Extensions for supersingular representations of $GL_2(Q_p)$

Details Extensions for supersingular representations of $GL_2(Q_p)$ Extensions for supersingular representations of $GL_2(Q_p)$

2007-10-04

arxiv.org/abs/0710.1053v3

Mathematics

Representation Theory

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

Scientific paper

Let $p>2$ be a prime number. Let $G:=GL_2(Q_p)$ 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 $Ext^1_G(\tau,\pi)\neq 0$ implies $\tau\cong \pi$. This answers affirmatively for $p>2$ a question of Colmez. We also determine $Ext^1_G(\tau,\pi)$, when $\pi$ is the Steinberg representation. As a consequence of our results combined with those already in the literature one knows $Ext^1_G(\tau,\pi)$ for all irreducible representations of $G$.

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