Dual Reductive Groups

The dual reductive group \(\hat G\) of a connected reductive group \(G\) is the unique connected complex reductive group whose root datum is dual to that of \(G\). It is the Langlands dual of \(G\) without Galois action; see L-groups for the Galois-twisted version.

Construction

Let \(G\) be a connected reductive group over a field \(k\), with root datum \((X^*(T), \Phi, X_*(T), \Phi^\vee)\) for a chosen maximal torus \(T \subseteq G\).

By the classification of split connected reductive groups (Chevalley- Demazure), there is a unique split connected reductive group \(\hat G\) over \(\mathbb{C}\) (or any algebraically closed field of characteristic 0) with root datum \[ (X_*(T), \Phi^\vee, X^*(T), \Phi), \] the dual root datum.

\(\hat G\) is well-defined up to non-canonical isomorphism (canonical when a pinning is chosen).

Examples

Functoriality

The construction \(G \mapsto \hat G\) is contravariant on isogenies and covariant at the level of pinned automorphisms:

• An isogeny \(G \to G'\) dualises to a (typically opposite-direction) isogeny \(\hat{G'} \to \hat G\).
• An outer automorphism \(\sigma \in \mathrm{Out}(G)\) lifts to \(\sigma \in \mathrm{Out}(\hat G)\) (preserving the dual structure).

Centre and fundamental group swap

Under duality, the centre and the algebraic fundamental group swap: \[ Z(\hat G) \;\cong\; X^*(\pi_1(G))^\vee, \qquad \pi_1(\hat G) \;\cong\; X^*(Z(G))^\vee. \]

Concretely:

• \(\mathrm{SL}_n\): \(Z = \mu_n\), \(\pi_1 = 0\) → \(\widehat{\mathrm{SL}_n} = \mathrm{PGL}_n\): \(Z = 1\), \(\pi_1 = \mathbb{Z}/n\).
• \(\mathrm{GL}_n\): \(Z = \mathbb{G}_m\), \(\pi_1 = \mathbb{Z}\) → \(\widehat{\mathrm{GL}_n}\): same (\(\mathrm{GL}_n\) self-dual, centre dual to fundamental group both \(\mathbb{Z}\)).

Functorial structure preservation

• Parabolics: \(P \subseteq G\) corresponds to "dual parabolics" \(\hat P \subseteq \hat G\), giving the Levi-conjugate decomposition: Levi \(L\) of \(P\) → Levi \(\hat L\) of \(\hat P\).
• Cuspidal data: depth-zero \(L\)-packet \(\sigma\) of \(L(F)\) corresponds to L-parameter into \(\hat L\); Langlands quotient passes from \(\sigma\) to \(\hat L \hookrightarrow \hat G\).
• Endoscopic groups \(H \subseteq G\) (in a derived sense) correspond to centralisers of semisimple elements in \(\hat G\).

Identification of the centre

\[ Z(\hat G) \;=\; \operatorname{Hom}(X_*(T)/Q^\vee, \mathbb{C}^\times) \;=\; \operatorname{Hom}(\pi_1(G), \mathbb{C}^\times). \]

This is the dual of \(\pi_1(G)\) as a finite abelian group, and is the relevant target for central characters of L-packets in the local Langlands correspondence.

Geometric Satake equivalence

Theorem (Mirković-Vilonen, Lusztig, Ginzburg). There is an equivalence of monoidal categories: \[ \mathrm{Perv}_{L^+ G}(\mathrm{Gr}_G) \;\simeq\; \mathrm{Rep}(\hat G), \] where \(\mathrm{Gr}_G\) is the affine Grassmannian and \(L^+ G = G(\mathcal{O})\) acts on it. This gives a purely geometric construction of \(\hat G\) intrinsic to the geometry of \(G\) — not from combinatorial duality but from the symmetry of perverse sheaves.

Role in Langlands

• L-parameters: continuous homomorphisms \(W_F \times \mathrm{SL}_2(\mathbb{C}) \to {}^L G\) up to \(\hat G\)-conjugacy parametrise L-packets of irreducible representations of \(G(F)\).
• L-functions: \(L(s, \pi, r)\) for an L-parameter \(\varphi : W_F \to \hat G\) and a finite-dimensional representation \(r : \hat G \to \mathrm{GL}_N\).
• Functoriality: morphisms \(\hat H \to \hat G\) conjecturally induce transfer of automorphic representations.
• Endoscopy and stable conjugacy: defined on the dual side via semisimple centralisers.

References

• Borel, Automorphic L-functions (Corvallis 1979).
• Mirković–Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings (Annals 2007).
• Kottwitz, Stable trace formula: cuspidal tempered terms (Duke 1984), §1.

Prerequisite Topics

• Dual Root Data
• Classification of Split Reductive Groups