Knowlpedia

Definition

Langlands LL-group

The complex dual group extended by the Weil or Galois action on its based root datum.

Definition

Let GG be a connected over a or FF, and let G^\widehat G be its pinned complex . The LL-group of GG is

LG=G^⋊WF,{}^LG=\widehat G\rtimes W_F,

where the acts on G^\widehat G through the on its . It comes with an exact sequence

1⟶G^⟶LG⟶WF⟶1.1\longrightarrow\widehat G\longrightarrow{}^LG \longrightarrow W_F\longrightarrow1.

For a split group the action is trivial, although the Weil-group factor and projection remain part of the LL-group.

Galois and global variants

Many sources write G^⋊ΓF\widehat G\rtimes\Gamma_F, especially when the action factors through a finite Galois quotient. For global questions, one may use an appropriate global Weil group or a conjectural global Langlands group. These versions carry different topology and extension data even when they induce the same finite action on the root datum.

LL-homomorphisms

An LL-homomorphism

LH⟶LG{}^LH\longrightarrow{}^LG

is continuous, is compatible with the projections to the Weil group, and is algebraic on the connected complex dual-group part. Such homomorphisms are considered up to conjugation by G^\widehat G and are the input to .

Dependence on pinning

A pinning turns the outer Galois action on the dual group into an actual action. Different compatible choices produce isomorphic LL-groups, canonically only up to . Statements about parameters are therefore made up to G^\widehat G-conjugacy.

References
  1. Armand Borel, “Automorphic LL-functions,” in Automorphic Forms, Representations and LL-Functions, Proceedings of Symposia in Pure Mathematics 33, part 2, 1979, §§2–3.
  2. Tasho Kaletha, “Representations of reductive groups over local fields,” §2.1, 2022. arXiv.