Local Langlands Conjectures - Representations of GL_n(F)

Representations of GL_n(F)

The representations of GL_n(F) appearing in the local Langlands correspondence are smooth irreducible complex representations.

• "Smooth" means that every vector is fixed by some open subgroup.
• "Irreducible" means that the representation is nonzero and has no subrepresentations other than 0 and itself.

Smooth irreducible representations are automatically admissible.

The Bernstein–Zelevinsky classification reduces the classification of irreducible smooth representations to cuspidal representations.

For every irreducible admissible complex representation π there is an L-function L(s,π) and a local ε-factor ε(s,π,ψ) (depending on a character ψ of F). More generally, if there are two irreducible admissible representations π and π' of general linear groups there are local Rankin–Selberg convolution L-functions L(s,π×π') and ε-factors ε(s,π×π',ψ).

Bushnell & Kutzko (1993) described the irreducible admissible representations of general linear groups over local fields.