Pinnings
A pinning of a split connected reductive group rigidifies its internal data so that the only remaining automorphisms are diagram automorphisms. This is the technical device that makes the classification of split reductive groups by based root data unique (not just up to isomorphism).
Definition
A pinning of a split connected reductive group \(G\) over \(k\) is a triple \[ (T, B, \{X_\alpha\}_{\alpha \in \Delta}) \] where:
- \(T \subseteq G\) is a split maximal torus;
- \(B \supseteq T\) is a Borel subgroup containing \(T\) (with corresponding set of simple roots \(\Delta\));
- For each simple root \(\alpha \in \Delta\), \(X_\alpha\) is a chosen non-zero element of the 1-dimensional root space \(\mathfrak{g}_\alpha\) (or equivalently, a chosen isomorphism \(\mathbb{G}_a \xrightarrow{\sim} U_\alpha\)).
The triple \((G, T, B, \{X_\alpha\})\) is a pinned reductive group.
Pinned isomorphisms
A morphism of pinned groups is a homomorphism \(f : (G, T, B, \{X_\alpha\}) \to (G', T', B', \{X'_{\alpha'}\})\) such that:
- \(f(T) \subseteq T'\);
- \(f(B) \subseteq B'\);
- \(f(X_\alpha) = X'_{\alpha'}\) under the induced map of simple roots.
A pinned isomorphism is determined by the underlying map of based root data.
Rigidification theorem
Theorem. Let \((G, T, B, \{X_\alpha\})\) be a pinned split connected reductive group. Then: \[ \mathrm{Aut}^{\mathrm{pin}}(G) \;=\; \mathrm{Aut}(\text{based root datum}) \;=\; \mathrm{Out}(G), \] the diagram automorphism group of \(G\).
Without pinning, \(\mathrm{Aut}(G) = G_{\mathrm{ad}}(k) \rtimes \mathrm{Out}(G)\) with the inner automorphisms forming a large piece. Pinning eliminates the inner factor.
Diagram automorphisms
\(\mathrm{Out}(G)\) is the automorphism group of the Dynkin diagram of \(G\):
| Type | \(\mathrm{Out}\) |
|---|---|
| \(A_n\), \(n \ge 2\) | \(\mathbb{Z}/2\) |
| \(B_n\), \(C_n\) | trivial |
| \(D_n\), \(n \ge 5\) | \(\mathbb{Z}/2\) |
| \(D_4\) | \(S_3\) (triality) |
| \(E_6\) | \(\mathbb{Z}/2\) |
| \(E_7\), \(E_8\) | trivial |
| \(F_4\), \(G_2\) | trivial |
| \(A_1\) | trivial |
These are exactly the symmetries of the Dynkin diagrams.
Twisted forms and pinnings
For a quasi-split form \(G\) of a split group \(G_0\): the Galois action on \(G_{\bar k}\) preserves a pinning of \(G_{\bar k}\), and the Galois action factors through \(\mathrm{Out}(G_{\bar k})\).
This is why pinnings are central to the classification of forms: quasi-split forms are exactly the Galois twists by pinned cocycles.
Examples
\(\mathrm{SL}_n\)
A pinning consists of:
- Diagonal maximal torus \(T\);
- Upper-triangular Borel \(B\);
- For each \(\alpha_i = \varepsilon_i - \varepsilon_{i+1}\), the root vector \(X_{\alpha_i} = E_{i, i+1}\) (the elementary matrix).
\(\mathrm{Out}(\mathrm{SL}_n) = \mathbb{Z}/2\) acts by "transpose-inverse," exchanging \(E_{i, i+1}\) with \(-E_{n-i, n-i+1}\) in pinned form.
\(D_4\) and triality
\(\mathrm{Out}(D_4) = S_3\) acts on the three "outer legs" of the Dynkin diagram. A pinning of \(\mathrm{Spin}_8\) makes this action explicit: \(S_3\) permutes the three minuscule representations of \(\mathrm{Spin}_8\) (the vector representation, and two half-spin representations).
\(G_2\) and \(F_4\)
No non-trivial pinned automorphisms (Dynkin diagrams have no symmetries).
Pinned automorphisms of Galois type
For \(G\) over \(k\) with separable closure \(\bar k\), the Galois group \(\Gamma = \mathrm{Gal}(\bar k / k)\) acts on \(\mathrm{Out}(G_{\bar k})\). A Galois-invariant pinning is a pinning of \(G_{\bar k}\) preserved by \(\Gamma\); equivalently, \(G\) is quasi-split over \(k\) (admits a Borel defined over \(k\)).
The classification of quasi-split forms reduces to: \[ \{\text{quasi-split forms of } G_0\} / \cong \;\cong\; \mathrm{Hom}(\Gamma, \mathrm{Out}(G_0))/\sim. \]
Pinned Chevalley basis
The pinned Lie algebra has a Chevalley basis \(\{X_\alpha, X_{-\alpha}, H_\alpha\}_{\alpha \in \Phi^+}\) with integer structure constants, where \(X_\alpha\) for \(\alpha\) positive non-simple are determined by iterated brackets from the simple root vectors.
This is the key technical device making Chevalley's construction of \(G_{\mathbb{Z}}\) work uniformly.
Role in Langlands
- L-group definition: requires a pinning of the dual group \(\hat G\) to make the Galois action canonical. The L-group is \({}^L G = \hat G \rtimes_{\rho_G} \Gamma\) with \(\rho_G\) acting by pinned automorphisms.
- Pinned dual \(\hat G\): the dual group with a chosen pinning, often denoted \(\hat G^*\) or similar.
- Endoscopy and stable conjugacy: pinnings are used to align reductive groups across inner forms.
References
- Demazure–Grothendieck, SGA 3, vol.\ III (1970): definitive source on pinnings and the Chevalley-Demazure existence theorem.
- Conrad, Reductive group schemes (Notes, 2014), §6.
- Borel, Linear Algebraic Groups (1991), §14.