Fourier transform of the additive group in algebraically closed valued fields

Yimu Yin

Published 2009-03-06, updated 2014-03-23Version 3

We continue the study of the Hrushovski-Kazhdan integration theory and consider exponential integrals. The Grothendieck ring is enlarged via a tautological additive character and hence can receive such integrals. We then define the Fourier transform in our integration theory and establish some fundamental properties of it. Thereafter a basic theory of distributions is also developed. We construct the Weil representations in the end as an application. The results are completely parallel to the classical ones.