Year: 2012
Type: dissertation
Abstract: This thesis contains a survey of homotopical group theory, integral and p-adic reflection group theory, and sets the stage to study characters of p-compact groups.Chapter 1 motivates and introduces p-compact groups as defined by Dwyer and Wilkerson [22] in their original setting, as homotopical analogs for compact Lie groups.In particular it explores how the structure of the maximal torus and the Weyl group of a connected compact Lie group pass to the corresponding structures in a finite loop space, and that these structures then pass via HF p -localization to the defined structures on the associated p-compact group.Chapter 2 is concerned with the connections between p-adic reflection groups and p-compact groups.It considers the topological realization of reflection groups over the p-adic integers, and goes through the construction of non-modular p-compact groups, as motivated by the spaces created by Clark and Ewing [18].It then explores the functorial connection between these reflection groups and p-compact groups.Chapters 3 and 4 look at studying character theory of compact connected Lie groups via equivariant K-theory and postulate that one can use K-theory with padic coefficients as studied by Jeanneret and Osse [32], and the stable transfer map of Bauer [10] to extend this approach to study characters of p-compact groups.In the end, we conjecture a Weyl formula for characters induced up from the maximal torus.
Author: James C. Peyton Jones
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2346573770