Dual Root Data

The dual root datum swaps characters with cocharacters and roots with coroots. It is the combinatorial source of Langlands duality — the operation that turns the root datum of \(G\) into the root datum of the Langlands dual group \(\hat G\).

Definition

For a root datum \(\Psi = (X, \Phi, X^\vee, \Phi^\vee)\) with the canonical pairing \(\langle -, - \rangle : X \times X^\vee \to \mathbb{Z}\), the dual root datum is \[ \Psi^\vee \;:=\; (X^\vee, \Phi^\vee, X, \Phi) \] with the pairing \(\langle -, - \rangle^\vee : X^\vee \times X \to \mathbb{Z}\), \(\langle x^\vee, x \rangle^\vee := \langle x, x^\vee \rangle\) (i.e.\ the swapped pairing).

The map \(\Psi \mapsto \Psi^\vee\) is a contravariant involution: \((\Psi^\vee)^\vee = \Psi\).

Properties

The bijection \(\Phi \leftrightarrow \Phi^\vee\) of the original root datum becomes part of the new root datum: roots of \(\Psi^\vee\) are coroots of \(\Psi\) and vice versa.
The pairing condition \(\langle \alpha, \alpha^\vee \rangle = 2\) is preserved (becomes \(\langle \alpha^\vee, \alpha \rangle = 2\) in \(\Psi^\vee\)).
The Weyl group of \(\Psi^\vee\) is canonically isomorphic to that of \(\Psi\) (same set of reflections, dual realisation).

Effects on Dynkin type

For irreducible root systems, duality acts on Dynkin types as follows:

Type \(G\)	Type \(\hat G\) (= dual)
\(A_n\)	\(A_n\) (self-dual)
\(B_n\)	\(C_n\)
\(C_n\)	\(B_n\)
\(D_n\)	\(D_n\) (self-dual)
\(E_6\)	\(E_6\) (self-dual)
\(E_7\)	\(E_7\) (self-dual)
\(E_8\)	\(E_8\) (self-dual)
\(F_4\)	\(F_4\) (long ↔ short swap)
\(G_2\)	\(G_2\) (long ↔ short swap)

Long and short roots swap their roles in non-simply-laced types.

Lattice gap inversion

The lattice gap \(\mathbb{Z}\Phi \subseteq X \subseteq P\) (weight lattice) and its dual \(Q^\vee = \mathbb{Z}\Phi^\vee \subseteq X^\vee \subseteq P^\vee\) swap under duality:

Simply connected \(G\) (\(X^\vee = Q^\vee\)) becomes adjoint \(\hat G\) (\(X = \mathbb{Z}\Phi\));
Adjoint \(G\) (\(X = \mathbb{Z}\Phi\)) becomes simply connected \(\hat G\).

So:

\((\mathrm{SL}_n)^\vee = \mathrm{PGL}_n\) (sc ↔ ad);
\((\mathrm{SO}_{2n+1})^\vee = \mathrm{Sp}_{2n}\) (sc-of-B_n ↔ sc-of-C_n);
\(\mathrm{GL}_n^\vee = \mathrm{GL}_n\) (self-dual, but with the determinant character swapped to the determinant cocharacter).

Canonical inclusions

The lattice inclusions \(Q \subseteq X \subseteq P\) dualise to inclusions \(Q^\vee \subseteq X^\vee \subseteq P^\vee\), giving: \[ \pi_1(\Psi) \;=\; X^\vee / Q^\vee \;\stackrel{\text{dual}}{\longleftrightarrow}\; X / Q \;=\; Z(\Psi^\vee). \]

This is the fundamental Kottwitz duality: \[ \pi_1(G) \;=\; X^*(Z(\hat G)). \]

Examples

\(\mathrm{GL}_n\) (self-dual)

\(X = X^\vee = \mathbb{Z}^n\); roots \(\Phi = \{\varepsilon_i - \varepsilon_j\} \subseteq X\) and coroots \(\Phi^\vee = \{\varepsilon_i^\vee - \varepsilon_j^\vee\} \subseteq X^\vee\).

Dualising swaps the two copies of \(\mathbb{Z}^n\). Under the identification, this is just the identity — \(\mathrm{GL}_n\) is self-dual.

\(\mathrm{SL}_n\) ↔ \(\mathrm{PGL}_n\)

\(\mathrm{SL}_n\): \(X = \mathbb{Z}^n/\mathbb{Z}(1, \ldots, 1)\), \(X^\vee = \{(a_i) : \sum a_i = 0\}\). Dualising swaps these, giving \(\mathrm{PGL}_n\)'s root datum.

Type \(C_2 = \mathrm{Sp}_4\)

Roots in \(X = \mathbb{Z}^2\): \(\Phi = \{\pm \varepsilon_1 \pm \varepsilon_2, \pm 2 \varepsilon_1, \pm 2 \varepsilon_2\}\). Coroots in \(X^\vee\): \(\Phi^\vee = \{\pm \varepsilon_1^\vee \pm \varepsilon_2^\vee, \pm \varepsilon_1^\vee, \pm \varepsilon_2^\vee\}\). Dualising gives \(B_2 = \mathrm{Spin}_5\) (simply connected of \(B_2\)).

Action on automorphism groups

Root-datum duality preserves the automorphism group: \[ \mathrm{Aut}(\Psi) \;=\; \mathrm{Aut}(\Psi^\vee). \]

For an outer form determined by a Galois representation \(\Gamma \to \mathrm{Out}(G)\), the dual outer form has the same representation \(\Gamma \to \mathrm{Out}(\hat G) = \mathrm{Out}(G)\).

Functoriality

Duality is contravariant on morphisms: for \(f : \Psi \to \Psi'\), the dual morphism \(f^\vee : (\Psi')^\vee \to \Psi^\vee\) is induced by the dual lattice maps.

This compatibility is what makes Langlands duality functorial: a homomorphism of L-groups corresponds to a "co-functoriality" in the forward direction.

Role in Langlands

The dual root datum is the algebraic underpinning of:

The dual group \(\hat G\): the unique split reductive group with root datum \(\Psi^\vee\).
Langlands duality: the principle that automorphic representations of \(G\) correspond to L-parameters into \({}^L G\), with the dual side built from \(\Psi^\vee\).
Symplectic-orthogonal correspondence: the duality \(\mathrm{Sp}_{2n} \leftrightarrow \mathrm{SO}_{2n+1}\), which makes Arthur's classification of automorphic representations symmetric between symplectic and odd-orthogonal groups, comes from \(\Psi^\vee\) for \(B_n \leftrightarrow C_n\).