MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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=$k$.

share|cite|improve this question
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
up vote 1 down vote accepted

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;)

share|cite|improve this answer

Your group has compact center (in fact trivial center). So supercuspidal representation indeed form a semi-simple category, as pm said (see 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 $L_k$

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.