Can an admissible SO(n) representation contain an SO(n-1) representation with infinite multiplicity? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T21:07:20Z http://mathoverflow.net/feeds/question/12663 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/12663/can-an-admissible-son-representation-contain-an-son-1-representation-with-inf Can an admissible SO(n) representation contain an SO(n-1) representation with infinite multiplicity? unknown (google) 2010-01-22T17:08:15Z 2010-01-22T18:50:34Z <p>For simplicity, just let G be GL(n) over real numbers, K=SO(n),K'=SO(n-1). Now if $\pi$ is an admissible representation of G with respect to K, i.e., any irreducible K-representation occurs with finite multiplicity.</p> <p>Now the question I want to know that: does any irreducible K'-representation occur with finite multiplicity?</p> http://mathoverflow.net/questions/12663/can-an-admissible-son-representation-contain-an-son-1-representation-with-inf/12665#12665 Answer by Emerton for Can an admissible SO(n) representation contain an SO(n-1) representation with infinite multiplicity? Emerton 2010-01-22T17:15:46Z 2010-01-22T17:15:46Z <p>Not necessarily. For example if $n = 2$, then $SO(1)$ is the trivial group, and so if $\pi$ is infinite dimensional, it is the trivial representation of $SO(1)$ with infinite multiplicity.</p> <p>Now suppose $n = 3$. The trivial representation of $SO(2)$ occurs with mult. one in every irrep. of $SO(3)$. So if $\pi$ is infinite dimensional, and so contains an infinite number of irreps. of $SO(3)$, then $\pi$ will contain the trivial representation of $SO(2)$ with infinite multiplicity.</p> http://mathoverflow.net/questions/12663/can-an-admissible-son-representation-contain-an-son-1-representation-with-inf/12675#12675 Answer by Ben Webster for Can an admissible SO(n) representation contain an SO(n-1) representation with infinite multiplicity? Ben Webster 2010-01-22T18:50:34Z 2010-01-22T18:50:34Z <p>Funnily enough, I wrote <a href="http://front.math.ucdavis.edu/0609.5846" rel="nofollow">a paper</a> about this question a few years ago.</p> <p>The takeaway is that there is a geometric method for understanding when such a restriction is admissible. For many pairs of groups it never is, but I think Matt found the only examples of the form SO(n) and SO(n-1). In general, each finite dimensional representation of SO(n) comes from quantizing a coadjoint orbit, and you want only finitely many of the coadjoint orbits that lie in the image of the moment map on the contangent bundle of the sphere $T^*S^{n-1}$. In particular, <code>$T^*S^1 \to \mathfrak{so}_2^*$</code> is surjective since $S^1$ is a regular action, and <code>$T^*S^2 \to \mathfrak{so}_3^*$</code> is surjective since the adjoint representation is covered by the orbit of any line.</p>