# Theorem of Cantor-Bernstein in the category of smooth representation of $G$

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 $\pi\leq\tau$.

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=$k$.

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I am not sure what $L_k$ is. So can you please give a definition. I would expect this to be a finite set, so why can you not take $\pi$ as the direct sum of $\sigma$'s? – Marc Palm Feb 27 '12 at 15:36
I mean $L_k$ is finite, since I guess you induce $GL(n, o / \p^k)$ reps there or something similar, which has only finitely many isos of reps anyway. – Marc Palm Feb 27 '12 at 15:39
How do you define your relation $\leq$ ? – Joël Feb 27 '12 at 18:36
I defined the set $L_{k}$ as follows : If $\pi$ is a generic irreducible representation of $GL(n,F)$, the theorem of Jaquet & Piateski-Chapiro & Shalika says : there is a positive integer $k$ such that the representation $\pi$ admits a non zero vector fixed by the compact open subgroup $\Gamma_{k}$ defined by : – Rajkarov Feb 28 '12 at 4:50
$$\Gamma_{k}=\left(\begin{array}{cccc|c} \mathcal{O}_{F}^{\times}&\mathcal{O}_{F}&\cdots&\mathcal{O}_{F}&\mathcal{O}_{F}{‌​}\\ \mathcal{O}_{F}&\ddots&\ddots&\vdots&\vdots\\ \vdots&{}&\ddots&\mathcal{O}_{F}&\vdots{}\\ \mathcal{O}_{F}&\cdots&\mathcal{O}_{F}&\mathcal{O}_{F}^{\times}&\mathcal{O}_{F}{‌​}\\ \hline \mathcal{P}_{F}^{k}& \cdots & \cdots &\mathcal{P}_{F}^{k}&\mathcal{O}_{F}^{\times}\\ \end{array}\right)$$ – Rajkarov Feb 28 '12 at 4:53

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;)

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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 $L_k$

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More precisely, this comes from the fact that (since the center is compact) supercuspidal representations are projective and injective objects of the category of smooth representations. – Paul Broussous Feb 28 '12 at 7:55