Langlands Dual Group

For split $G$ with maximal torus $T$, its root datum is $(X^*(T),\Phi,X_*(T),\Phi^\vee)$, where $X_*(T):=\mathrm{Hom}_k(\mathbb{G}_m,T)$ and $\Phi^\vee$ are coroots.

The dual root datum is $(X_*(T),\Phi^\vee,X^*(T),\Phi)$.

The Langlands dual group $\widehat G$ is the connected complex reductive group with this dual root datum (the letter writes $\widehat G$ as “$cG$”, and uses $cL$ for dual lattices).

Key property (for Satake):

• Unramified Hecke eigencharacters correspond to semisimple conjugacy classes in $\widehat G$ (more precisely in the $L$-group ).

Example: $\widehat{\mathrm{GL}_n}=\mathrm{GL}_n(\mathbb{C})$, and $\widehat{\mathrm{SL}_n}=\mathrm{PGL}_n(\mathbb{C})$.