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The representation theory of the Lie group $SU(2)$ is well understood, easy to describe, and in fact central to the classical theory of representation theory of compact Lie groups.

But what happens if we ignore the smooth structure and even the topology on $SU(2)$ and simply see it as a discrete group? (I will write $SU(2)^{\delta}$ for this group in the following.) Three related questions one might ask are the following.

(1) Is there a 2-dimensional representation $SU(2)^{\delta} \to SL_2(\mathbb C)$ that is not conjugate to the inclusion $SU(2)^{\delta} \to SU(2) \subset SL_2(\mathbb C)$?

(2) Are there conditions on a representation that guarantee that it will be continuous/smooth?

(3) Is it possible to describe the complex irreducible representations of $SU(2)^{\delta}$?

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    $\begingroup$ What do you call "conjugate"? if you mean conjugate by an inner automorphism, there are more reps. If you mean by an automorphism of $SL_2(SL_2(\mathbf{C}))$, probably there's none. Additional reps are indeed by "conjugating" the standard rep by an automorphism of the field $\mathbf{C}$ (extended to $SL_2(\mathbf{C})$). For homomorphisms between Lie groups, continuous and smooth are the same; continuous reps of $SU(2)$ into $SL_2(\mathbf{C}$ are indeed all conjugate to the inclusion by an inner automorphism of the target group. $\endgroup$
    – YCor
    Oct 12, 2017 at 11:51
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    $\begingroup$ Finally (3) is probably tackled in Borel-Tits "homomorphismes abstraits de groupes algébriques simples", see jstor.org/stable/1970833?seq=1#page_scan_tab_contents, or numdam.org/item/SB_1972-1973__15__307_0 for Steinberg's Bourbaki seminar account (the latter is in English). $\endgroup$
    – YCor
    Oct 12, 2017 at 11:57
  • $\begingroup$ Hmm, okay, that is kind of cheap. I did mean conjugate by an inner automorphism. And what about non-trivial endomorphisms of $SU(2)^{\delta}$. Are there any? $\endgroup$
    – Jens Reinhold
    Oct 12, 2017 at 11:59
  • $\begingroup$ you should check in the given references, as it should answer your latter question. $\endgroup$
    – YCor
    Oct 12, 2017 at 12:05
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    $\begingroup$ @YemonChoi Yes; so it implies that $\mathrm{SU}(2)^\delta$ is not Type I, hence in a sense a “no” answer to question (3) for unitary representations. $\endgroup$
    – Francois Ziegler
    Oct 13, 2017 at 2:27

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