one knows $Ex{t}_G^1(\tau ,\pi )$ for all irreducible representations of $G$.

arXiv commentThis version contains more details. Sections 3 and 9 containing some background information are new. An appendix is added

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