Similarity classes of integral $p$-adic matrices and representation zeta functions of groups of type $A_2$

Abstract

We compute explicitly Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of various $p$-adic analytic and adelic profinite groups of type $\mathsf{A}_2$. This has consequences for the representation zeta functions of arithmetic groups $Γ\subset \mathbf{H}(k)$, where $k$ is a number field and $\mathbf{H}$ a $k$-form of $\mathsf{SL}_3$: assuming that $Γ$ possesses the strong Congruence Subgroup Property, we obtain precise, uniform estimates for the representation growth of $Γ$. Our results are based on explicit, uniform formulae for the representation zeta functions of the $p$-adic analytic groups $\mathsf{SL}_3(\mathfrak{o})$ and $\mathsf{SU}_3(\mathfrak{o})$, where $\mathfrak{o}$ is a compact discrete valuation ring of characteristic $0$. These formulae build on our classification of similarity classes of integral $\mathfrak{p}$-adic $3\times3$ matrices in $\mathfrak{gl}_3(\mathfrak{o})$ and $\mathfrak{gu}_3(\mathfrak{o})$, where $\mathfrak{o}$ is a compact discrete valuation ring of arbitrary characteristic. Organising the similarity classes by invariants which we call their shadows allows us to combine the Kirillov orbit method with Clifford theory to obtain explicit formulae for representation zeta functions. In a different direction we introduce and compute certain similarity class zeta functions. Our methods also yield formulae for representation zeta functions of various finite subquotients of groups of the form $\mathsf{SL}_3(\mathfrak{o})$, $\mathsf{SU}_3(\mathfrak{o})$, $\mathsf{GL}_3(\mathfrak{o})$, and $\mathsf{GU}_3(\mathfrak{o})$, arising from the respective congruence filtrations; these formulae are valid in case that the characteristic of $\mathfrak{o}$ is either $0$ or sufficiently large. Analysis of some of these formulae leads us to observe $p$-adic analogues of `Ennola duality'.