Merits and demerits of the orbit method. (English) Zbl 0940.22013
The paper under review is an excellent survey of the orbit method in representation theory written by the father of this method. This paper is the expanded version of the author’s talk at the AMS meeting in 1997. As the author says, he explains to non-experts how to use the orbit method, discusses its strong and weak points and advertises some open problems. Not only beginners but experts too can find plenty of interesting and useful things in this paper.
The orbit method is based on that there is a connection between representations of Lie groups and orbits of their coadjoint representations. An ideal situation for this method is nilpotent Lie groups. In this case there is a one-to-one correspondence between unitary irreducible representations and coadjoint orbits, and the orbit method gives an adequate and transparent language to answer main questions of representation theory: to decompose restrictions of representations and induced representations, to determine the topological structure of the unitary dual, to write formulae for characters etc. Understood in a wider sense, the orbit method (an ideology) is the unification of harmonic analysis with symplectic geometry and can also be considered as a part of the more general idea of the unification of mathematics and physics.
The paper is very rich in ideas, examples, problems. Headings of chapters give a good idea of contents (and style) of this paper: 0. Introduction: merits versus demerits; 1. Geometry of coadjoint orbits; 2. The triumph of the orbit method: the case of solvable Lie groups; 3. The first obstacle: compact Lie groups; 4. More troubles: non-compact semisimple groups; 5. Beyond the Lie groups; 6. Explanation of why the orbit method works; 7. Byproducts, side effects and related topics; 8. Some open problems and subjects for meditation.
The orbit method is based on that there is a connection between representations of Lie groups and orbits of their coadjoint representations. An ideal situation for this method is nilpotent Lie groups. In this case there is a one-to-one correspondence between unitary irreducible representations and coadjoint orbits, and the orbit method gives an adequate and transparent language to answer main questions of representation theory: to decompose restrictions of representations and induced representations, to determine the topological structure of the unitary dual, to write formulae for characters etc. Understood in a wider sense, the orbit method (an ideology) is the unification of harmonic analysis with symplectic geometry and can also be considered as a part of the more general idea of the unification of mathematics and physics.
The paper is very rich in ideas, examples, problems. Headings of chapters give a good idea of contents (and style) of this paper: 0. Introduction: merits versus demerits; 1. Geometry of coadjoint orbits; 2. The triumph of the orbit method: the case of solvable Lie groups; 3. The first obstacle: compact Lie groups; 4. More troubles: non-compact semisimple groups; 5. Beyond the Lie groups; 6. Explanation of why the orbit method works; 7. Byproducts, side effects and related topics; 8. Some open problems and subjects for meditation.
Reviewer: V.F.Molchanov (Tambov)
MSC:
| 22E47 | Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) |
| 22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |
| 20C35 | Applications of group representations to physics and other areas of science |
Keywords:
Unitary representations; Lie groups; coadjoint orbits; symplectic geometry; geometric quantization; orbit method; representation theoryReferences:
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