Langlands Dual Groups
The Langlands dual group \(\hat G\) (also called the Langlands–Deligne dual or simply the dual group) of a connected reductive group \(G\) is the complex reductive group whose root datum is dual to that of \(G\). It is the recipient of L-parameters on the "Galois side" of the Langlands correspondence.
Definition
Let \(G\) be a connected reductive group over a field \(k\), with root datum \((X^*, \Phi, X_*, \Phi^\vee)\). The Langlands dual group \(\hat G\) is the connected reductive group over \(\mathbb{C}\) (or any algebraically closed field of characteristic \(0\)) whose root datum is \[ (X_*, \Phi^\vee, X^*, \Phi), \] i.e.\ the dual root datum with characters and cocharacters swapped.
By the classification of split reductive groups, \(\hat G\) is unique up to isomorphism. A pinning of \(\hat G\) — a choice of Borel, torus, and root vectors — makes \(\hat G\) functorial in \(G\).
Basic examples
| \(G\) | \(\hat G\) | Type swap |
|---|---|---|
| \(\mathrm{GL}_n\) | \(\mathrm{GL}_n\) | self-dual |
| \(\mathrm{SL}_n\) | \(\mathrm{PGL}_n\) | \(A_{n-1}\) sc ↔ \(A_{n-1}\) ad |
| \(\mathrm{Sp}_{2n}\) | \(\mathrm{SO}_{2n+1}\) | \(C_n \leftrightarrow B_n\) |
| \(\mathrm{SO}_{2n+1}\) | \(\mathrm{Sp}_{2n}\) | \(B_n \leftrightarrow C_n\) |
| \(\mathrm{SO}_{2n}\) | \(\mathrm{SO}_{2n}\) | self-dual (\(D_n\)) |
| \(\mathbb{G}_m\) | \(\mathbb{G}_m\) | self-dual torus |
| \(\mathrm{U}(n)\) (\({}^2 A_{n-1}\)) | \(\mathrm{GL}_n\) with Galois twist | outer form |
| \(G_2\) | \(G_2\) | self-dual |
| \(F_4\) | \(F_4\) | self-dual |
| \(E_6\) sc | \(E_6\) ad | non-trivial centre dualises |
| \(E_7\) sc | \(E_7\) ad | non-trivial centre dualises |
| \(E_8\) | \(E_8\) | self-dual |
Why the swap?
Root data are inherently \"asymmetric\" between characters and cocharacters of the maximal torus. The swap exchanges:
- characters \(\leftrightarrow\) cocharacters;
- roots \(\leftrightarrow\) coroots;
- long roots \(\leftrightarrow\) short roots (in non-simply-laced types, this gives \(B_n \leftrightarrow C_n\) and \(F_4 \leftrightarrow F_4\) via a non-trivial swap);
- \(Z(G) \leftrightarrow \pi_1(\hat G)\) and vice versa.
The last point is the Kottwitz duality: \(Z(\hat G)\) is the Pontryagin dual of \(\pi_1(G)\): \[ Z(\hat G) \;=\; \operatorname{Hom}(\pi_1(G), \mathbb{C}^\times). \]
The L-group
For \(G\) not split, the Galois group \(\Gamma = \operatorname{Gal}(\bar k/k)\) acts on the root datum of \(G\), hence (via the swap) on \(\hat G\) as a group of pinned automorphisms. The L-group is the semidirect product: \[ {}^L G \;=\; \hat G \rtimes \Gamma \] (or its Weil-form variant \(\hat G \rtimes W_F\) for local fields).
This is the receptacle of L-parameters and the natural target for functoriality.
Geometric Satake
The Langlands dual group has a purely geometric characterisation in the geometric Langlands programme:
Theorem (Mirković–Vilonen, Ginzburg). The category of perverse sheaves on the affine Grassmannian \(\mathrm{Gr}_G\) (with the \(L^+ G\)-equivariant structure) is equivalent (as a tensor category) to \(\mathrm{Rep}(\hat G)\): \[ \mathrm{Perv}_{L^+ G}(\mathrm{Gr}_G) \;\simeq\; \mathrm{Rep}(\hat G). \]
This gives a Tannakian construction of \(\hat G\) intrinsic to the geometry of \(G\) — not as a combinatorial dual but as the symmetry group of a perverse-sheaf category.
Role in Langlands
- L-parameters: continuous homomorphisms \(\varphi : W_F \times \mathrm{SL}_2(\mathbb{C}) \to {}^L G\) up to \(\hat G\)-conjugacy.
- L-packets: fibres of the conjectural map \(\{\text{irreps of } G(F)\} \to \{\text{L-parameters}\}\).
- L-functions: \(L(s, \pi, r)\) for an L-parameter \(\varphi_\pi\) and a finite-dimensional representation \(r : {}^L G \to \mathrm{GL}_N\).
- Functoriality: morphisms \({}^L H \to {}^L G\) conjecturally transfer automorphic representations of \(H\) to \(G\).
- Endoscopy: endoscopic groups arise as centralisers of semisimple elements in \(\hat G\).
References
- Borel, Automorphic L-functions (Proc.\ Sympos.\ Pure Math.\ 33, Corvallis 1979).
- Kottwitz, Stable trace formula: cuspidal tempered terms (Duke 1984).
- Mirković–Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings (Annals 2007).
- Bezrukavnikov, On two geometric realizations of an affine Hecke algebra (Publ.\ IHES 2016).