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" Twisted Conjugation, Quasi-Hamiltonian Geometry, and Duistermaat-Heckman Measures "
Zerouali, Ahmed Jihad
Meinrenken, Eckhard
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Latin Dissertation
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English
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| Record Number
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1105771
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TLpq2323127826
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| Main Entry
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Meinrenken, Eckhard
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Zerouali, Ahmed Jihad
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| Title & Author
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Twisted Conjugation, Quasi-Hamiltonian Geometry, and Duistermaat-Heckman Measures\ Zerouali, Ahmed JihadMeinrenken, Eckhard
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| College
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University of Toronto (Canada)
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| Date
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2019
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2019
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| Degree
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Ph.D.
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| Page No
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101
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| Abstract
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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).
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Mathematics
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