# Representation growth and rational singularities of the moduli space of local systems

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## Abstract

*G*be a semisimple algebraic group defined over \(\mathbb {Q}_p\), and let \(\Gamma \) be a compact open subgroup of \(G(\mathbb {Q}_p)\). We relate the asymptotic representation theory of \(\Gamma \) and the singularities of the moduli space of

*G*-local systems on a smooth projective curve, proving new theorems about both:

- (1)
We prove that there is a constant

*C*, independent of*G*, such that the number of*n*-dimensional representations of \(\Gamma \) grows slower than \(n^{C}\), confirming a conjecture of Larsen and Lubotzky. In fact, we can take \(C=3\cdot {{\text {dim}}}(E_8)+1=745\). We also prove the same bounds for groups over local fields of large enough characteristic. - (2)
We prove that the coarse moduli space of

*G*-local systems on a smooth projective curve of genus at least \(\lceil C/2\rceil +1=374\) has rational singularities.

## Mathematics Subject Classification

Primary 20F69 14B05 Secondary 20G25 14B07 14B07 53D30## Notes

### Acknowledgments

We thank Karl Schwede, Angelo Vistoli, Sandor Kovacs, and Laurent Moret-Baily for answering questions related to rational singularities and algebraic geometry on MathOverflow as well as the site’s administrators for the platform. We thank Vladimir Hinich for answering many questions about Grothendieck duality and rational singularities. We benefitted from conversations with Joseph Bernstein, Roman Bezrukavnikov, Alexander Braverman, Vladimir Drinfeld, Pavel Etingoff, Victor Ginzburg, David Kazhdan, Michael Larsen, Alex Lubotzky, Tony Pantev, and Yakov Varshavsky. We thank them all. A.A. was partially supported by NSF grant DMS-1100943 and ISF grant 687/13; N.A. was partially supported by NSF grants DMS-0901638 and DMS-1303205. Both authors were also partially supported by BSF grant 2012247.

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