Twisted Conjugation, Quasi-Hamiltonian Geometry, and Duistermaat-Heckman Measures
General Material Designation
[Thesis]
First Statement of Responsibility
Zerouali, Ahmed Jihad
Subsequent Statement of Responsibility
Meinrenken, Eckhard
.PUBLICATION, DISTRIBUTION, ETC
Name of Publisher, Distributor, etc.
University of Toronto (Canada)
Date of Publication, Distribution, etc.
2019
PHYSICAL DESCRIPTION
Specific Material Designation and Extent of Item
101
DISSERTATION (THESIS) NOTE
Dissertation or thesis details and type of degree
Ph.D.
Body granting the degree
University of Toronto (Canada)
Text preceding or following the note
2019
SUMMARY OR ABSTRACT
Text of Note
Let usdGusd be a Lie group, and let usd\kappa\in\mathrm{Aut}(G)usd. Let usdG\kappausd denote the group usdGusd equipped with the usd\kappausd-twisted conjugation action, usd\mathrm{Ad}_{g}^{\kappa}(h)=gh\kappa(g^{-1})usd. A twisted quasi-Hamiltonian manifold is a triple (M,\omega,\Phi)usd, where usdMusd is a usdGusd-space, the equivariant map usd\Phi:M\to G\kappausd is called the moment map, and usd\omegausd is a certain invariant 2-form with properties generalizing those of a symplectic structure. The first topic of this work is a detailed study of usd\kappausd-twisted conjugation, for usdGusd compact, connected, simply connected and simple, and for usd\kappausd induced by a Dynkin diagram automorphism of usdGusd. After recovering the classification of usd\kappausd-twisted conjugacy classes by elementary means, we highlight several properties of the so-called \textit{twining characters} usd\tilde{\chi}^{(\kappa)}:G\rightarrow\mathbb{C}usd. We show that as elements of usdL^{2}(G\kappa)^{G}usd, the twining characters generalize several properties of the usual characters in a natural way. We then discuss usd\kappausd-twisted representation and fusion rings, in relation to recent work of J. Hong. This discussion is taken from the preprint "Twisted conjugation on simply connected Lie groups and twining characters" (Ahmed J. Zerouali, arXiv:1811.06507, 2018), and is presented in Chapter 2 of this thesis. The second topic of this work is the study of the Duistermaat-Heckman (DH) measure \mathrm{DH}_{\Phi}\in\mathcal{D}'(G\kappa)^{G}usd of a twisted quasi-Hamiltonian manifold usd(M,\omega,\Phi)usd. After developing the necessary background, we prove a localization formula for the Fourier coefficients of the measure usd\mathrm{DH}_{\Phi}usd, and we illustrate the theory with several examples of twisted moduli spaces. These character varieties parametrize a class of local systems on bordered surfaces, for which the transition functions take values in usdG\rtimes\mathrm{Aut}(G)usd instead of usdGusd. This material is covered in Chapters 3 and 4, which constitute an expanded version of the preprint "Twisted moduli spaces and Duistermaat-Heckman measures" (Ahmed J. Zerouali, in preparation, 2018).