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 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.
Cite
@article{arxiv.1706.05525,
title = {On representation zeta functions for special linear groups},
author = {Nero Budur and Michele Zordan},
journal= {arXiv preprint arXiv:1706.05525},
year = {2018}
}
Comments
ERRATUM added: a mistake in the p-adic group theoretic argument from Lemma 4.3 invalidates the proof of the main results. In a new preprint arXiv:1809.05180, Corollary 1.3 is proved by geometric methods, so that all results in the Introduction are still true, except Proposition 1.5 which remains open