On Representation Zeta Functions for Special Linear Groups

Abstract We prove that the numbers of irreducible n-dimensional complex continuous representations of the special linear groups over p-adic integers grow slower than $n^{2}$. We deduce that the abscissas of convergence of the representation zeta functions of the special linear groups over the ring of integers are bounded above by 2. In order to show these results we prove also that if G is a connected, simply connected, semi-simple algebraic group defined over the field of rational numbers, then the G-representation variety of the fundamental group of a compact Riemann surface of genus n has rational singularities if and only if the G-character variety has rational singularities.