### 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) |
---|---|

Pages (from-to) | 245-316 |

Number of pages | 72 |

Journal | Inventiones Mathematicae |

Volume | 204 |

Issue number | 1 |

DOIs | |

State | Published - Apr 1 2016 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Inventiones Mathematicae*,

*204*(1), 245-316. https://doi.org/10.1007/s00222-015-0614-8

}

*Inventiones Mathematicae*, vol. 204, no. 1, pp. 245-316. https://doi.org/10.1007/s00222-015-0614-8

**Representation growth and rational singularities of the moduli space of local systems.** / Aizenbud, Avraham; Avni, Nir.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Representation growth and rational singularities of the moduli space of local systems

AU - Aizenbud, Avraham

AU - Avni, Nir

PY - 2016/4/1

Y1 - 2016/4/1

N2 - 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.

AB - 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.

KW - 14B05

KW - 14B07

KW - 14B07

KW - 53D30

KW - Primary 20F69

KW - Secondary 20G25

UR - http://www.scopus.com/inward/record.url?scp=84939627839&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84939627839&partnerID=8YFLogxK

U2 - 10.1007/s00222-015-0614-8

DO - 10.1007/s00222-015-0614-8

M3 - Article

VL - 204

SP - 245

EP - 316

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 1

ER -