Merits and demerits of the orbit method

DOI10.1090/S0273-0979-99-00849-6zbMath0940.22013WikidataQ55920167 ScholiaQ55920167MaRDI QIDQ4269107

Alexandre Kirillov

Publication date: 31 October 1999

Published in: Bulletin of the American Mathematical Society (Search for Journal in Brave)

zbMATH Keywords

symplectic geometryLie groupsrepresentation theorycoadjoint orbitsgeometric quantizationorbit methodUnitary representations

Mathematics Subject Classification ID

Applications of group representations to physics and other areas of science (20C35) Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) (22E47) Research exposition (monographs, survey articles) pertaining to topological groups (22-02)

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Cites Work

• Théorie de Mackey pour les groupes de Lie algébriques
• Non-Abelian bosonization in two dimensions
• Orbital theory for affine Lie algebras
• Poisson Lie groups, dressing transformations, and Bruhat decompositions
• The solution to a generalized Toda lattice and representation theory
• Representations of finite classical groups. A Hopf algebra approach
• Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups
• Addendum to ``On the variation in the cohomology of the symplectic form of the reduced phase space
• Geometric quantization and multiplicities of group representations
• On quantum gauge theories in two dimensions
• Kirillov's character formula for reductive Lie groups
• The Campbell-Hausdorff formula and invariant hyperfunctions
• Numbers of classes and chains of subgroups in finite groups
• Geometric approach to discrete series of unirreps for Vir
• Unitary representation theory of exponential Lie groups
• Representations of exponential Lie groups
• Spherical functions on affine Lie groups
• A unified representation-theoretic approach to special functions
• Flat connections and polyubles
• Decomposition of unitary representations defined by discrete subgroups of nilpotent groups
• The characters of unitary representations of Lie groups
• Polarization and unitary representations of solvable Lie groups. Appendix by Calvin C. Moore
• Irreducible unitary representations of the Lorentz group
• Sur Les Représentations Unitaires Des Groupes De Lie Nilpotents. III
• Opérateurs différentiels bi-invariants sur un groupe de Lie
• Minimal representations, geometric quantization, and unitarity.
• The Yang-Mills equations over Riemann surfaces
• Dual topology of a nilpotent Lie group
• Weak Containment and Induced Representations of Groups
• On the Theory of Exponential Groups
• On Frobenius Reciprocity for Unipotent Algebraic Groups Over Q
• Unitary representations of solvable Lie groups
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