Question

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=$k$.

Answer 1

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;)

Answer 2

Your group has compact center (in fact trivial center). So supercuspidal representation indeed form a semi-simple category, as pm said (see http://www.math.uchicago.edu/~mitya/langlands/Bernstein/Bernstein93new.dvi pg. 22-25, 36).

Your Set $L_k$ is finite set of irreducible representations. So the representation that you are looking for is the direct sum of all the representations in $L_k$