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Title: Representations of topological groups and Kac-Moody groups
Author: Hristova, Katerina
ISNI:       0000 0004 8505 2837
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2019
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This thesis studies continuous representations of topological groups over a field F. We form three categories of continuous representations of such a group G: the category of discrete representations MdpGq, category of linearly topologized complete representations MltcpGq and the category of (linearly) compact representations McpGq. We obtain a version of Frobenius reciprocity in these categories. We then study categories MApGq and MA;_pGq of discrete A-semisimple smooth representations of a locally compact totally disconnected group G, and A-semisimple smooth representations of G coming from a fixed character X, where A ¤ G is a closed central subgroup. We establish an equivalence between the categories above and certain subcategories of the category of smooth modules over the Hecke algebra of G. Our main results encompass an upper bound on the projective dimension of MApGq and MA;_pGq, as well as construction of explicit projective resolutions of objects in these categories, in the case when G acts continuously on a simplicial set X with a contractible geometric realisation |X|. We also study categories of G-equivariant sheaves and cosheaves on the simplicial set X. We prove a localisation result relating those to the category of smooth representations. We also prove existence of finitely generated projective resolutions of Schneider-Stuhler type when |X| is a tree. We finish with an investigation of complete locally compact totally disconnected Kac-Moody groups. We define a simplicial complex with a contractible geometric realisation on which they act. We conclude with a study of cocompact lattices in locally pro-p-complete Kac-Moody groups.
Supervisor: Not available Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics