Unnormalized parabolic induction (${\mathrm{Ind}}_B^G\chi$)

For a smooth linear character $\chi$ of a parabolic subgroup $B$, unnormalized induction consists of smooth functions satisfying $f(bg)=\chi (b)f(g)$, with right-translation action. The quotient $B\backslash G$ is compact for the local general linear groups in question. Normalized induction inserts a square root of the parabolic modulus in the covariance, changing formulas for the inducing linear character.

Evaluation of an intertwiner at the identity gives a $B$-equivariant functional. Conversely a functional $\ell$ gives $A(v)(g)=\ell (\pi (g)v)$, which has the required covariance and is fixed on the right by any compact-open subgroup fixing $v$. These inverse constructions prove the displayed adjunction. If $\xi$ is trivial on the unipotent radical, it is equivalent to a character-valued functional on the Jacquet module.

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.

The local Steinberg representation is the quotient of locally constant functions on the projective line by constant functions, with the action induced from GL2. In the unnormalized convention, $V(1)$ fits into $0\to 1\to V(1)\to \mathrm{St}\to 0$. Its scalar endomorphism algebra obstructs a splitting. This local infinite-dimensional representation differs from the finite-field Steinberg representation.

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.