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It is a basic theorem of Harish-Chandra that an irreducible unitary representation $\pi$ of a reductive group $G$ over $\mathbb{R}$ on a Hilbert space is admissible, meaning that every irreducible representation of a maximal compact $K$ that occurs in $\pi|_K$ does so with finite multiplicity.

What is the standard example of an irreducible representation of $G$ that fails to be admissible, say for $G=\mathrm{SL}(2,\mathbb{R})$ or $\mathrm{GL}(2,\mathbb{R})$?

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  • $\begingroup$ Apparently this is not so standard; apparently it was a conjecture of Harish-Chandra that there wouldn't be any! The first counter-example in Banach spaces is due to Soergel in jstor.org/stable/2047636 in 1988, though he gives much credit to Vogan. Soergel uses Enflo's invariant subspace counter-example, so could it be that at present no examples for Hilbert spaces are actually known? $\endgroup$
    – Stefan Dawydiak
    Commented Apr 18 at 8:46

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