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When $\ell = 1$ I know that the smallest non-trivial irreducible complex representations of $SL_n(\mathbb{Z}/p\mathbb{Z})$ has dimension $\frac{p^n - 1}{p-1} - 1$ (with maybe some exceptions for small values of $n$ or $p$).

For $\ell > 1$ we have the map $SL_n(\mathbb{Z}/p^\ell \mathbb{Z}) \to SL_n(\mathbb{Z}/p\mathbb{Z})$ so we still get a representation of this size. I was wondering if this is still the smallest possible dimension (again maybe with a finite number of exceptions for small $p$ and $n$), and if not are there known lower bounds for how small such a representation can be?

I found some calculations of characters for small values of $n$ that suggest this could be true, but otherwise couldn't find much. Is there anywhere in the literature that addresses this sort of question? I'm also interested in similar results for $Sp_{2n}(\mathbb{Z}/p^\ell \mathbb{Z})$.

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    $\begingroup$ Look at this paper, (Theorem 1): arxiv.org/pdf/1202.4194.pdf $\endgroup$
    – MO B
    Commented Oct 24, 2018 at 2:15
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    $\begingroup$ Maybe you mean "some exceptions for small $n$ OR $p$"? (The minimal dimension for $n=2$ is $(p-1)/2$ for any large $p$.) $\endgroup$
    – user148212
    Commented Oct 24, 2018 at 5:48
  • $\begingroup$ @user148212 Ah good catch, the context where this came up for me has $n>2$, but you are absolutely correct. $\endgroup$
    – Nate
    Commented Oct 24, 2018 at 20:18

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For $n=2$ and $p$ odd:

The minimal possible dimension of a non-trivial irrep of $G_l=\mathrm{SL}_2(\mathbb{Z}/p^l)$ is always $(p-1)/2$, and the minimal possible dimension of a non-trivial primitive irrep of $G_l$ is $p^{l-2}(p^2-1)/2$.

This can be found in Shalika's paper Representation of the two by two unimodular group over local fields.

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