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

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.