Non-Commutative Kirillov Method and Coadjoint Orbits

Gou, Tianpeng

Non-Commutative Kirillov Method and Coadjoint Orbits

Gou, Tianpeng

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Publication Type:: Thesis
Issue Date:: 2024

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Field	Value	Language
dc.contributor.advisor	au.edu.uts.lib/cph
dc.contributor.author	Gou, Tianpeng
dc.date.accessioned	2026-03-23T02:29:34Z
dc.date.available	2026-03-23T02:29:34Z
dc.date.issued	2024
dc.identifier.uri	http://hdl.handle.net/10453/194414
dc.description	University of Technology Sydney. Faculty of Science.	en_US.UTF-8
dc.description.abstract	Elie Cartan and Hermann Weyl’s theorem of highest weight states that every irreducible highest weight representation of a compact real form of a complex semisimple Lie algebra, integrates to an irreducible highest weight representation of a compact, simply connected, semisimple Lie group. However, there is no explicit computation method for this correspondence. Based on the work of Raed Raffoul, we combine the Kirillov orbit method, the sum of adjoint orbits, the convexity theorem for moment maps, Nelson’s formula for Weyl calculus, and the transversality condition and induced differential operators, to develop a non-commutative Kirillov method. This framework allows us to explicitly compute the exponential of irreducible highest weight representations of compact semisimple Lie algebras. It also enables the lift of invariant vector fields, arising from root vectors in the complex semisimple Lie algebras, to differential operators induced by the adjoint action of the group, with respect to the lift of matrix coefficients. Moreover, we show that the Euclidean Fourier transform of the lift of highest weight representations consists of polynomials of transversal differential operators acting on an invariant measure supported on the image of the moment map of the representation, in the dual Lie algebra.	en_US.UTF-8
dc.format	Thesis (PhD)
dc.language.iso	en_US	en_US.UTF-8
dc.relation	https://opus.lib.uts.edu.au/bitstream/10453/194414/1/thesis.pdf
dc.rights	info:eu-repo/semantics/openAccess
dc.rights	The author owns the copyright in this thesis including all reproduction and reuse rights for the work. The work may not be altered without the permission of the copyright owner. Attribution is essential when quoting or paraphrasing from this thesis.
dc.rights	© 2024 Tianpeng Gou
dc.rights	au.edu.uts.lib/cph
dc.title	Non-Commutative Kirillov Method and Coadjoint Orbits	en_US.UTF-8
dc.type	Thesis
utslib.copyright.status	open_access	*

Abstract:

Elie Cartan and Hermann Weyl’s theorem of highest weight states that every irreducible highest weight representation of a compact real form of a complex semisimple Lie algebra, integrates to an irreducible highest weight representation of a compact, simply connected, semisimple Lie group. However, there is no explicit computation method for this correspondence. Based on the work of Raed Raffoul, we combine the Kirillov orbit method, the sum of adjoint orbits, the convexity theorem for moment maps, Nelson’s formula for Weyl calculus, and the transversality condition and induced differential operators, to develop a non-commutative Kirillov method. This framework allows us to explicitly compute the exponential of irreducible highest weight representations of compact semisimple Lie algebras. It also enables the lift of invariant vector fields, arising from root vectors in the complex semisimple Lie algebras, to differential operators induced by the adjoint action of the group, with respect to the lift of matrix coefficients. Moreover, we show that the Euclidean Fourier transform of the lift of highest weight representations consists of polynomials of transversal differential operators acting on an invariant measure supported on the image of the moment map of the representation, in the dual Lie algebra.

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http://hdl.handle.net/10453/194414