Langlands dual group

Let $G$ be a split connected reductive group with split maximal torus $T$ and based root datum

\Psi(G)= \bigl( X^*(T),\Delta, X_*(T),\Delta^\vee \bigr).

The Langlands dual group $\widehat G$ is the connected complex reductive group with dual based root datum

\Psi(\widehat G)= \bigl( X_*(T),\Delta^\vee, X^*(T),\Delta \bigr).

A pinning makes $\widehat G$ and automorphisms of its based root datum usable coherently, although the unpinned group is intrinsically determined only up to inner isomorphism.

Nonsplit groups

For a nonsplit $G/F$ , the absolute Galois group or Weil group acts on the based root datum and hence by pinned automorphisms on $\widehat G$ . The $L$ -group combines these data as an extension or semidirect product. The dual group alone does not record the $F$ -form.

Examples

$\widehat{\operatorname{GL}_n}=\operatorname{GL}_n(\mathbb C)$ .
$\widehat{\operatorname{SL}_n}=\operatorname{PGL}_n(\mathbb C)$ .
$\widehat{\operatorname{PGL}_n}=\operatorname{SL}_n(\mathbb C)$ .
$\widehat{\operatorname{Sp}_{2n}}=\operatorname{SO}_{2n+1}(\mathbb C)$ .

Duality exchanges simply connected and adjoint semisimple isogeny forms.

Relation to the letter

The letter writes $\widehat G$ as $cG$ and uses $cL$ for the dual lattice, now represented by the dual-lattice construction. Its construction is the root-datum origin of the modern dual group; the later $L$ -group adds the Galois action needed for nonsplit groups and local parameters.

Rank-one scope warning

Over $\mathbb C$ , $\operatorname{PGL}_2$ is also the automorphism group of the projective line. That action does not define Langlands duality: $\widehat{\operatorname{SL}_2}=\operatorname{PGL}_2$ because roots and coroots, with their character and cocharacter lattices, are exchanged.

References

A. Borel, “Automorphic $L$ -functions,” Proc. Sympos. Pure Math. 33, part 2, 1979.
Robert P. Langlands, “Problems in the theory of automorphic forms,” 1970. DOI.