Representation theory of Lie groups
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| Representation theory of Lie groups | |
|---|---|
| Name | Representation theory of Lie groups |
| Field | Mathematics |
| Notable figures | Élie Cartan, Hermann Weyl, Harish-Chandra, George Mackey, Roger Howe |
- Introduction
- Basic Definitions and Examples
- Finite-dimensional Representation Theory
- Unitary and Infinite-dimensional Representations
- Structure Theory and Classification (Highest Weight, Harish-Chandra)
- Induced Representations and Parabolic Induction
- Applications and Connections (Harmonic Analysis, Number Theory, Physics)
Representation theory of Lie groups is the study of homomorphisms from Lie groups to groups of linear transformations on vector spaces, connecting structures from Élie Cartan and Sophus Lie to modern work in Harish-Chandra theory and Langlands program. The subject unites methods from Hermann Weyl's harmonic analysis on Felix Klein's geometric settings, bridges results of Harish-Chandra with applications in John von Neumann's functional analysis, and informs developments in Andrew Wiles-adjacent number theory and Richard Feynman-inspired physics.
Introduction
The field traces origins to Sophus Lie and Ferdinand Georg Frobenius and matured through contributions of Élie Cartan, Hermann Weyl, and Harish-Chandra, forming links between Felix Klein's Erlangen program, David Hilbert's invariant theory, and the Langlands program. Central themes include classification of irreducible representations, character theory influenced by Atle Selberg and Harish-Chandra, and tools from John von Neumann's operator algebras and George Mackey's imprimitivity theorem.
Basic Definitions and Examples
One studies a Lie group G (examples: SO(3), SU(2), SL(2,R), GL(n,C)) and a representation rho: G -> GL(V) on a vector space V, with finite-dimensional cases treated by Hermann Weyl and infinite-dimensional cases by Harish-Chandra and I. M. Gelfand. Key examples include the defining representations of GL(n,R), the spin representations related to Élie Cartan's spin groups, and principal series for SL(2,R) developed by Harish-Chandra and George Mackey.
Finite-dimensional Representation Theory
The finite-dimensional theory for compact groups (e.g. SU(n), SO(n), Sp(n)) follows the highest weight classification of Hermann Weyl and the Weyl character formula, drawing on roots and weights from Élie Cartan's classification of simple Lie algebras and the Dynkin diagram program of Eugène Dynkin. Semisimple algebraic methods from Claude Chevalley, Jean-Pierre Serre, and Armand Borel give complete reducibility and tensor product decompositions, with computational frameworks advanced by Roger Howe and applications to Peter-Weyl theorem contexts influenced by Marshall Stone.
Unitary and Infinite-dimensional Representations
Unitary representation theory, shaped by John von Neumann and George Mackey, treats Hilbert space representations of noncompact groups like SL(2,R), SO(1,n), and GL(n,R). The work of Harish-Chandra on admissible representations, distributions, and characters, together with the Plancherel formula of Atle Selberg and harmonic analysis of Israel Gelfand, yields classification of tempered and discrete series, with further development by Anthony Knapp and Wilfried Schmid.
Structure Theory and Classification (Highest Weight, Harish-Chandra)
Highest weight theory, originating with Hermann Weyl and formalized via Élie Cartan's root systems and Eugène Dynkin diagrams, classifies irreducibles for compact and complex semisimple groups; foundational contributors include Claude Chevalley and Armand Borel. Harish-Chandra's work on the infinite-dimensional setting for real reductive groups (e.g. GL(n,R), SL(2,R), SO(p,q)) introduces infinitesimal characters, the Langlands classification formulated with input from Robert Langlands, and the character theory with analytic contributions by Atle Selberg and Harish-Chandra himself.
Induced Representations and Parabolic Induction
Induction methods, developed by Frobenius in finite groups and extended by George Mackey and Anthony Knapp to Lie groups, produce principal series and parabolic induction from Levi subgroups associated to parabolic subgroups studied by Armand Borel and Jacques Tits. The theory of Eisenstein series in the Langlands program uses parabolic induction to relate automorphic representations of GL(n), SL(n), and reductive groups to L-functions pioneered by Robert Langlands and applied by Andrew Wiles in arithmetic contexts.
Applications and Connections (Harmonic Analysis, Number Theory, Physics)
Representation theory underpins harmonic analysis on symmetric spaces of Élie Cartan and spectral theory used by Atle Selberg in trace formulas, informs the Langlands program and automorphic forms central to Andrew Wiles and Robert Langlands, and supplies symmetry descriptions in quantum mechanics via Paul Dirac and Richard Feynman. Interactions with algebraic geometry appear through the geometric Langlands work influenced by Alexander Beilinson and Edward Frenkel, while connections to mathematical physics are advanced by Edward Witten and Michael Atiyah in topological quantum field theory and index theory originating with Atiyah–Singer index theorem-related developments. Computational and categorical expansions involve contributions from Pierre Deligne, Joseph Bernstein, and David Kazhdan.