## Departmental PhD Thesis Exam – Jack Ding

Thursday, August 19, 2021
10:00 a.m.

PhD Candidate: Jack Ding
Supervisor: Lisa Jeffrey
Thesis title:  The Atiyah-Bott Lefschetz formula applied to the based loops on
SU(2)

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We prove two generalizations of localization formulae for finite-dimensional spaces to the infinite-dimensional based loop group $\Omega G$.

The Atiyah-Bott-Lefschetz Formula is a well-known formula for computing the equivariant index of an elliptic operator on a compact smooth manifold. We provide an analogue of this formula for the based loop group $\Omega SU(2)$ with respect to the natural $(T \times S^1)$-action. This is accomplished by computing certain equivariant multiplicities in the K-theory of affine Schubert varieties. From this result we also derive an effective formula for computing characters of certain Demazure modules.

The based loop group for a compact Lie group $G$ has been studied intensively since the work of Atiyah and Pressley and the book of Pressley and Segal. It is an infinite-dimensional symplectic manifold equipped with a Hamiltonian torus action, where the torus is the product of a circle and the maximal torus of $G$. When $G = SU(2)$,
the fixed points for this action are in bijective correspondence with the integers. Our final result is a Duistermaat-Heckman type oscillatory integral over the based loop group, expanded around the fixed points of the torus action. To accomplish this we use Frenkel’s results (1984) on pinned Wiener measure for orbital integrals on the affine Lie algebra, as well as the results of Urakawa (1975) on the heat kernel for a compact Lie group and Fegan’s inversion formula (1978) for orbital integrals.

A copy of the thesis can be found here: thesis