W&M ScholarWorks Undergraduate Honors Theses Theses, Dissertations, & Master Projects 5-2019 Minimal Principal Series Representations of SL(3,R) Jacopo Gliozzi Follow this and additional works at: https://scholarworks.wm.edu/honorstheses Part of the Analysis Commons, Geometry and Topology Commons, and the Harmonic Analysis and Representation Commons Recommended Citation Gliozzi, Jacopo, "Minimal Principal Series Representations of SL(3,R)" (2019). Undergraduate Honors Theses. Paper 1308. https://scholarworks.wm.edu/honorstheses/1308 This Honors Thesis is brought to you for free and open access by the Theses, Dissertations, & Master Projects at W&M ScholarWorks. It has been accepted for inclusion in Undergraduate Honors Theses by an authorized administrator of W&M ScholarWorks. For more information, please contact
[email protected]. Contents Acknowledgments ii Abstract v 1 Introduction 1 2 Background 5 3 Induced Representations 11 3.1 FiberBundles.......................... 11 3.2 Induced Representations on Vector Bundles . 13 3.3 Density Bundles . 16 3.4 NormalizedInduction. 20 4 Principal Series 23 4.1 Structure Theory . 23 4.2 Parabolic Induction . 29 4.3 Principal Series of SL(3, R)................... 35 5 K-Types 43 5.1 K-Type Decomposition . 43 5.2 K-Type Multiplicities . 45 6 Conclusion 54 i Acknowledgments Iwouldliketothankfirstandforemostmyadvisor,ProfessorPierreClare,forhis endless patience and guidance throughout this project. Not only was he always willing to help me on a moment’s notice, he also expanded my horizons from a purely physics- oriented point of view into the realm of math for its own sake. I cannot overstate how instrumental he was to my growth in understanding and attempting to do math.