 Local principal series of GL2

The local principal series of GL2 induces a smooth linear character of its diagonal torus, inflated to its upper-triangular subgroup. Its unnormalized Jacquet module has two one-dimensional composition factors, $\chi$ and ${\chi }^w{\delta }^{-1}$, where $\delta (\mathrm{diag}(a,d))=|d/a|$. When these coincide, the module is a nonsplit self-extension, so one must not count the linear character twice as two independent quotients.