Strong Inner Forms
A strong inner form (Adams-Barbasch-Vogan) is an inner form equipped with an additional rigidifying datum (a "strong real form" in the real case). Strong inner forms provide a Vogan-style parametrisation of L-packets that is canonical and includes a "central character" refinement.
Definition (real case)
A strong real form of a complex reductive group \(G_\mathbb{C}\) is a pair \((G_\mathbb{R}, \delta)\) where:
- \(G_\mathbb{R}\) is a real form of \(G_\mathbb{C}\);
- \(\delta \in G_\mathbb{R}^*\) (a section of complex conjugation, i.e.\ a specific anti-holomorphic involution of \(G_\mathbb{C}\) realising the form).
Two strong real forms \((G_\mathbb{R}, \delta)\) and \((G_\mathbb{R}', \delta')\) are equivalent if there is an inner automorphism of \(G_\mathbb{C}\) intertwining them.
Strong inner forms for general fields
For a connected reductive \(G\) over an arbitrary field \(k\):
A strong inner form is a triple \((G^*, \delta, \omega)\) where:
- \(G^*\) is an inner form of \(G\);
- \(\delta : G \to G^*\) is a specific inner-twisting isomorphism over \(\bar k\);
- \(\omega\) is a "central character" lift specifying the central data of the cocycle.
The triple data is what makes the inner form strong: the central character lifts the cocycle from \(H^1(k, G_{\mathrm{ad}})\) to a finer set parametrising both forms and central data.
Adams-Barbasch-Vogan parametrisation
Theorem (ABV). For a real reductive Lie group \(G_\mathbb{R}\), the L-packet of an L-parameter \(\varphi\) on a strong real form \((G_\mathbb{R}, \delta)\) is canonically parametrised by irreducible representations of the component group \(S_\varphi^+\) of the centraliser of \(\varphi\) in \(\hat G\).
The bijection \[ \Pi_\varphi(G_\mathbb{R}, \delta) \;\stackrel{\sim}{\longleftrightarrow}\; \mathrm{Irr}(S_\varphi^+) \] depends on the strong-form data \((G_\mathbb{R}, \delta)\); changing \(\delta\) changes the basepoint of the bijection.
Examples
\(\mathrm{SL}_2(\mathbb{R})\) and \(\mathrm{SU}(2)\)
The two real forms of \(\mathrm{SL}_2(\mathbb{C})\); each has strong- form refinements determined by the choice of involution and a sign \(\pm\) for the central character lift.
Discrete series of \(\mathrm{SL}_2(\mathbb{R})\)
L-packets for \(\mathrm{SL}_2(\mathbb{R})\) contain pairs of discrete series; strong-form data distinguishes them. The "holomorphic" vs.\ "anti-holomorphic" discrete series correspond to the two strong-form choices.
Relation to other refinements
| Refinement | Data | Reference |
|---|---|---|
| Inner form | \(H^1(k, G_{\mathrm{ad}})\) | classical Vogan |
| Pure inner twist | \(H^1(k, G)\) | Vogan 1993 |
| Strong inner form | + central character lift | ABV 1992 |
| Rigid inner twist | Galois gerbe | Kaletha 2016 |
Each refinement is a finer parametrisation of L-packet members; rigid inner twists give the most canonical version.
Role in Langlands
- Real Langlands: ABV parametrisation is the foundation of the real local Langlands correspondence.
- Vogan's \(\mathrm{PSL}_2(\mathbb{C})\)-trick: passing between strong forms and ordinary forms via Galois cohomology of the centre.
- Arthur's parameters: Arthur's classification for orthogonal and symplectic groups uses strong-form data implicitly.
References
- Adams–Barbasch–Vogan, The Langlands Classification and Irreducible Characters for Real Reductive Groups (Progress in Math.\ 1992).
- Vogan, The local Langlands conjecture (in: Representation Theory of Groups and Algebras, AMS 1993).
- Kaletha, Rigid inner forms of real and p-adic groups (Annals 2016).