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$\DeclareMathOperator\SO{SO}$Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $\mathfrak{o}$ denote the ring of integers, with maximal ideal $\mathfrak{p}$. Let $G_l$ denote the finite group $\SO_3(\mathfrak{o}/\mathfrak{p}^l)$.

Question: Is there a formula for the number of irreducible representations of $G_l$ in a given dimension $d$?

Remarks: 1. I am aware of the results of Aizenbud–Avni - Representation growth and rational singularities of the moduli space of local systems (which gives an estimate $C d^{22}$ for the number I'm interested in, over $\mathfrak{o}$) and of Jaikin-Zapirain - Zeta function of representations of compact $p$-adic analytic groups (which gives a qualitative result on the representation zeta function). Both hold for more general algebraic groups. My question is whether there exists a quantitative, explicit (possibly recursive in $l$) formula in the special case of $\SO_3$.

  1. For $\operatorname{GL}_2$ in place of $\SO_3$, such a formula can be found in Onn - Representations of automorphism groups of finite O-modules of rank two (Thm. 1.4).

Remark: This question was originally posed for $\SO_2$. As Paul Broussous pointed out, it is trivial in this case since $\SO_2$ is abelian.

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  • $\begingroup$ ${\rm SO}_2$ is an abelian algebraic group, isn't it ? So is ${\rm SO}_2 ({\mathfrak o}/{\mathfrak p}^l )$ and this group is isomorphic to its dual ... $\endgroup$ – Paul Broussous Sep 7 at 15:44
  • $\begingroup$ Oops you are of course right. I reposted the question for $SO_3$. Shame on me for not noticing this. (I am eventually interested in higher $SO_n$.) $\endgroup$ – nikola karabatic Sep 9 at 10:02
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As far as I know, there is currently no such explicit formula in the literature. In fact, even for $\mathrm{SO}_3(\mathbb{F}_q)$ (i.e., the case $l=1$), I have not seen a neat table of the dimensions and multiplicities of all irreps, although this could certainly be obtained from the work of Lusztig, Irreducible representations of finite classical groups, Invent. Math. 43, 125-175 (1977).

For some groups of type A of low rank over $\mathfrak{o}_l$, the representation zeta function has been determined explicitly, as long as the residue characteristic of $\mathfrak{o}$ is not too small: Avni, Klopsch, Onn, Voll, Similarity classes of integral (\mathfrak p)-adic matrices and representation zeta functions of groups of type (\mathsf{A}_{2}), Proc. Lond. Math. Soc. (3) 112, No. 2, 267-350 (2016); see Theorem C.

In principle, one can extract the number of irreps of dim $n$ as the $n$-th coefficient in the Dirichlet series expansion of the representation zeta function, but in practice this can be difficult.

Note that the bound $Cd^{22}$ of Aizenbud--Avni is an upper bound on the number of irreps of dimension at most $d$, so it's not exactly the number you are asking about. Also, for particular types, this bound can be made stronger. In any case, an upper bound is something rather different than an exact formula.

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