# On representation zeta functions for special linear groups

## TL;DR

This paper investigates the growth of representation counts for special linear groups over p-adic integers, establishing bounds on their zeta functions' convergence and relating algebraic properties of G-representation varieties.

## Contribution

It proves bounds on the growth of irreducible representations and the convergence of their zeta functions, linking algebraic geometry and representation theory for special linear groups.

## Key findings

- Representation growth is slower than quadratic in n.
- Abscissas of convergence are bounded above by 2.
- G-representation varieties have rational singularities under certain conditions.

## 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 the square of n. 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.