Quantization of Coadjoint Orbits via Positivity of Kirillov's Character Formula

Open Access
Author:
Khanmohammadi, Ehssan
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
August 17, 2015
Committee Members:
  • Nigel David Higson, Dissertation Advisor
  • Nigel David Higson, Committee Chair
  • Nathanial Patrick Brown, Committee Member
  • Ping Xu, Committee Member
  • Murat Gunaydin, Committee Member
Keywords:
  • Kirillov's Character Formula
  • Positivity
  • The orbit method
  • Quantization
Abstract:
Kirillov proved his character formula for simply connected nilpotent Lie groups in 1962 and conjectured its universality. The validity of this conjecture has been verified for some other classes of Lie groups, most notably for the case of tempered representations of reductive Lie groups by Rossmann. In this dissertation we explain how Kirillov's character formula can be used in the quantization of coadjoint orbits. First we prove a positivity property of Kirillov's character formula for some classes of Lie groups, including nilpotent Lie groups, which possess real polarizing subgroups. Then we use this positivity property to construct group representations following the ideas of Gelfand, Naimark, and Segal. Finally we discuss several approaches to proving positivity in the absence of real polarizations.