Inner Forms
An inner form of a reductive group \(G\) over a field \(k\) is a twist of \(G\) by a 1-cocycle valued in the adjoint group \(G_{\mathrm{ad}}\). Inner forms preserve the "shape" of \(G\) (root datum, Dynkin diagram) but can change its arithmetic properties (splitting, anisotropy, division algebras).
Setup
Let \(G\) be a connected reductive group over \(k\). The inner automorphisms form a normal subgroup \(\mathrm{Inn}(G) = G_{\mathrm{ad}} \subseteq \mathrm{Aut}(G)\), with quotient the outer automorphism group \(\mathrm{Out}(G) = \mathrm{Aut}(\text{based root datum})\): \[ 1 \;\to\; G_{\mathrm{ad}} \;\to\; \mathrm{Aut}(G) \;\to\; \mathrm{Out}(G) \;\to\; 1. \]
An inner form is a \(k\)-form \(G'\) of \(G\) (i.e.\ \(G'_{\bar k} \cong G_{\bar k}\)) such that the Galois action on \(\mathrm{Aut}(G_{\bar k})\) projects trivially to \(\mathrm{Out}\). Equivalently, the twisting cocycle lies in \(G_{\mathrm{ad}}\).
Cohomological classification
The set of inner forms of \(G\) over \(k\) (up to \(k\)-isomorphism) is in bijection with the pointed set \[ H^1(k, G_{\mathrm{ad}}). \]
The marked point is \(G\) itself (the quasi-split inner form when \(G\) is quasi-split).
Examples
\(\mathrm{GL}_n\) and central simple algebras
For \(G = \mathrm{GL}_n\): \(G_{\mathrm{ad}} = \mathrm{PGL}_n\), \(H^1(k, \mathrm{PGL}_n) = \mathrm{Br}(k)[n]\) — the \(n\)-torsion of the Brauer group. So inner forms of \(\mathrm{GL}_n\) are \[ G^c \;=\; D^\times \;=\; (M_r(D'))^\times, \] where \(D\) is a central simple \(k\)-algebra of dimension \(n^2\), \(D'\) the underlying division algebra, and \(r \cdot \deg D' = n\).
The split form (\(D = M_n(k)\)) corresponds to the marked class \(0 \in \mathrm{Br}(k)[n]\).
Real forms of \(\mathrm{SL}_n\)
\(H^1(\mathbb{R}, \mathrm{PGL}_n)\) gives:
- the split form \(\mathrm{SL}_n(\mathbb{R})\);
- the compact form \(\mathrm{SU}_n\);
- intermediate forms \(\mathrm{SU}(p, q)\) for \(p + q = n\).
These are all the "inner forms" of \(\mathrm{SL}_n\) over \(\mathbb{R}\). (Outer forms include \(\mathrm{SL}_n(\mathbb{H})\) for \(n\) even.)
Inner forms of \(\mathrm{Sp}_{2n}\)
\(H^1(k, \mathrm{Sp}_{2n,\mathrm{ad}})\) is non-trivial only when \(k\) admits non-split central simple algebras of exponent 2. For \(k\) local, this is the Brauer group \(\mathrm{Br}(k)[2]\), and the non-trivial element gives a unique non-split inner form.
Inner forms of orthogonal groups
\(H^1(k, \mathrm{SO}_n^{\mathrm{ad}})\) classifies non-degenerate quadratic forms of dimension \(n\) with the same discriminant (and Hasse-Witt invariant constraints).
Inner twists and rigid inner forms
In the modern formulation (Vogan, Kaletha):
- An inner twist \((G^c, c)\) of a quasi-split \(G\) is a pair of an inner form \(G^c\) and a chosen cocycle representative \(c\).
- A rigid inner twist strengthens the cocycle: it carries additional data (a \(z\)-extension lift, e.g.) that rigidifies the L-packet classification.
- Two inner twists are equivalent if they differ by a Galois cohomology trivialisation.
This refinement is essential for stating local Langlands cleanly: the L-packet of a parameter is not naturally indexed by representations of one inner form, but by representations across all inner forms parametrised by rigid inner twists.
Local Langlands and inner forms
For a local field \(F\), the conjectural local Langlands correspondence groups irreducible representations into L-packets: \[ \Pi_\varphi \;\longleftrightarrow\; \varphi : W_F' \to {}^L G. \]
Each L-packet \(\Pi_\varphi\) is divided across all inner forms of \(G\). The component group \(S_\varphi = \pi_0(Z_{\hat G}(\varphi)/Z(\hat G)^\Gamma)\) acts as a refinement, and: \[ \Pi_\varphi \;\leftrightarrow\; \mathrm{Irr}(S_\varphi). \]
The choice of bijection depends on a rigid inner twist (Kaletha's formalism).
Role in endoscopy
The Langlands–Shelstad transfer between inner forms uses:
- transfer factors on \(G\) and \(G^c\);
- character relations between L-packets on the two groups;
- Kaletha's refinement: rigid inner twists make the transfer canonical.
References
- Borel, Automorphic L-functions (Corvallis 1979).
- Kottwitz, Stable trace formula: elliptic singular terms (Math.\ Ann.\ 1986).
- 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) and Rigid inner forms vs isocrystals (J.\ EMS 2018).