• Tags
  • #lie-theory #MOC
• Refs:
  • Basic linear algebra: http://perso.ens-lyon.fr/sophie.morel/MAT_217_notes.pdf#page=1 #resources/course-notes
  • Basic homological algebra: http://perso.ens-lyon.fr/sophie.morel/notes540.pdf#page=5 #resources/course-notes
  • https://services.math.duke.edu/~jdr/papers/building.pdf #resources/papers
  • Ernest B. Vinberg: Linear Representations of Groups #resources/books
  • Reps of finite groups: https://www2.math.upenn.edu/~zvihr/courses/RTNotes.pdf#page=15 #resources/course-notes
  • Lots of notes https://ivganev.github.io/notes/
  • Princetone course notes: http://perso.ens-lyon.fr/sophie.morel/rep_theory_notes.pdf#page=7
  • Reps of topological groups: http://perso.ens-lyon.fr/sophie.morel/notes_449.pdf#page=5 #resources/course-notes
• Links:
  • algebraic group
  • induction
  • depth
  • parahoric
  • parabolic
  • Weyl group
  • Unsorted/Kazhdan Lusztig conjecture
  • Unsorted/BGG resolution
  • Pontryagin dual
  • perverse sheaf

representation theory

Groups

• Cartan
• Levi
• Borel
• maximal torus
• split torus
• Regular representation
• Maschke’s theorem
• Schur’s lemma

Algebraic Groups

• algebraic group
• Hopf algebra
• unipotent radical
• Lie algebra
• Verma module
• Category O
• Wall-crossing morphisms

Groups

• Basic definitions and properties of representations, including Schur’s Lemma and Maschke’s Theorem.
• The representation theory of finite groups, including Schur orthogonality.
• Fundamental constructions such as tensor product, dual representations and induced representations.
• Representation theory of compact groups, including the Peter-Weyl Theorem.
• Description of the irreducible representations of \(S_n, {\operatorname{SU}}_2, {\operatorname{SO}}_3, {\mathfrak{sl}}_2({\mathbf{C}})\).

Notation

Notes

Smooth and admissible reps

Supercuspidal reps

Classification

Topics

Basics

• simple
  • semisimple
• irreducibles
• indecomposable objects of a category
• classification of representations of compact Lie groups
• characters
• weight of a representation
• Frobenius reciprocity

Finite Groups

• Maschke’s theorem
  • completely reducible
• Schur’s Lemma
• sign representation
• permutation representation

Lie Groups

• root system
• Weyl group
  • Coxeter groups
  • Bruhat order

Advanced

• Verma module
• Category O
• Hecke algebra
  • categorification
• Engel’s theorem

Algebras, representations, Schur’s lemma. Representations of SL(2). Representations of finite groups, Maschke’s theorem, characters, applications. Induced representations, Burnside’s theorem, Mackey formula, Frobenius reciprocity. Representations of quivers.

Results

An irreducible representation of \(G\) is completely determined by its character.