## Abstract

Let G be a semisimple algebraic group defined over (Formula presented.) , and let (Formula presented.) be a compact open subgroup of (Formula presented.). We relate the asymptotic representation theory of (Formula presented.) 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 (Formula presented.) grows slower than (Formula presented.) , confirming a conjecture of Larsen and Lubotzky. In fact, we can take (Formula presented.). 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 (Formula presented.) has rational singularities. 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.

Original language | English (US) |
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Pages (from-to) | 245-316 |

Number of pages | 72 |

Journal | Inventiones Mathematicae |

Volume | 204 |

Issue number | 1 |

DOIs | |

State | Published - Apr 1 2016 |

### Funding

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. 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 sites 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.

## Keywords

- 14B05
- 14B07
- 14B07
- 53D30
- Primary 20F69
- Secondary 20G25

## ASJC Scopus subject areas

- General Mathematics