Let $G=GL(n,F)$, where $F$ is a non-archimedean local field. If we consider a smooth representation $\pi$ of $G$ such that every irreducible generic representation of $G$ embeds in $\pi$, is it true that the representation $Ind\_U^G\chi$ embeds in $\pi$, where $U$ is the standard unipotent subgroup of $G$ and $\chi$ is any fixed non-degenerate caracter of $U$?



Generic representations of $GL(n,F)$
No. At least I think not.


I assume that $Ind\_U^G \chi$ means the space of all functions $f:G\to \mathbb C$ which satisfy (1) $f(ug) = \chi(u) f(g)$ and (2) there is an open compact subgroup $K$ of $G$ such that $f(gk) = f(g)$ for all $g \in G$ and $k \in K.$


What I want to do is construct a subrepresentation of this space which is reducible but not decomposable. To do that, I take any irreducible subspace $V$ and multiply all of the functions in it by $\log|\det g|.$ Let's see that this works. Since $V$ is irreducible, it has some central character $\omega.$ So, if $f \in V$ then $[\rho(z)f](g) = f(zg) = \omega(z)\cdot f(g)$ for all $z \in Z$ and $g \in G.$ Here $Z$ is the center of $G.$ Define $f\_1(g) := \log|\det g|f(g)$ and consider $[\rho(z) f\_1].$ We have
$$
[\rho(z)f\_1] (g) = f\_1(zg) = f(zg) \log|\det zg| =\omega(z) f(g) \log|\det zg|
=\omega(z) f\_1(g) + \omega(z) \log|z| f(g). 
$$
In other words, $\rho(z)$ acts on the two dimensional space spanned by $f$ and $f\_1$ via the matrix $$
\begin{pmatrix} \omega(z) & \log|z|\omega(z) \\ 0& \omega(z) \end{pmatrix}.
$$
Let $V\_1 = \{ f\_1 \mid f \in V\}$ where $f\_1$ is defined in terms of $f$ as above. Then it follows from the previous discussion that $V + V\_1$ is a subrepresentation of $Ind\_U^G \chi$ which is not a sum of irreducibles.


Now take $\pi$ to be the direct sum of all the irreducible subrepresentations in $Ind\_U^G\chi.$ Then $V+V\_1$ can not map into $\pi,$ and therefore $Ind\_U^G\chi$ can't either.


