Pure Inner Twists
A pure inner twist of a connected reductive group is an inner twist together with a chosen cocycle in \(G\) itself (rather than just \(G_{\mathrm{ad}}\)). The "pure" refinement (due to Vogan) keeps track of the \(Z(G)\)-part of the cocycle, making the L-packet parametrisation cleaner.
Definition
Let \(G\) be a connected reductive group over a field \(k\). A pure inner twist of \(G\) is a pair \((G^*, c)\) where:
- \(c \in Z^1(k, G)\) is a 1-cocycle valued in \(G\) (not just \(G_{\mathrm{ad}}\));
- \(G^* = G^c\) is the inner-twisted form of \(G\) by the image of \(c\) in \(G_{\mathrm{ad}}\).
Two pure inner twists are equivalent if their cocycles are cohomologous in \(Z^1(k, G)\).
Vs.\ ordinary inner twists
| Object | Cocycle in | Equivalence | Classifies |
|---|---|---|---|
| Inner form | \(Z^1(k, G_{\mathrm{ad}})\) | cohomology in \(G_{\mathrm{ad}}\) | inner forms of \(G\) |
| Pure inner twist | \(Z^1(k, G)\) | cohomology in \(G\) | inner forms + central character refinement |
The natural map \[ H^1(k, G) \;\to\; H^1(k, G_{\mathrm{ad}}) \] sends pure inner twists to ordinary inner forms; the fibres are torsors under the central character data of \(Z(G)\).
Long exact sequence
The exact sequence \(1 \to Z(G) \to G \to G_{\mathrm{ad}} \to 1\) induces: \[ H^1(k, Z(G)) \;\to\; H^1(k, G) \;\to\; H^1(k, G_{\mathrm{ad}}) \;\to\; H^2(k, Z(G)). \]
So:
- The map \(H^1(k, G) \to H^1(k, G_{\mathrm{ad}})\) has fibres parametrised by \(H^1(k, Z(G))\).
- The image is the kernel of the boundary \(H^1(k, G_{\mathrm{ad}}) \to H^2(k, Z(G))\); only inner forms whose central character data "lifts" come from pure inner twists.
Vogan's L-packet refinement
Vogan's bijection (1993, refined by Kaletha). For an L-parameter \(\varphi : W_F \to {}^L G\), the L-packet \(\Pi_\varphi\) (across all inner forms of \(G\)) is in bijection with irreducible representations of the enhanced \(S_\varphi\)-group: \[ S_\varphi^+ \;:=\; \pi_0(Z_{\hat G}(\varphi)) / \text{centre}, \] where the parametrisation is canonical when one uses pure inner twists (rather than just inner forms).
This is the basis of the modern formulation of local Langlands across inner forms.
Example: Real \(\mathrm{SL}_2\)
For \(G = \mathrm{SL}_2 / \mathbb{R}\):
- \(Z(G) = \{\pm 1\}\);
- \(H^1(\mathbb{R}, Z(G)) = \{\pm 1\}\);
- \(H^1(\mathbb{R}, G) = \{\pm 1\}\) (split form \(\mathrm{SL}_2(\mathbb{R})\) and compact form \(\mathrm{SU}(2)\));
- \(H^1(\mathbb{R}, G_{\mathrm{ad}}) = \{\pm 1\}\) (split form \(\mathrm{PGL}_2(\mathbb{R})\) and compact form \(\mathrm{PSU}(2)\)).
Pure inner twists: 4 classes (split or compact \(\times\) trivial or non-trivial central character lift), with the trivial lift recovering the standard inner-form classification.
Limitations of pure inner twists
Pure inner twists work cleanly when \(G\) has appropriate central extensions — specifically when the long exact sequence cleanly parametrises the fibres of \(H^1(k, G) \to H^1(k, G_{\mathrm{ad}})\).
For more general settings (e.g.\ non-quasi-split inner forms with "big" central character data), the rigid inner twist formalism of Kaletha (using a Galois gerbe) is needed; see rigid inner twists.
Role in Langlands
- Local Langlands across inner forms: pure inner twists give Vogan's parametrisation of L-packets.
- Endoscopic transfer: the transfer between inner forms is cleanest with pure inner twists.
- Adjoint vs.\ pure groups: distinguishing the central data matters for genuine vs.\ non-genuine representations of metaplectic covers.
References
- Vogan, The local Langlands conjecture (in: Representation Theory of Groups and Algebras, AMS 1993).
- Kottwitz, Stable trace formula: elliptic singular terms (Math.\ Ann.\ 1986).
- Kaletha, Genericity and contragredience in the local Langlands correspondence (Algebra Number Theory 2013).