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Let $G$ be a connected semi simple Lie group with finite center. Fix a maximal compact subgroup $K$. An irreducible representation $(\pi,V)$ of $G$ is called a "class-one representation", if it contains non-zero $K$-fixed vectors. It is often treated as well-known, at least for splitrank one groups, that every irreducible unitary class-one representation is either trivial or infinitesimally equivalent to a representation induced from a minimal parabolic (and not only a sub representation of the latter as the sub representation theorem says).

Where can I find a proof of this "well-known"?

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  • $\begingroup$ As stated, it si false: the trivil representation of $G$ is an example, which is not induced from a minimal parabolic subgroup $\endgroup$
    – Venkataramana
    Commented Sep 2, 2014 at 7:27
  • $\begingroup$ Yes ok, sorry, let's add the conditions of $G$ being connected and exclude the trivial representation. I update the question. $\endgroup$
    – Doug
    Commented Sep 2, 2014 at 9:14
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    $\begingroup$ The paper by Barbasch and Baldoni Silva (see the math reviews ams.org/mathscinet-getitem?mr=696689) answers this for all rank one groups; special cases were done by Hirai earlier $\endgroup$
    – Venkataramana
    Commented Sep 2, 2014 at 13:38
  • $\begingroup$ It's probably not trivial to extract the class-one case from the paper (or its references), but anyway it's freely available via GDZ: gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002099276 $\endgroup$
    – Jim Humphreys
    Commented Sep 2, 2014 at 15:25
  • $\begingroup$ P.S. Are you only asking about split rank 1, or is your "well known" supposed to apply more generally? $\endgroup$
    – Jim Humphreys
    Commented Sep 2, 2014 at 15:27

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