We relate the asymptotic representation theory of SL(d,Zp​) 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,Zp​) grows slower than n22, 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.Comment: preliminary version, comments are welcome. v2. Revised version, now
covering all semi simple group