Pinned Dual Groups
The pinned dual group \((\hat G, \hat T, \hat B, \{X_{\alpha^\vee}\})\) is the dual group equipped with a chosen Borel, maximal torus, and simple-coroot vectors. The pinning is the canonical data making the Galois action on the dual root datum lift to an action on \(\hat G\), so that the L-group \({}^L G = \hat G \rtimes \Gamma\) is well-defined.
Definition
For a connected reductive group \(G\) over a field \(k\), a pinned dual group consists of:
- A connected reductive group \(\hat G\) over \(\mathbb{C}\) with the dual root datum of \(G\);
- A maximal torus \(\hat T \subseteq \hat G\);
- A Borel subgroup \(\hat B \supseteq \hat T\);
- For each simple coroot \(\alpha^\vee \in \Delta^\vee\) (= simple root of the dual), a chosen non-zero element \(X_{\alpha^\vee} \in \hat{\mathfrak{g}}_{\alpha^\vee}\).
This pinning is the dual of a pinning of \(G\): the role of simple roots in \(G\) is played by simple coroots in \(\hat G\).
Functoriality
The pinned dual is functorial with respect to pinned isomorphisms: if \((G, T, B, \{X_\alpha\}) \cong (G', T', B', \{X'_\alpha\})\), then \((\hat G, \hat T, \hat B, \{X_{\alpha^\vee}\}) \cong (\hat G', \hat T', \hat B', \{X'_{\alpha^\vee}\})\) canonically.
For a pinning-preserving outer automorphism \(\sigma : G \to G\) (\(\sigma \in \mathrm{Out}(G)\)), the dual \(\hat \sigma : \hat G \to \hat G\) preserves the dual pinning.
Why the pinning matters
Without a pinning, the dual group \(\hat G\) is well-defined up to non-canonical isomorphism — multiple isomorphisms differ by inner automorphisms. This causes problems when:
- Defining the Galois action on \(\hat G\): different choices of \(\hat G\) give different Galois actions.
- Composing L-morphisms: the composition of pinned-preserving maps is canonical.
- Defining the L-group: \({}^L G = \hat G \rtimes \Gamma\) needs a canonical \(\Gamma\)-action.
The pinning rigidifies \(\hat G\) so all these constructions are canonical (not just up to inner conjugacy).
Galois action on pinned dual
For \(G\) defined over \(k\) (not necessarily split): the Galois action on the based root datum of \(G_{\bar k}\) dualises to an action on the based root datum of \(\hat G\). This lifts (via the rigidity of pinned groups) to a Galois action on the pinned dual: \[ \rho_G : \Gamma \;\longrightarrow\; \mathrm{Aut}^{\mathrm{pin}}(\hat G). \]
For quasi-split \(G\): the action factors through \(\mathrm{Out}(\hat G)\) (diagram automorphisms).
For inner forms of a quasi-split \(G\): the action on the pinned dual is the same as for the quasi-split form.
Examples
\(\mathrm{GL}_n\) (split)
Pinned dual: \(\hat G = \mathrm{GL}_n(\mathbb{C})\), \(\hat T\) = diagonal, \(\hat B\) = upper-triangular, \(\{X_{\alpha^\vee}\} = \{E_{i, i+1}\}\). Galois action trivial.
\(\mathrm{U}(n)\) (unitary, quasi-split)
Pinned dual: \(\hat G = \mathrm{GL}_n(\mathbb{C})\) with non-trivial Galois action by the "transpose-inverse" diagram automorphism. \[ {}^L \mathrm{U}(n) \;=\; \mathrm{GL}_n(\mathbb{C}) \rtimes \mathrm{Gal}(K/F), \] where the non-trivial Galois element acts by the outer automorphism.
\({}^3 D_4\), the triality form
Pinned dual: \(\hat G = \mathrm{Spin}_8(\mathbb{C})\) with \(S_3\) action by triality. \({}^L({}^3 D_4) = \mathrm{Spin}_8 \rtimes S_3\) (via the cubic Galois group).
The L-group with pinning
\[ {}^L G \;:=\; \hat G(\mathbb{C}) \;\rtimes\; \Gamma, \] where the semi-direct product uses the Galois action \(\rho_G\) on the pinned dual.
The pinning is essential: without it, the semi-direct product would depend on a non-canonical choice of inner automorphism, and the L-group would only be well-defined up to inner conjugacy.
Relation with quasi-split forms
A connected reductive group \(G\) over \(k\) is quasi-split iff its pinned dual has a \(\Gamma\)-fixed pinning (equivalently, the Galois action on the pinning lifts trivially).
Every group has a quasi-split inner form (unique up to inner-form twist), so the pinned dual of \(G\) coincides with that of its quasi- split form, with the same Galois action.
Mathlib
Not formalised.
References
- Borel, Automorphic L-functions (Corvallis 1979).
- Kottwitz, Stable trace formula: cuspidal tempered terms (Duke 1984), §1.
- Conrad, Reductive group schemes (Notes, 2014), §7.