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Integration on valuation fields over local fields. (English) Zbl 1203.11081

Let \(F\) be a valuation field with valuation group \(\Gamma\) and integers \({\mathcal O}_F\) whose residue field \(\overline{F}\) is a non-discrete, locally compact field (i.e. a local field: \({\mathbb R},{\mathbb C}\) or non-archimedean). Given a Haar integrable function \(f:\overline{F}\rightarrow{\mathbb C}\), consider the lift \(f^{0,0}\) of \(f\) to \({\mathcal O}_F\) together with its scalings \(f^{0,0}(\alpha x+a), a\in F, \alpha\in F^{\times}\). An integral is defined on the space spanned by such \(f^{0,0}\), with values in \({\mathbb C}\Gamma\) (the complex group algebra of \(\Gamma\)). The paper proceeds to develop some harmonic analysis for fields that are self-dual in a suitable sense.
The main results generalise results of I. Fesenko [Doc. Math., J. DMV Extra Vol., 261–284 (2003; Zbl 1130.11335); Proceedings of the St. Petersburg Mathematical Society. Vol. XII. Transl. from the Russian. Providence, RI: American Mathematical Society (AMS). Translations. Series 2. American Mathematical Society 219, 149–165 (2006; Zbl 1203.11080)]. The final sections of the paper are devoted to zeta integrals, including interesting connections with quantum physics.

MSC:

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
11S40 Zeta functions and \(L\)-functions
12J10 Valued fields
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References:

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