Langlands classification

Notation

• g is the Lie algebra of a real reductive Lie group G in the Harish-Chandra class.
• K is a maximal compact subgroup of G, with Lie algebra k.
• ω is a Cartan involution of G, fixing K.
• p is the −1 eigenspace of a Cartan involution of g.
• a is a maximal abelian subspace of p.
• Σ is the root system of a in g.
• Δ is a set of simple roots of Σ.

  Classification

• F is a subset of Δ
• Q is the standard parabolic subgroup of F, with Langlands decomposition Q = MAN
• σ is an irreducible tempered representation of the semisimple Lie group M
• λ is an element of Hom with α>0 for all simple roots α not in F.

Variations

• Instead of taking an irreducible quotient, one can take an irreducible submodule.
• Since tempered representations are in turn given as certain representations induced from discrete series or limit of discrete series representations, one can do both inductions at once and get a Langlands classification parameterized by discrete series or limit of discrete series representations instead of tempered representations. The problem with doing this is that it is tricky to decide when two irreducible representations are the same.