Capabilities of the Atlas Software

Here is a description of the main features of the software, Version 1.0 (February 2017).

The groups:

  • a general connected complex reductive group G;
  • an inner class of strong real forms of G;
  • a real form of G.
Structure theory:
  • the component group of G;
  • the conjugacy classes of Cartan subgroups of G;
  • the relative Weyl group of each Cartan subgroup;
  • K-orbits on G/B and G/P
  • Parabolic subgroups; real and theta-stable parabolics
  • A parameter space {p|pϵS} for the standard and irreducible representations of G
  • An arbitrary standard representation I(p) (pϵS)
  • An arbitrary irreducible representation J(p) (the unique irreducible quotient of I(p))
  • The block B of J(p)
  • The Kazhdan-Lusztig-Vogan polynomials of B
  • The Character formula of any irreducible representation J(p)
  • The composition series of any standard module I(p)
  • The Jantzen filtration of I(p)
  • Twisted versions all these objects (following Lusztig and Vogan)
  • Vogan duality for B
Branching to K:
  • Parametrization of the irreducible representations of (possibly disconnected) K
  • K-types of I(p) and J(p)
  • Induction from an arbitrary representation on a real parabolic subgroup
  • Cohomological induction in the weakly fair range
  • Euler characteristic of cohomological induction in general
Hermitian Forms:
  • Signature of the c-invariant form on I(p) and J(p)
  • Determine if J(p) admits an invariant Hermitian form
  • Signature of invariant Hermitian form on J(p)
  • Determine if the invariant Hermitian form on J(p) is definite, i.e. J(p) is unitary

Here is some mathematical background.