Measures on higher dimensional local fields and algebraic groups on them - van Urk, Wester (2020)
This 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))).