= Equal-character Jacquet self-extension for GL2
At $\chi=\chi^w\delta^{-1}$, a determinant-character twist reduces to $\chi(\operatorname{diag}(a,d))=|a|$. In the open Bruhat coordinate, a section $g_0(x)=1$ for $|x|\le1$ and $g_0(x)=|x|^{-1}$ otherwise has $\pi(\operatorname{diag}(\varpi,1))g_0-q^{-1}g_0=(1-q^{-1})1_{\varpi\mathfrak o}$. 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>.
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