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Is there any reference where I can find the character table of $\mathrm{SL}_2(\mathbb{Z}/p^n\mathbb{Z})$? A simple search in google gave me this paper of Philip C. Kutzko on "The characters of the binary modular congruence group". But this is just an announcement and I couldn't find the complete paper. I also know that Kloosterman has a paper which discuss representation theory of $\mathrm{SL}_2(\mathbb{Z}/p^n\mathbb{Z})$ but as far as I see he hasn't computed the characters. I am looking for a reference where I can find the actual character table.

I would be thankful if you please let me any reference about this.

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  • $\begingroup$ I suppose Kutzko's PhD thesis has the proofs, etc. $\endgroup$ – Dima Pasechnik Apr 1 '15 at 19:06
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    $\begingroup$ I asked a similar question at one point, perhaps the answers there can be useful to you: mathoverflow.net/questions/87254 $\endgroup$ – Dan Petersen Apr 1 '15 at 20:27
  • $\begingroup$ Also, if you send me an e-mail I can send you a copy of Kutzko's PhD thesis. $\endgroup$ – Dan Petersen Apr 1 '15 at 20:27
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Unlike the well-known case of these groups over a prime field, it's probably asking too much to exhibit full character tables over all such finite rings. (Note too that Kutzko limits his discussion to odd primes.)

There have been related discussions over the years of representations of the finite groups coming from $\mathrm{GL}_2$ and $\mathrm{SL}_2$, typically in the context of $p$-adic rings and representations of the resulting infinite groups. Maybe it would help to look at older papers by Alexandre Nobs and his collaborators, such as here. There is also some more recent work in this direction by Alexander Stasinski here. I'm unaware of any published "tables", though I'm not a specialist in this field.

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