In the book “The Local Langlands Conjecture for GL(2)” by Bushnell and Henniart, they prove the Schur’s lemma for groups G that G/K is countable where K is any open compact subgroup. Is Schur’s lemma valid for general groups? Are there counter examples?
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$\begingroup$ There is certainly a version of Schur's lemma that is true for strongly continuous unitary irreps of arbitrary locally compact groups - does that sound like what you are after? Or do you want irreps that might not be unitary, such as SL(2,R) acting on R^2? $\endgroup$– Yemon ChoiCommented Oct 23, 2017 at 15:05
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$\begingroup$ Sorry，I made mistake in my title,I mean smooth irreducible representations of locally profinite groups. $\endgroup$– wuzxCommented Oct 23, 2017 at 15:09
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7$\begingroup$ Let $G = \mathbf{C}(t)^{\times}$ with the discrete topology. This is trivially a locally compact and locally profinite group with no reasonable countability property, and it acts on the $\mathbf{C}$-vector space $V = \mathbf{C}(t)$ by the evident scaling irreducibly but violates Schur's Lemma over $\mathbf{C}$. This (counter)example comes from considering the way countability is used in the proof of the result you mention. $\endgroup$– nfdc23Commented Oct 23, 2017 at 15:10
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$\begingroup$ Yes, it is even locally profinite with discrete topology. Thanks! $\endgroup$– wuzxCommented Oct 23, 2017 at 15:17
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1$\begingroup$ @nfdc23, would you be willing to post that as an answer, so that this problem doesn't show up as unanswered? $\endgroup$– LSpiceCommented Oct 23, 2017 at 22:30
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@nfdc23's counterexample (posted as CW to avoid reputation):
Let $G=\mathbb C(t)^\times$ with the discrete topology. This is trivially a locally compact and locally profinite group with no reasonable countability property, and it acts on the $\mathbb C$-vector space $V=\mathbb C(t)$ by the evident scaling irreducibly but violates Schur's Lemma over $\mathbb C$. This (counter)example comes from considering the way countability is used in the proof of the result you mention.