Realising The Smooth Representations Of GL2(O_F) Using p-adic Geometric Methods
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University of Cambridge
Department of Pure Mathematics and Mathematical Statistics
Department of Pure Mathematics and Mathematical Statistics
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Let p be an odd prime number and F be a finite extension of the p-adic numbers Q_p with valuation ring O_F and residue field k. In this thesis, we study the smooth representation theory of the profinite group GL_2(OO_F) using the theory of dagger analytic geometry developed by Grosse-Kl\"onne [GK00].
For a locally profinite group G that acts continuously on a smooth dagger space X we adapt the techniques of Ardakov and Wadsley from [AW24] in order to develop a theory of G-equivariant vector bundles with a flat connection on X. In the case that X is affinoid and connected, the de Rham cohomology groups of these vector bundles on X are a source of smooth representations of G. By applying this theory to GL_2(O_F)-stable (dagger) affinoid subdomains of the Drinfeld upper half plane, and related spaces, we are able to provide p-adic geometric realisations of several families of smooth representations of GL_2(O_F). For example, we reconstruct the cuspidal representations of GL_2(k) and provide a complete classification of the so-called principal split representations of GL_2(O_F).
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