= Local principal series of GL2
{title2=$V(\chi)=\operatorname{Ind}_B^{\operatorname{GL}_2(F)}\chi$}
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 $\delta(\operatorname{diag}(a,d))=|d/a|$. When these coincide, the module is a nonsplit self-extension, so one must not count the <linear character> twice as two independent quotients.
Back to article page
