Persistent Link:
http://hdl.handle.net/10150/284342
Title:
Conjugacy classes, characters and coadjoint orbits of Diff⁺S¹
Author:
Dai, Jialing
Issue Date:
2000
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
The principal motivation of this dissertation is to understand the unitary irreducible representations and characters of Dif f⁺S¹-the group of all orientation-preserving diffeomorphisms of S¹ by studying conjugacy classes of Dif f⁺S¹ and its coadjoint orbits. For this purpose, we mainly focus on the following two topics. The first is to study the relation between a real Lie group G and its associated complex semigroup S(G), which was initiated by Oshansky. We consider two particular examples: (1) PSU(1.1) and PSL(2, C)⁺ (Chapter 1); (2) D and A (Chapter 2). We have shown that (a) The equivalence classes determined by the function q on PSL(2, C)⁺ (resp. A) are the same as the conjugacy classes in PSL(2, C)⁺ (resp. A ). In fact the restriction of q to PSL(2, C)⁺ equals the square of the "smaller" of the two eigenvalues of an element in PSL(2, C)+. (b) The fact that the representation of PSL(2, C)⁺ is of trace class makes the character of PSL(2, C)⁺ well-defined. Moreover the character of PSL(2, C)⁺ has analytic continuation onto PSU(1,1) except on a set of measure zero. Surprisingly, the extended characters are exactly Harish-Chandra global characters Xᵐ(PSU)₍₁.₁₎ =Θm. Secondly, we investigate the coadjoint orbits of Virasoro group-the central extension of Dif f⁺S¹ (Chapter 3), which has been considered before by Segal, Kirillov and (later) Witten. We improved Segal's result by parameterizing coadjoint orbits precisely in terms of following conjugacy classes in P͂S͂U͂(1,1): Par⁺₀,{n,n ∈ N},{Elln, n∈N},{Par⁽⁺/⁻⁾(n),n > 0}, {Hyp(n),n ≥ }0. We also completed Kirillov's list of representatives of coadjoint orbits, and we fleshed out the connection between Segal's and Kirillov's and Witten's work by giving the correspondence between conjugacy classes in P͂S͂U͂ (1,1) and the representatives of coadjoint orbits and stabilizers.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Mathematics.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
Pickrell, Doug

Full metadata record

DC FieldValue Language
dc.language.isoen_USen_US
dc.titleConjugacy classes, characters and coadjoint orbits of Diff⁺S¹en_US
dc.creatorDai, Jialingen_US
dc.contributor.authorDai, Jialingen_US
dc.date.issued2000en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractThe principal motivation of this dissertation is to understand the unitary irreducible representations and characters of Dif f⁺S¹-the group of all orientation-preserving diffeomorphisms of S¹ by studying conjugacy classes of Dif f⁺S¹ and its coadjoint orbits. For this purpose, we mainly focus on the following two topics. The first is to study the relation between a real Lie group G and its associated complex semigroup S(G), which was initiated by Oshansky. We consider two particular examples: (1) PSU(1.1) and PSL(2, C)⁺ (Chapter 1); (2) D and A (Chapter 2). We have shown that (a) The equivalence classes determined by the function q on PSL(2, C)⁺ (resp. A) are the same as the conjugacy classes in PSL(2, C)⁺ (resp. A ). In fact the restriction of q to PSL(2, C)⁺ equals the square of the "smaller" of the two eigenvalues of an element in PSL(2, C)+. (b) The fact that the representation of PSL(2, C)⁺ is of trace class makes the character of PSL(2, C)⁺ well-defined. Moreover the character of PSL(2, C)⁺ has analytic continuation onto PSU(1,1) except on a set of measure zero. Surprisingly, the extended characters are exactly Harish-Chandra global characters Xᵐ(PSU)₍₁.₁₎ =Θm. Secondly, we investigate the coadjoint orbits of Virasoro group-the central extension of Dif f⁺S¹ (Chapter 3), which has been considered before by Segal, Kirillov and (later) Witten. We improved Segal's result by parameterizing coadjoint orbits precisely in terms of following conjugacy classes in P͂S͂U͂(1,1): Par⁺₀,{n,n ∈ N},{Elln, n∈N},{Par⁽⁺/⁻⁾(n),n > 0}, {Hyp(n),n ≥ }0. We also completed Kirillov's list of representatives of coadjoint orbits, and we fleshed out the connection between Segal's and Kirillov's and Witten's work by giving the correspondence between conjugacy classes in P͂S͂U͂ (1,1) and the representatives of coadjoint orbits and stabilizers.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectMathematics.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorPickrell, Dougen_US
dc.identifier.proquest9972080en_US
dc.identifier.bibrecord.b40638273en_US
All Items in UA Campus Repository are protected by copyright, with all rights reserved, unless otherwise indicated.