Langlands group

In mathematics, the Langlands group is a conjectural group

L

F

{\displaystyle L_{F}}

attached to each local or global field

F

{\displaystyle F}

that satisfies properties similar to those of the Weil group. It was named after Robert Langlands by Robert Kottwitz. In Kottwitz's formulation, the Langlands group should be an extension of the Weil group by a compact group. When

F

{\displaystyle F}

is local archimedean,

L

F

{\displaystyle L_{F}}

is the Weil group of

F

{\displaystyle F}

, when

F

{\displaystyle F}

is local non-archimedean,

L

F

{\displaystyle L_{F}}

is the product of the Weil group of

F

{\displaystyle F}

with SU(2). When

F

{\displaystyle F}

is global, the existence of

L

F

{\displaystyle L_{F}}

is still conjectural, though James Arthur gives a conjectural description of it. The Langlands correspondence for

F

{\displaystyle F}

is a "natural" correspondence between the irreducible

n

{\displaystyle n}

-dimensional complex representations of

L

F

{\displaystyle L_{F}}

and, in the global case, the cuspidal automorphic representations of

GL

n

⁡

(

A

F

)

{\displaystyle \operatorname {GL} _{n}(\mathbb {A} _{F})}

, where

A

F

{\displaystyle \mathbb {A} _{F}}

denotes the adeles of

F

{\displaystyle F}

.