Pinning and pinned automorphisms

A pinning of a split connected reductive group $G$ is data

(B,T,\{X_\alpha\}_{\alpha\in\Delta}),

where $B$ is a Borel subgroup, $T\subset B$ a maximal torus, and $X_\alpha$ a nonzero vector, or equivalently a root-group parametrization, for every simple root.

A pinned automorphism preserves $B$ and $T$ and carries the chosen simple-root parametrizations according to its induced permutation of the based root datum. If the simple roots are regarded as individually labeled and each $X_\alpha$ is required to be fixed pointwise, the resulting automorphism is usually the identity; diagram automorphisms require the permutation-compatible formulation.

Splitting outer automorphisms

For a pinned reductive group there is a canonical splitting

\operatorname{Out}(G) \simeq \operatorname{Aut}(\Psi_0(G)) \longrightarrow \operatorname{Aut}(G)

of the passage from automorphisms to the based root datum. This realizes Dynkin-diagram automorphisms as actual group automorphisms without an inner-automorphism ambiguity.

L-group role

For a reductive group over a nonsplit field, the absolute Galois group acts on its based root datum. A pinning of the dual group $\widehat G$ lifts this action to automorphisms and defines the semidirect-product presentation of the $L$ -group. Changing the pinning changes the presentation by an inner isomorphism.

Relation to the letter

The letter's group $\Omega$ acts on pinned root data and hence on the dual construction. Modern terminology makes explicit whether an automorphism fixes labels pointwise or permutes them through a diagram automorphism.

References

T. A. Springer, Linear Algebraic Groups, second edition, Birkhäuser, 1998.
Kevin Buzzard and Toby Gee, “The conjectural connections between automorphic representations and Galois representations,” §2.1. arXiv.