Theorem of Cantor-Bernstein in the category of smooth representation of $G$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T15:35:21Z http://mathoverflow.net/feeds/question/89669 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89669/theorem-of-cantor-bernstein-in-the-category-of-smooth-representation-of-g Theorem of Cantor-Bernstein in the category of smooth representation of $G$ Rajkarov 2012-02-27T14:28:05Z 2012-02-28T04:47:49Z <p>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 $F$. </p> <p>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). </p> <p>If $L$ a non empty subset of $Irr(G)$, we define a $L$-minimal representation as a smooth representation $\pi$ of $G$ such that :</p> <p>1) For every $\sigma\in L$, $\sigma \leq \pi$. </p> <p>2) For every $\tau \in R(G)$, if $\sigma\leq\tau$ for every $\sigma\in L$, then $\pi\leq\tau$.</p> <p>I ask the following questions: </p> <p>Q1) An $L$-minimal representation exits ?</p> <p>Q2) unicity ? </p> <p>Q3) If $\pi$ an $L$-minimal (if there exist) representation, $dim\mathbf{Hom}_{G}(\sigma,\pi)$, where $\sigma\in L$, is minimal ?</p> <p>I'm interested of this question for the set $L_{k}$ of equivalence classes of irreducible supercuspidal representation of $PGL(n,F)$ with conductor=$k$.</p> http://mathoverflow.net/questions/89669/theorem-of-cantor-bernstein-in-the-category-of-smooth-representation-of-g/89674#89674 Answer by Marc Palm for Theorem of Cantor-Bernstein in the category of smooth representation of $G$ Marc Palm 2012-02-27T15:34:26Z 2012-02-27T15:52:50Z <p>Here are some observations, too long for a comment:</p> <p>1) Note that cuspidal irreducible representation are compactly induced</p> <p>$\sigma = c-ind_K^G \tau = Ind_K^G \tau$</p> <p>2) You have the second adjointness theorem (proposition on pg. 20 Bushnell-Henniart "Local Langlands for GL(2)".)</p> <p>$Hom_G( c-ind_K^G \tau, \pi) = Hom_K( \tau, Res_K \pi)$</p> <p>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 $GL(n)$.</p> <p>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).</p> <p>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;)</p> http://mathoverflow.net/questions/89669/theorem-of-cantor-bernstein-in-the-category-of-smooth-representation-of-g/89730#89730 Answer by Rami for Theorem of Cantor-Bernstein in the category of smooth representation of $G$ Rami 2012-02-28T04:47:49Z 2012-02-28T04:47:49Z <p>Your group has compact center (in fact trivial center). So supercuspidal representation indeed form a semi-simple category, as pm said (see <a href="http://www.math.uchicago.edu/~mitya/langlands/Bernstein/Bernstein93new.dvi" rel="nofollow">http://www.math.uchicago.edu/~mitya/langlands/Bernstein/Bernstein93new.dvi</a> pg. 22-25, 36). </p> <p>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 $L_k$</p>