Software

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.

• 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 | p ∈ S} for the standard and irreducible representations of G
• An arbitrary standard representation I(p) (p ∈ S)
• 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

• 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

• 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.