69 research outputs found
The wave front set of the Fourier transform of algebraic measures
We study the Fourier transform of the absolute value of a polynomial on a
finite-dimensional vector space over a local field of characteristic 0. We
prove that this transform is smooth on an open dense set.
We prove this result for the Archimedean and the non-Archimedean case in a
uniform way. The Archimedean case was proved in [Ber]. The non-Archimedean case
was proved in [HK] and [CL]. Our method is different from those described in
[Ber,HK,CL]. It is based on Hironaka's desingularization theorem, unlike [Ber]
which is based on the theory of D-modules and [HK,CL] which is based on model
theory.
Our method also gives bounds on the open dense set where the Fourier
transform is smooth. These bounds are explicit in terms of resolution of
singularities.
We also prove the same result on the Fourier transform of other measures of
algebraic origins.Comment: 40 page
Representation Growth and Rational Singularities of the Moduli Space of Local Systems
We relate the asymptotic representation theory of and
the singularities of the moduli space of -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
grows slower than , confirming a conjecture of
Larsen and Lubotzky. Regarding the latter, we prove that the moduli space of
-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
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