Representations of compact groups
Thirteen lessons from unitarity and Peter–Weyl to maximal tori, highest weights, Weyl formulas, Laplacians, tensor products and homogeneous spaces. Includes complete proofs, worked examples and 52 exercises with complete solutions, together with profinite characters, Schur–Weyl duality, trace moments and positive spherical kernels.
English · GPT-6.1 Sol (OpenAI), Ultra · self-checked by the writing AI · original course text CC0 1.0
Download the reader and editable sources · Sources, authorship and component terms · Prerequisite readings · Proof foundations and further reading
- Representations of compact groups: unitarity, complete reducibility and finite dimension · source
- Matrix coefficients and the Peter–Weyl theorem · source
- Fourier analysis and class functions on compact groups · source
- SU(2), SO(3) and spherical harmonics · source
- Compact Lie groups, their Lie algebras and the adjoint representation · source
- Tori and the maximal torus theorem · source
- Roots and the Weyl group of a compact Lie group · source
- The Weyl integration formula · source
- Unitary groups: characters, dimensions and branching · source
- Weights and the theorem of the highest weight · source
- The Weyl character formula for compact connected Lie groups · source
- Casimir operators, Laplacians and tensor products · source
- Homogeneous spaces, induced representations and Gelfand pairs · source