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# Representations with Weighted Frames and Framed Parabolic Bundles

Published:2000-12-01
Printed: Dec 2000
• J. C. Hurtubise
• L. C. Jeffrey
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## Abstract

There is a well-known correspondence (due to Mehta and Seshadri in the unitary case, and extended by Bhosle and Ramanathan to other groups), between the symplectic variety $M_h$ of representations of the fundamental group of a punctured Riemann surface into a compact connected Lie group~$G$, with fixed conjugacy classes $h$ at the punctures, and a complex variety ${\cal M}_h$ of holomorphic bundles on the unpunctured surface with a parabolic structure at the puncture points. For $G = \SU(2)$, we build a symplectic variety $P$ of pairs (representations of the fundamental group into $G$, weighted frame'' at the puncture points), and a corresponding complex variety ${\cal P}$ of moduli of framed parabolic bundles'', which encompass respectively all of the spaces $M_h$, ${\cal M}_h$, in the sense that one can obtain $M_h$ from $P$ by symplectic reduction, and ${\cal M}_h$ from ${\cal P}$ by a complex quotient. This allows us to explain certain features of the toric geometry of the $\SU(2)$ moduli spaces discussed by Jeffrey and Weitsman, by giving the actual toric variety associated with their integrable system.
 MSC Classifications: 58F05 - unknown classification 58F0514D20 - Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13}

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