PUB - Publikationen an der Universität Bielefeld

We compute explicitly Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of various p-adic analytic and ad` elic profinite groups of type A(2). This has consequences for the representation zeta functions of arithmetic groups Gamma subset of H(k), where k is a number field and H is a k-form of SL3: assuming that Gamma possesses the strong congruence subgroup property, we obtain precise, uniform estimates for the representation growth of Gamma. Our results are based on explicit, uniform formulae for the representation zeta functions of the p-adic analytic groups SL3(o) and SU3(o), where o is a compact discrete valuation ring of characteristic 0. These formulae build on our classification of similarity classes of integral p-adic 3 x 3 matrices in gl(3)(o) and gu(3)(o), where 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 SL3(o), SU3(o), GL(3)(o), and GU(3)(o), arising from the respective congruence filtrations; these formulae are valid in case that the characteristic of o is either 0 or sufficiently large. Analysis of some of these formulae leads us to observe p-adic analogues of 'Ennola duality'.