Outer Forms
An outer form of a connected reductive group \(G\) is a form twisted by a cocycle valued in \(\mathrm{Out}(G)\) — equivalently, a non-trivial Galois action on the based root datum of \(G_{\bar k}\). Outer forms are the "non-trivial diagram twists" that arise from Dynkin diagram automorphisms.
Definition
Let \(G_0\) be a split connected reductive group over \(k\). An outer form of \(G_0\) is a form whose isomorphism class in \(H^1(k, \mathrm{Aut}(G_0))\) projects non-trivially to \(H^1(k, \mathrm{Out}(G_0))\) under the exact sequence \[ 1 \to G_{0, \mathrm{ad}} \to \mathrm{Aut}(G_0) \to \mathrm{Out}(G_0) \to 1. \]
Equivalently: \(G\) is an outer form iff the Galois action on the based root datum of \(G_{\bar k}\) is non-trivial.
Classification
The set of outer-form classes is \[ H^1(k, \mathrm{Out}(G_0)) \;=\; \mathrm{Hom}_{\mathrm{cont}}(\Gamma_k, \mathrm{Out}(G_0)) / \mathrm{Out}(G_0)\text{-conj}. \]
Within each outer-form class, the inner-twist refinement is by \(H^1(k, G_{0, \mathrm{ad}}^c)\) (where \(c\) is the outer twist).
Out(G) for irreducible types
| Type | \(\mathrm{Out}(G)\) | Generators |
|---|---|---|
| \(A_n\), \(n \ge 2\) | \(\mathbb{Z}/2\) | reverse Dynkin diagram |
| \(B_n, C_n\) | trivial | — |
| \(D_4\) | \(S_3\) | triality (3-fold) |
| \(D_n\), \(n \ge 5\) | \(\mathbb{Z}/2\) | swap outer leaves |
| \(E_6\) | \(\mathbb{Z}/2\) | reflection |
| \(E_7, E_8\) | trivial | — |
| \(F_4, G_2\) | trivial | — |
So outer forms exist only for types \(A_n\) (\(n \ge 2\)), \(D_n\) (\(n \ge 4\)), and \(E_6\).
Examples
Outer forms of \(\mathrm{GL}_n\) (type \(A_{n-1}\))
\(\mathrm{Out}(\mathrm{GL}_n) = \mathbb{Z}/2\); the non-trivial outer form is the unitary group \(\mathrm{U}(n, K/k)\) for a quadratic extension \(K/k\). The Galois action on the Dynkin diagram of \(A_{n-1}\) is the reflection sending \(\alpha_i \mapsto \alpha_{n-i}\).
Inner forms within the unitary class: \(\mathrm{U}(p, q)\) for \(p + q = n\), classified by the Hermitian form's signature.
\({}^3 D_4\) (triality form)
For a cubic Galois extension \(K/k\) with \(\mathrm{Gal}(K/k) = \mathbb{Z}/3\): the outer form \({}^3 D_4\) is the unique form of \(\mathrm{Spin}_8\) over \(k\) whose Galois action on the Dynkin diagram is the 3-fold cyclic rotation of the outer leaves.
\({}^3 D_4\) shows up over function fields and is foundational for "triality" phenomena in number theory.
Outer forms of \(D_n\) (\(n \ge 5\))
Over \(\mathbb{R}\): the outer forms include \(\mathrm{SO}(p, q)\) for \((p, q)\) different from \((n, n)\) — these are inner forms of \(\mathrm{SO}_{2n}\) for the trivial outer twist.
The non-trivial outer twist gives \({}^2 D_n\), realised by \(\mathrm{SO}^*(2n)\) over the reals (quaternionic form).
\({}^2 E_6\)
Outer form of \(E_6\) determined by a quadratic extension. Over \(\mathbb{R}\): the compact form of \(E_6\) is the quasi-split outer form; \(E_{6(-26)}\) is another real form.
Quasi-split outer forms
Every outer-form class contains a unique quasi-split form (a form with a Borel defined over \(k\)). The quasi-split outer form is the "largest" in its class.
For \(\mathrm{GL}_n\) with non-trivial outer twist: quasi-split form = \(\mathrm{U}(n)\) (the unitary group of the standard split Hermitian form).
Galois representations on the Dynkin diagram
For each Dynkin type and each base field \(k\), classifying outer forms reduces to classifying continuous homomorphisms \[ \Gamma_k \;\to\; \mathrm{Out}(G_0). \]
This is a tractable problem: \(\mathrm{Out}(G_0)\) is a finite group, so the classification reduces to listing the finite Galois extensions of \(k\) with appropriate Galois groups.
For \(k\) a finite field: \(\Gamma_k = \hat{\mathbb{Z}}\), so \(\mathrm{Hom}(\Gamma_k, \mathrm{Out}(G_0)) = \mathrm{Out}(G_0)\) — one outer form per element.
Role in Langlands
- L-group: \({}^L G = \hat G \rtimes \Gamma_k\) with the Galois action coming from the outer twist.
- Endoscopic groups of an outer form may themselves be outer forms — endoscopic transfer must be Galois-compatible.
- Theta correspondence for \(\mathrm{U}(n)\): a major topic of modern automorphic forms, using outer forms as the natural setting.
- Functoriality: lifting from \(\mathrm{U}(n)\) to \(\mathrm{GL}_n\) is base change for the quadratic extension, an instance of L-morphism reversal.
References
- Tits, Classification of algebraic semisimple groups (Proc.\ AMS Symp.\ 1966).
- Borel–Serre, Théorèmes de finitude en cohomologie galoisienne (Comment.\ Math.\ Helv.\ 1964).
- Platonov–Rapinchuk, Algebraic Groups and Number Theory (1994).