Sunday, 25 October 2009

rt.representation theory - Can an admissible SO(n) representation contain an SO(n-1) representation with infinite multiplicity?

Funnily enough, I wrote a paper about this question a few years ago.



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 TSn1. In particular, TS1tomathfrakso2 is surjective since S1 is a regular action, and TS2tomathfrakso3 is surjective since the adjoint representation is covered by the orbit of any line.

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