Measure and integration on non-locally compact spaces and representation theory

Waller, Raven (2019) Measure and integration on non-locally compact spaces and representation theory. PhD thesis, University of Nottingham.

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Abstract

The problem of studying representations of algebraic groups over higher dimensional local fields has already attracted a lot of attention. While many of the advances in this area involve very technical machinery and produce highly abstract results, in this thesis we approach this problem from a much more explicit angle, where we encounter phenomena which do not present themselves in the earlier approaches.

Along the way, we come to several developments which are interesting in their own right. These include a generalisation of the "higher measure" of Fesenko, and the study of topological spaces which are "not too far" from being (locally) compact.

By following Fesenko’s explicit approach to defining a shift-invariant measure on a higher-dimensional local field F, we construct such a measure instead on the general linear group GL2(F). The integral against this measure coincides with an integral previously defined by Morrow using lifts from the residue field, but its explicit nature makes the measure constructed here rather versatile.

Motivated by several patterns that appear while constructing the higher measures, we then study a new class of objects consisting of a group together with a topological framework called a level structure. These groups not only generalise the groups F and GL2(F) associated to higher-dimensional local fields, but several standard topological concepts as well. In particular, the level structure gives rise to a very natural generalisation of compactness.

Following the more general digression, we then return to the concrete example of GL2(F), using the new tools. We use the level structure to study its principal series representations, and unlike in previous, very technical incarnations of this theory, our methodology makes this much more simple.

Item Type: Thesis (University of Nottingham only) (PhD)
Supervisors: Fesenko, Ivan
Keywords: Higher-dimensional local fields; Representation theory
Subjects: Q Science > QA Mathematics
Faculties/Schools: UK Campuses > Faculty of Science > School of Mathematical Sciences
Item ID: 55905
Depositing User: Waller, Matthew
Date Deposited: 19 Jul 2019 13:11
Last Modified: 19 Jul 2019 13:11
URI: http://eprints.nottingham.ac.uk/id/eprint/55905

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