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Let $k$ be a local field of residue characteristic $p$, and let D be a central division algebra over $k$ of index $n>2$. How to determine the irreducible complex representations of the group $SL_1(D)$? Suggest some reference regarding this.

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    $\begingroup$ In addition to @PaulBroussous's very apposite answer, there is the general answer that all these representations, which are necessarily supercuspidal, will arise (at least for $p > n$) from the construction of Yu (MR). It may help to read this in parallel with, say, CMS (MR), which I think gives an approachable introduction to the $\mathrm{GL}_1(D)$ case. $\endgroup$
    – LSpice
    Commented Jul 6, 2017 at 11:07

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An article by Shai Shechter recently appeared on Math ArXiv:

"Characters of the Norm-One Units of Local Division Algebras of Prime Degree"

https://arxiv.org/abs/1512.02448

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The representation theory of ${\rm SL}_1 (D)$ was the topics of Göran Kirchner's PhD (defended in 2007, under the supervision of E.-W. Zink, Berlin). To my knowledge it has not been published.

https://www.amazon.de/Zur-Darstellungstheorie-von-SL1-D/dp/3868059970

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  • $\begingroup$ Thank you @Paul Broussous for your reference. But it's not available. $\endgroup$
    – sampath
    Commented Jul 7, 2017 at 9:48
  • $\begingroup$ Thank you for your suggestion. I received a copy (PhD dissertation) from Kirchner. $\endgroup$
    – sampath
    Commented Sep 24, 2017 at 8:51

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