Quasi-Split Forms
A quasi-split form of a reductive group is one with a Borel subgroup defined over the base field. Quasi-split forms are the "largest" forms in their outer-twist class — every form differs from the quasi-split one only by an inner twist. They are the natural "canonical representatives" in the classification of reductive groups.
Definition
Let \(G\) be a connected reductive group over a field \(k\). \(G\) is quasi-split if it has a Borel subgroup defined over \(k\): \[ \exists \, B \subseteq G, \; B \text{ a Borel subgroup with } B = B_k. \]
Equivalently: \(G\) admits a \(k\)-rational pinning, i.e.\ a maximal torus, Borel, and root vectors all defined over \(k\) (modulo sign).
Existence and uniqueness
Theorem. Every outer-form class contains a unique quasi-split form (up to \(k\)-isomorphism).
The quasi-split form is the trivial inner twist within its outer-form class: \(H^1(k, G_{\mathrm{ad}})\) measures the inner twists from the quasi-split form.
Theorem. Every connected reductive \(k\)-group \(G\) is an inner form of a unique quasi-split form.
Examples
Quasi-split forms of \(\mathrm{GL}_n\)
For \(\mathrm{GL}_n\) (which has \(\mathrm{Out} = \mathbb{Z}/2\)):
- Trivial outer twist: quasi-split form = \(\mathrm{GL}_n(k)\) itself (split). Inner forms: \(\mathrm{GL}_r(D)\) for \(D/k\) of degree \(n/r\).
- Non-trivial outer twist (via quadratic extension \(K/k\)): quasi-split form = \(\mathrm{U}(n)\) (= unitary group of the standard Hermitian form). Inner forms: \(\mathrm{U}(p, q)\) with \(p + q = n\) for Hermitian forms of signature \((p, q)\).
Quasi-split form of \(D_4\): triality
For the triality form of \(D_4\) (over a cubic Galois extension \(K/k\)), the quasi-split form is \({}^3 D_4\): the unique form with a Borel defined over \(k\) and Galois action permuting the three outer "leaves" of the Dynkin diagram cyclically.
Quasi-split form of \(E_6\)
\(E_6^{\mathrm{qs}}\) is the unique outer form of \(E_6\) over \(k\) admitting a Borel; over \(\mathbb{R}\) it is the split form \(E_{6(6)}\) (in Cartan's notation).
Borel-Serre criterion
Theorem. A connected reductive \(G/k\) is quasi-split iff \(H^1(k, B)\) is trivial for some Borel \(B \subseteq G\) defined over \(\bar k\) (equivalently: there is a Galois-equivariant Borel of \(G_{\bar k}\)).
This is equivalent to the Galois action factoring through \(\mathrm{Out}(G)\), preserving a chosen pinning.
Quasi-split reductive groups have a Whittaker model
For a quasi-split \(G\) over a local field \(F\) with a Borel \(B = T U\): there is a non-degenerate character \(\psi : U(F) \to \mathbb{C}^\times\) (in particular generic), and the Whittaker model \(\mathrm{Hom}_{U(F)}(\pi, \psi)\) is non-trivial for many \(\pi\).
In contrast: for an inner form that is not quasi-split, the Whittaker model can be zero for all irreducible \(\pi\) (non-generic representations are then the rule).
Role in Langlands
- L-packets: structured around the quasi-split form. The L-packet of an L-parameter \(\varphi\) contains a unique generic member on the quasi-split form (= a member with non-trivial Whittaker model).
- Endoscopy: defined relative to the quasi-split form as the reference; inner forms enter via the rigid-inner-form formalism.
- Trace formula: Arthur's stable trace formula compares trace formulas on \(G\) and its endoscopic groups; the quasi-split form serves as the "stable" reference.
- Galois representations: for quasi-split groups, L-parameters match cleanly with Galois representations (without ambiguity from inner twists).
Mathlib
Not yet formalised.
References
- Borel–Tits, Groupes réductifs (Publ.\ IHES 1965).
- Tits, Classification of algebraic semisimple groups (Proc.\ AMS Symp.\ 1966).
- Kottwitz, Stable trace formula: cuspidal tempered terms (Duke 1984).
- Borel, Linear Algebraic Groups (1991), §16.