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The Atiyah-Bott Lefschetz Formula Applied to the Based Loops on SU(2)

C. R. Math. Rep. Acad. Sci. Canada Vol. 42 (3) 2020, pp. 42-62
Vol.42 (3) 2020
Jack Ding Details
(Received: 2020-07-23 , Revised: 2020-10-01 )
(Received: 2020-07-23 , Revised: 2020-10-01 )

Jack Ding, Department of Mathematics, University of Toronto, Toronto, Canada M5S 2E4; e-mail: jding@math.toronto.edu

Abstract/Résumé:

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. From this result we also derive an effective formula for computing characters of certain Demazure modules.

La formule d’Atiyah-Bott-Lefschetz est une formule bien connue pour l’indice équivariante d’un opérateur elliptique sur une variété lisse compacte. Nous donnons une analogue de cette formule pour le groupe de lacets basés \(\Omega SU(2)\) par rapport à l’action naturelle de \(T \times S^1\). Avec ce résultat nous démontrons aussi une formule effective pour les caractères de certains modules de Demazure.

Keywords: Atiyah-Bott, Loop groups, character formula, fixed point theorem, localization

AMS Subject Classification: Infinite-dimensional Lie groups and their Lie algebras, Geometric quantization 22E65, 53D50

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