Equal-character Jacquet self-extension for GL2

At $\chi ={\chi }^w{\delta }^{-1}$, a determinant-character twist reduces to $\chi (\mathrm{diag}(a,d))=|a|$. In the open Bruhat coordinate, a section $g_0(x)=1$ for $|x|\le 1$ and $g_0(x)=|x{|}^{-1}$ otherwise has $\pi (\mathrm{diag}(\varpi ,1))g_0-q^{-1}{g}_0=(1-q^{-1})1_{\varpi \mathfrak{o}}$. Its additive Haar integral is nonzero, whereas every unipotent translation difference has zero such integral. The difference therefore survives in the Jacquet module, proving that this two-dimensional self-extension is nonsplit and has just one quotient functional of the common linear character.