Atlas of Lie Groups and Representations

Software

Main features of the atlas software

Here is a summary of the main features of the atlas software. Also see the documentation on the atlas library, and the files atlas.help and atlas-functions.help in the atlas-scripts directory.

The basics

  • Complex reductive groups, root systems
  • Real reductive groups, inner classes and forms
  • Maximal compact subgroup K(ℝ) and its complexification K
  • K-orbits on G/B
  • Langlands parameters
  • Irreducible representations of K
  • Branching to K
  • Kazhdan–Lusztig–Vogan polynomials
  • Composition series and character formulas

Hermitian forms and unitarity

  • Hermitian representations, invariant Hermitian forms
  • The c-invariant form
  • Signatures of forms
  • Unitarity
  • Jantzen filtration

Induction

  • Parabolic subgroups, Levi subgroups
  • Real parabolic induction and cohomological induction

Structure theory

  • Levi subgroups
  • Pseudo-Levi subgroups
  • Equal rank reductive subgroups and their poset

The Weyl group

  • The Weyl group W
  • Conjugacy classes, characters of W
  • Character table of W
  • Representations of W
  • Coherent continuation

Nilpotent orbits

  • Nilpotent orbits of a complex group
  • Nilpotent orbits of a real group
  • Component group of the centralizer of a nilpotent element
  • Special nilpotent orbits
  • Duality of nilpotent orbits
  • Springer correspondence
  • Associated variety of ideals in the universal enveloping algebra
  • Gelfand–Kirillov dimension

Lusztig’s theory of the Weyl group

  • Generic degree and fake degree
  • Cells of representations of the Weyl group
  • Lusztig’s parametrization of representations of W

Unipotent representations

  • Unipotent representations of a real reductive group
  • Weak Arthur packets of unipotent representations

Miscellaneous

  • Structure constants
  • The Tits group
  • Extended groups