## Departmental PhD Thesis Exam – Jihad Zerouali

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Friday, July 5, 2019
11:10 a.m.
BA6183

PhD Candidate:  Jihad Zerouali
Supervisor:   Eckhard Meinrenken
Thesis title:  Twisted conjugation, quasi-Hamiltonian geometry, and Duistermaat-Heckman measures

***

Let $G$ be a Lie group, and let $\kappa\in\mathrm{Aut}(G)$. Let  $G\kappa$ denote the group $G$ equipped with the $\kappa$-twisted conjugation action, $\mathrm{Ad}_{g}^{\kappa}(h)=gh\kappa(g^{-1})$. A twisted quasi-Hamiltonian manifold is a triple $(M,\omega,\Phi)$, where $M$ is a $G$-space, the equivariant map $\Phi:M\to G\kappa$ is called the moment map, and $\omega$ 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 $\kappa$-twisted conjugation, for $G$ compact, connected, simply connected and simple, and for $\kappa$ induced by a Dynkin diagram automorphism of $G$. After recovering the classification of $\kappa$-twisted conjugacy classes by elementary means, we highlight several properties of the so-called \textit{twining characters} $\tilde{\chi}^{(\kappa)}:G\rightarrow\mathbb{C}$.

We show that as elements of $L^{2}(G\kappa)^{G}$, the twining characters generalize several properties of the usual characters in a natural way. We then discuss $\kappa$-twisted representation and fusion rings, in relation to recent work of J. Hong. The second topic of this work is the study of the Duistermaat-Heckman (DH) measure $\mathrm{DH}_{\Phi}\in\mathcal{D}'(G\kappa)^{G}$ of a twisted quasi-Hamiltonian manifold $(M,\omega,\Phi)$. After developing the necessary background, we prove a localization formula for the Fourier coefficients of the measure $\mathrm{DH}_{\Phi}$, 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 $G\rtimes\mathrm{Aut}(G)$ instead of $G$.

A copy of the thesis can be found here:

Thesis_AJ_Zerouali_Draft_V20190702