Measures on higher dimensional local fields and algebraic groups on themTools van Urk, Wester (2020) Measures on higher dimensional local fields and algebraic groups on them. MPhil thesis, University of Nottingham.
AbstractThis thesis is about defining finitely additive measures on sets. The prototype for what we’re doing is defining a R((X))-valued measure on a 2dimensional local field (such as Qp{{t}}). The thesis consists of three main parts. The first part consists of defining finitely additive measures and integration in relatively high degree of generality so that we can not only integrate over 2-dimensional local fields but also higher dimensional local fields, C((t)) and over algebraic groups. The second part consists of applying this theory to obtain a sequence of more refined measures µn on a 2-dimensional local field which allow us to define the Fourier transform intrinsically. The third and final part consists of applying the theory to coset measures on GL(2) and SL(2), including rigorously defining a local Hecke operator on GL(2,C((t))).
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