Submission date: 13 March 2009.

Abstract:

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 (without differential operators) is also developed. We construct the Weil representation in the end as an application. The results are completely parallel to the classical ones.

Mathematics Subject Classification: 03C60, 11S80, 20C08

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