Let $G$ be a real, connected, non-compact, semisimple Lie group with finite center and real rank $1$. Let $G=KAN$ be an Iwasawa decomposition, then $K\subset G$ is a maximal compact subgroup, and $\dim A=1$ (which is the definition of the real rank of $G$ being $1$). Let $M:=Z_K(A)$ be the centralizer of $A$ in $K$. Note that $M$ is a compact subgroup of $G$, being a closed subgroup of $K$.

Let $\tau:M\to \mathrm{End}(V)$ be a finite-dimensional complex representation of $M$. Then I would like to know: Under which circumstances does there exist a finite-dimensional complex representation of $K$ that extends $\tau$?

I searched for quite a long time now, and all literature about extension of representations of compact Lie groups only deals with normal subgroups. However, $M$ need not be normal in $K$ in general.

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    $\begingroup$ One of the older references which might (or might not) be helpful here is the set of lecture notes David Collingwood wrote after his lecture series in Argentina: Representations of rank one Lie groups (Pitman, 1985). These notes often discuss special cases, which can be illuminating in view of the sophisticated underlying general theory here. $\endgroup$ – Jim Humphreys Mar 3 '17 at 21:39
  • $\begingroup$ In case I haven't misunderstood your question, you are looking for branching laws from $K$ to $M$. When $\dim A=1$, these branching laws (depending of $G=SO(n,1), SU(n,1), Sp(n,1), F_4$) should be known. $\endgroup$ – emiliocba May 15 '17 at 20:02

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