$\DeclareMathOperator\SO{SO}$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 $\SO_3$.

- 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.