Generic representations of $GL(n,F)$

Question

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 $U$?

Answer 1

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.

Answer 2

It seems to me that the universal property that you want for an "$L$-minimal" representation is precisely the universal property of the direct sum. In other words, the direct sum of the elements of $L$ (technically, of representatives for the elements of $L,$ which are isomorphism classes) is an $L$ minimal representation, and is isomorphic to any other.

Answer 3

Okay, this is beautiful cons example. But sincerely, i'm interested for a verey general question. If we note $Irr(G)$ the set equivalence classes of irreducible smooth repesentations of $G$ and we fix a non empty subset $L$ of $Irr(G)$. We define an $L$-minimal representation as a representation $\pi$ of $G$ such that : 1) For every $\sigma\in L$, $\sigma$ embeds in $\pi$.\ 2) For evrey smooth representation $\tau$ of $G$, if for evrey $\sigma\in L$, $\sigma$ embeds in $\tau$, then $\pi$ embeds in $\tau$.\

I ask the following questions :

1) An $L$-minimal representation exists ? 2) uniqueness up to isomorphism ?

I ask the question for the set $L_{k}$ of equivalence classes of irreducible supercuspidal representation of $G$ with conductor $k$.