Representation Growth and Rational Singularities of the Moduli Space of Local Systems

TL;DR

This paper connects the growth of representations of $SL(d,\mathbb{Z}_p)$ with the singularities of the moduli space of $SL(d)$-local systems, proving new bounds and singularity properties using algebraic and analytic techniques.

Contribution

It proves the growth rate of $SL(d,\mathbb{Z}_p)$ representations confirms a conjecture, and establishes rational singularities of the moduli space for genus at least 12.

Findings

Representation growth slower than $n^{22}$ for all d

Moduli space of $SL(d)$-local systems has rational singularities for genus ≥ 12

Push forwards of smooth measures have continuous density under certain conditions

Abstract

We relate the asymptotic representation theory of $SL(d,\mathbb{Z}_p)$ and the singularities of the moduli space of $SL(d)$-local systems on a smooth projective curve, proving new theorems about both. Regarding the former, we prove that, for every d, the number of n-dimensional representations of $SL(d,\mathbb{Z}_p)$ grows slower than $n^{22}$, confirming a conjecture of Larsen and Lubotzky. Regarding the latter, we prove that the moduli space of $SL(d)$-local systems on a smooth projective curve of genus at least 12 has rational singularities. Most of our results apply more generally to semi-simple algebraic groups. For the proof, we study the analytic properties of push forwards of smooth measures under algebraic maps. More precisely, we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities.