Question

There are many references for the representation theory (say over $\mathbf C$) of $SL_2(\mathbf{F}_q)$ and $GL_2(\mathbf{F}_q)$, for instance lecture 5 in Fulton--Harris "Representation theory" and section 4.1 in Bump's "Automorphic forms and representations". Is there anywhere I can find a similarly explicit description of the representation theory of $SL_2$ and $GL_2$ over the ring $\mathbf Z/n$, where $n$ is not a prime? (By the Chinese remainder theorem, it suffices to consider the case when $n$ is a prime power.)

Answer 1

By now there is a fairly long paper trail dealing with this kind of question, which is usually a byproduct of the study of representation theory over rings of $p$-adic integers, etc. I'm not aware of any specific book that would be helpful for your question, but you might benefit from looking at recent literature and perhaps tracing some of the references. For example, a paper by A. Stasinski on the arXiv here, and another by Onn here (both later published formally).

I can add more references of this sort, but I also want to mention the recent text by Cedric Bonnafe (published by Springer) Representations of $SL_2(\mathbb{F}_q)$ which studies the problem over finite fields in the wider Deligne-Lusztig context applicable to all finite groups of Lie type.

P.S. An older research announcement by Kutzko here is probably most directly relevant to your question.

Answer 2

The case of $\mathrm{GL}_2$ over a finite local principal ideal ring was treated in a paper by Nagornyj in the late 70s under some (unnecessary) restrictions, and Kutzko obtained similar results independently, but never published them.

The case of $\mathrm{SL}_2$ over a finite local principal ideal ring with odd residue characteristic was treated by Kutzko and Shalika respectively, in the late 70s. These works remain little known, even though Shalika's results were published in the volume dedicated to him. A write-up of these results appears in the end of Jaikin-Zapirain's paper here.

The case of $\mathrm{SL}_2$ where the residue characteristic is even is much more difficult, and has as far as I know only been done for the rings $\mathbb{Z}/(2^r)$ in a series of papers by Nobs or Nobs and Wolfart (which also treats the odd residue characteristic case). The method of Nobs and Wolfart, using Weil representations, is quite different from that of Kutzko and Shalika, and it remains an open problem to construct the representations of $\mathrm{SL}_2(\mathbb{F}_q[[t]]/(t^r))$, where $q$ is a power of $2$.

Answer 3

Another reference that goes along the lines of what Professor Humphreys mentioned in his answer is a paper by Bill Casselman "the restriction of a representation of $GL_2(k)$ to $GL_2(O)$"Math. Ann. 206, 311- 318 (1973). Also with respect Kutzko's results, mentioned in previous answers, even though they were not published one can find them in his Ph.D thesis at Madison. On convenient feature about Kutzko's thesis is that it contains at the end very explicit character tables for the groups in question. If you are interested I have a scan copy of such tables.