If we fix a locally profinite group $G$ , we note $R(G)$ the category of smooth representations of $G$, $\mathcal{E}$ the set of equivalence classes of $R(G)$, and finaly $Irr(G)$ the set of irreducible equivalence calasses. I recall that the theorem of Cantor-Bernstein says : If $E$ and $F$ two sets. If there is an injection from $E$ to $F$ and an injection from $F$ to $E$, then there is a bijection between $E$ and $F$. This enables us to define an ordering relation in the set of equipotence classes of sets : If $E$ and $F$ two sets, we note $E\leq F$ if $E$ injects in $F$. 


Similarly, we define a relation $\leq$ in $\mathcal{E}$, but in general is not an ordering relation, I think that is an ordering relation if $R(G)$ is semisimple (for example, for compact locally profinite group). 


If $L$ a non empty subset of $Irr(G)$, we define a $L$-minimal representation as a smooth representation $\pi$ of $G$ such that :


1) For every $\sigma\in L$, $\sigma \leq \pi$. 


2) For every $\tau \in R(G)$, if $\sigma\leq\tau$ for every $\sigma\in L$, then $\pi\leq\tau$.


I ask the following questions: 


Q1) An $L$-minimal representation exits ?


Q2) unicity ? 


Q3) If $\pi$ an $L$-minimal (if there exist) representation, $dim\mathbf{Hom}\_{G}(\sigma,\pi)$, where $\sigma\in L$, is minimal ?


I'm interested of this question for the set $L\_{k}$ of equivalence classes of irreducible supercuspidal representation of $PGL(n,F)$ with conductor=$k$.



Theorem of Cantor-Bernstein in the category of smooth representation of $G$
Here are some observations, too long for a comment:


1) Note that cuspidal irreducible representation are compactly induced


$\sigma = c-ind\_K^G \tau = Ind\_K^G \tau$


2) You have the second adjointness theorem (proposition on pg. 20 Bushnell-Henniart "Local Langlands for GL(2)".)


$Hom\_G( c-ind\_K^G \tau, \pi) = Hom\_K( \tau, Res\_K \pi)$


3) Silberger PGL(2) over the $p$ adics assect that $Res\_K \pi$ is essentially $Ind\_{B(o)}^{GL(2, o)} 1$ except for a finite dimensional part. I expect this to be true for $GL(n)$.


Hence classify the $\tau$ needed for $L\_k$ ( I am not sure what your definition is here). $Res\_K \pi$ has been described for cuspidal $\pi$ (Bushnell-Kutzko).


In fact, I think that the supercuspidal representation form a semisimple category, so there the question might really reduce to something trivial, very much like for profinite groups. (profinite groups are actually exactly the compact locally profinite groups;)


