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.

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