Atlas of Lie Groups and Representations

Here is a summary of the main features of the Atlas software. Also see See documenation on the atlas library, and the files and in the atlas-scripts directory.

The basics

  1. Complex reductive groups, root systems
  2. Real Reductive groups, inner classes and forms
  3. Maximal compact subgroup K(R) and its complexification K
  4. K orbits on G/B
  5. Langlands parameters
  6. Irreducible representations of K
  7. Branching to K
  8. Kazhdan-Lusztig-Vogan polynomials
  9. Composition series and Character formulas

Hermitian Forms and unitarity

  1. Hermitian representations, invariant Hermitian forms
  2. The c-invariant form
  3. Signatures of forms
  4. Unitarity
  5. Jantzen filtration


  1. Parabolic subgroups, Levi subgroups
  2. Real parabolic induction and cohomological induction

Structure Theory

  1. Levi subgroups
  2. Pseudo-Levi subgroups
  3. Equal rank reductive subgroups and their poset
  4. Levi subgroups

The Weyl group

  1. The Weyl group W
  2. Conjugacy classes, characters of W
  3. Character table of W
  4. Representations of W
  5. Coherent Continuation

Nilpotent Orbits

  1. Nilpotent orbit of a complex group
  2. Nilpotent orbits of a real group
  3. Component group of centralizer of a nilpotent element
  4. Special nilpotent orbits
  5. Duality of nilpotent orbits
  6. Springer Correspondence
  7. Associated variety of ideals in the Universal enveloping algebra
  8. Gelfand Kirillov dimension

Lusztig's theory of the Weyl group

  1. Generic degree and fake degree
  2. cells of representations of the Weyl group
  3. Lusztig's parametrization of representations of W

Unipotent Representations

  1. Unipotent representations of a real reductive group
  2. Weak Arthur packets of unipotent representations


  1. structure constants
  2. The Tits group
  3. Extended groups