12.4 Representation theory and coadjoint orbits

Representation theory and coadjoint orbits bridge abstract algebra and geometry. They connect Lie groups, their algebras, and symplectic structures on orbits. This link reveals deep insights into group actions and their geometric representations.

The orbit method ties coadjoint orbits to irreducible representations, while geometric quantization builds Hilbert spaces from symplectic manifolds. These techniques illuminate the interplay between algebraic structures and their geometric realizations in symplectic spaces.

Representation Theory and Symplectic Geometry

Foundational Concepts

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• Representation theory examines abstract algebraic structures represented as linear transformations of vector spaces
• Symplectic geometry focuses on manifolds equipped with a closed, non-degenerate 2-form called a symplectic form
• Connection arises through study of Lie groups and associated Lie algebras with natural symplectic structures on coadjoint orbits
• Moment map bridges symplectic action of Lie group on manifold and dual of its Lie algebra
• Symplectic reduction constructs new symplectic manifolds from those with symmetries, deeply connecting to Lie group representation theory

Key Methods and Correspondences

• Orbit method (developed by Kirillov) establishes correspondence between coadjoint orbits of Lie group and unitary irreducible representations
• Geometric quantization constructs Hilbert spaces and operators from symplectic manifolds, providing geometric approach to Lie group representations
• Kirillov-Kostant-Souriau form defines symplectic structure on coadjoint orbits using Lie bracket and natural pairing between Lie algebra and dual
• Poisson structure on Lie algebra dual induces symplectic structure on coadjoint orbits, linking Poisson and symplectic geometry

Symplectic Structure on Coadjoint Orbits

Definitions and Properties

• Coadjoint orbits result from coadjoint action of Lie group on dual space of Lie algebra
• Kirillov-Kostant-Souriau form equips coadjoint orbits with natural symplectic structure
• Symplectic structure remains invariant under coadjoint action, making orbits homogeneous symplectic manifolds
• Coadjoint orbit dimensions always even, consistent with symplectic manifold nature
• For matrix Lie groups, coadjoint action often computed explicitly using matrix operations (SU(2), SO(3))

Construction and Analysis

• Symplectic reduction constructs coadjoint orbits as reduced spaces of simpler symplectic manifolds
• Approach provides alternative perspective on symplectic structure of coadjoint orbits
• Analysis of coadjoint orbits reveals geometric properties of Lie group representations
• Symplectic volume of coadjoint orbits relates to dimensions of corresponding representations

Coadjoint Orbits for Lie Group Representations

Applications to Representation Theory

• Orbit method classifies unitary irreducible representations for nilpotent and solvable Lie groups
• Character formula for compact Lie groups derived using symplectic structure of coadjoint orbits and equivariant cohomology localization techniques
• Borel-Weil-Bott theorem connects representation theory of compact Lie groups to geometry of line bundles over flag varieties (viewed as coadjoint orbits)
• Coadjoint orbits provide geometric realization of highest weight modules for semisimple Lie algebras

Advanced Topics

• Study of degenerate coadjoint orbits leads to theory of unipotent representations for reductive groups over local fields
• Quantization of coadjoint orbits (geometric or deformation) constructs representations of Lie groups
• Coadjoint orbit method extends to study of branching laws and restrictions of representations
• Approach provides geometric insights into algebraic problems in representation theory

Coadjoint Orbits vs Geometric Quantization

Geometric Quantization Procedure

• Constructs Hilbert space and operators from symplectic manifold, with coadjoint orbits as fundamental examples
• Prequantization step builds line bundle with connection over coadjoint orbit
• Curvature of line bundle relates to symplectic form on orbit
• Choice of polarization in geometric quantization of coadjoint orbits often corresponds to different realizations of resulting representation
• Metaplectic correction crucial for obtaining correct half-form pairing and constructing unitary representations

Specific Applications and Methods

• Borel-Weil construction for compact Lie groups viewed as special case of geometric quantization applied to coadjoint orbits
• Kostant's operator method provides algebraic approach to geometric quantization of coadjoint orbits
• Connects representation theory with differential geometry
• Relationship between coadjoint orbits and geometric quantization extends to study of branching laws and restrictions of representations
• Provides geometric insights into algebraic problems in representation theory