Automorphisms of Root Data

The automorphism group of a root datum is a finite group capturing the "rigid" symmetries of a split reductive group. It coincides with the outer automorphism group \(\mathrm{Out}(G)\) and with the automorphism group of the Dynkin diagram (for based data).

Definitions

For a root datum \(\Psi = (X, \Phi, X^\vee, \Phi^\vee)\):

An automorphism of \(\Psi\) is a pair \((\phi, \phi^\vee)\) of \(\mathbb{Z}\)-linear isomorphisms \(\phi : X \to X\) and \(\phi^\vee : X^\vee \to X^\vee\) compatible with the pairing \(\langle \phi(x), \phi^\vee(x^\vee) \rangle = \langle x, x^\vee \rangle\) and preserving \(\Phi\) and \(\Phi^\vee\) (with the coroot bijection).

\(\mathrm{Aut}(\Psi)\) is finite (since \(\Phi\) is finite and generates \(X \otimes \mathbb{Q}\)).

For a based root datum \((X, \Phi, X^\vee, \Phi^\vee; \Delta)\):

An automorphism must additionally preserve the base \(\Delta\) as a set.

\(\mathrm{Aut}(\Psi; \Delta) \subseteq \mathrm{Aut}(\Psi)\) is the subgroup fixing \(\Delta\).

Decomposition

The Weyl group \(W = W(\Phi)\) sits inside \(\mathrm{Aut}(\Psi)\) as a normal subgroup, with quotient \[ \mathrm{Aut}(\Psi) \;/\; W \;\cong\; \mathrm{Aut}(\Psi; \Delta). \]

In particular: every automorphism of a root datum is uniquely \(W \cdot \sigma\) for some \(\sigma \in \mathrm{Aut}(\Psi; \Delta)\).

Dynkin diagram automorphisms

For a based root datum, \(\mathrm{Aut}(\Psi; \Delta)\) coincides with the automorphism group of the Dynkin diagram (permutations of vertices preserving edge multiplicities and arrow directions).

For irreducible types:

Dynkin type	\(\mathrm{Aut}(\text{diagram})\)	Generator
\(A_1\)	trivial	—
\(A_n\) (\(n \ge 2\))	\(\mathbb{Z}/2\)	reverse order of vertices
\(B_n\), \(C_n\)	trivial	—
\(D_4\)	\(S_3\)	triality (3 outer legs)
\(D_n\) (\(n \ge 5\))	\(\mathbb{Z}/2\)	swap two outer leaves
\(E_6\)	\(\mathbb{Z}/2\)	reflection of diagram
\(E_7\)	trivial	—
\(E_8\)	trivial	—
\(F_4\)	trivial in Dynkin; \(\mathbb{Z}/2\) in unfolded form	—
\(G_2\)	trivial in Dynkin; \(\mathbb{Z}/2\) in unfolded form	—

For \(F_4, G_2\): there is no outer automorphism preserving the Dynkin diagram-with-arrows, but the underlying unweighted graph has a non-trivial symmetry — relevant in tales of "twisted" duality.

Connection with \(\mathrm{Out}(G)\)

Theorem. For a connected reductive group \(G\) with based root datum \(\Psi\): \[ \mathrm{Out}(G) \;:=\; \mathrm{Aut}(G) / \mathrm{Inn}(G) \;\cong\; \mathrm{Aut}(\Psi; \Delta). \]

The full automorphism group satisfies \[ \mathrm{Aut}(G) \;=\; G_{\mathrm{ad}}(k) \;\rtimes\; \mathrm{Out}(G), \] i.e.\ the inner automorphisms form a group of \(k\)-points of the adjoint quotient, and the outer ones lift the Dynkin symmetries.

Examples

\(\mathrm{SL}_n\) (\(n \ge 2\)) and \(\mathrm{PGL}_n\)

\(\mathrm{Out}(\mathrm{SL}_n) = \mathbb{Z}/2\); generator = "transpose- inverse": \(g \mapsto (g^T)^{-1}\). For \(n = 2\): \((g^T)^{-1} = w g w^{-1}\) for the longest Weyl element \(w\), so the outer automorphism is "inner" in this case — \(\mathrm{Out}(\mathrm{SL}_2) = 1\).

\(\mathrm{Spin}_8\) (\(D_4\)) and triality

\(\mathrm{Out}(\mathrm{Spin}_8) = S_3\). The three outer "leaves" of the \(D_4\) diagram are the three minuscule representations: vector, half-spin, and the other half-spin. Triality permutes them.

\(E_6\)

\(\mathrm{Out}(E_6) = \mathbb{Z}/2\); the generator is the reflection of the \(E_6\) diagram (swapping the two ends).

Galois twists

For an outer form \(G\) of a split group \(G_0\) over a field \(k\): the Galois action factors through \(\mathrm{Out}(G_0)\), giving a homomorphism \[ \rho : \Gamma_k \;\longrightarrow\; \mathrm{Out}(G_0). \]

Quasi-split outer forms are classified by such homomorphisms up to conjugacy: \[ \{\text{quasi-split outer forms of } G_0\} \;\cong\; \mathrm{Hom}(\Gamma_k, \mathrm{Out}(G_0)) / \mathrm{Out}(G_0). \]

Automorphisms via lattices

For a root datum \((X, \Phi, X^\vee, \Phi^\vee)\), \(\mathrm{Aut}(\Psi)\) is a subgroup of \(\mathrm{Aut}(X)\), and is computable by:

Listing automorphisms of the \(\mathbb{Q}\)-vector space \(X \otimes \mathbb{Q}\) that preserve \(\Phi\);
Restricting to those preserving \(X\) and inducing an isomorphism on the coroot side.

For a semisimple \(\Psi\), \(\mathrm{Aut}(\Psi) = W \rtimes \mathrm{Aut}(\text{diagram})\).

Role in Langlands

L-group construction: \(\mathrm{Out}\) appears as the Galois action on the dual root datum, giving the semi-direct product in \({}^L G = \hat G \rtimes \Gamma\).
Outer forms: classified by \(\mathrm{Hom}(\Gamma, \mathrm{Out}(G_0))\); inner forms are further refined by \(H^1(\Gamma, G_{\mathrm{ad}})\).
Diagonal forms in functoriality: the diagonal embedding \(G \hookrightarrow G \times G^\sigma\) for \(\sigma \in \mathrm{Out}\) is a basic L-morphism.