Abstract
Let G be a semisimple compact connected Lie group. An N-fold reduced product of G is the symplectic quotient of the Hamiltonian system of the Cartesian product of N coadjoint orbits of G under diagonal coadjoint action of G. Under appropriate assumptions, it is a symplectic orbifold. Using the technique of nonabelian localization and the residue formula of Jeffrey and Kirwan, we investigate the symplectic volume of an N-fold reduced product of G. Suzuki and Takakura gave a volume formula for the N-fold reduced product of \( \mathbf {SU}(3) \) in [25] by using geometric quantization and the Riemann–Roch formula. We compare our volume formula with theirs and prove that our volume formula agrees with theirs in the case of triple reduced products of \( \mathbf {SU}(3) \).
Résumé
Soit G un groupe de Lie compact connexe semi-simple. Un produit réduit de G (N fois) est la réduction sympectique du système Hamiltonien du produit de N orbites coadjointes de G sous l’action coadjointe diagonale de G. Vu certaines hypothèses, c’est un orbi-espace symplectique. Nous utilisons la localisation non-abélienne et la formule de résidus de Jeffrey et Kirwan pour étudier le volume symplectique du produit réduit de SU(3). Suzuki et Takakura ont donné une formule pour le volume du produit réduit (N-fois) de \(\mathbf {SU}(3)\) dans [25] en se servant de la quantification géométrique et la formule de Riemann–Roch. Nous considérons leur formule par rapport à la notre. Nous démontrons que notre formule accorde avec leur formule dans le cas \(N=3\).
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L. Jeffrey is partially supported by an NSERC Discovery Grant. J. Ji is partially supported by the University of Toronto.
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Jeffrey, L., Ji, J. Volume formula for N-fold reduced products. Ann. Math. Québec 47, 263–294 (2023). https://doi.org/10.1007/s40316-021-00171-9
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DOI: https://doi.org/10.1007/s40316-021-00171-9