Strong Real Forms
A strong real form of a complex reductive group is a real form together with a chosen anti-holomorphic involution realising it. Strong real forms are the basic data for the Adams-Barbasch-Vogan (ABV) parametrisation of L-packets at real places.
Definition
Let \(G_\mathbb{C}\) be a connected reductive complex algebraic group.
A strong real form of \(G_\mathbb{C}\) is a pair \((G_\mathbb{R}, \delta)\) where:
- \(G_\mathbb{R}\) is a real form of \(G_\mathbb{C}\) (i.e.\ \(G_\mathbb{R} \otimes \mathbb{C} \cong G_\mathbb{C}\));
- \(\delta : G_\mathbb{C} \to G_\mathbb{C}\) is the anti-holomorphic involution whose fixed point group is \(G_\mathbb{R}(\mathbb{R})\).
Two strong real forms \((G_\mathbb{R}, \delta)\) and \((G_\mathbb{R}', \delta')\) are equivalent if there is an inner automorphism \(\mathrm{Int}(g)\) of \(G_\mathbb{C}\) such that \(\delta' = \mathrm{Int}(g) \circ \delta \circ \mathrm{Int}(g)^{-1}\).
Equivalence with cocycles
A strong real form is equivalent to a 1-cocycle \(c : \mathrm{Gal}(\mathbb{C}/\mathbb{R}) \to G_\mathbb{C}(\mathbb{C})\) (not just into \(G_{\mathrm{ad}}(\mathbb{C})\), as for ordinary inner forms). Two cocycles are equivalent iff cohomologous in \(H^1(\mathbb{R}, G_\mathbb{C})\).
So strong real forms correspond to elements of \[ H^1(\mathbb{R}, G_\mathbb{C}) \;=\; H^1(\mathbb{Z}/2, G_\mathbb{C}(\mathbb{C})). \]
(Cf.\ ordinary inner forms = \(H^1(\mathbb{R}, G_{\mathrm{ad}})\); strong forms include the central character data.)
Examples
\(\mathrm{SL}_2(\mathbb{C})\)
Strong real forms = elements of \(H^1(\mathbb{R}, \mathrm{SL}_2(\mathbb{C})) = \{\pm 1\}\):
- \(c = 1\) (trivial): strong real form = \(\mathrm{SL}_2(\mathbb{R})\) (split).
- \(c = -1\): strong real form = \(\mathrm{SU}(2)\) (compact).
These match the ordinary real forms (only two), but the strong- form data records the central character lift of the inner form data.
\(\mathrm{GL}_n(\mathbb{C})\)
Real forms are \(\mathrm{GL}_n(\mathbb{R})\), \(\mathrm{U}(n)\), \(\mathrm{U}(p, q)\), \(\mathrm{GL}_n(\mathbb{H})\) (for \(n\) even). Strong real forms refine these by central character data.
Outer real forms
For \(D_n\), \(E_6\): the outer real forms have strong-form refinements with further central data; the triality strong forms of \(D_4\) are an example.
Relation to Cartan involutions
A strong real form determines a Cartan involution of \(G_\mathbb{R}\) up to inner conjugacy: take \(\theta = \delta \cdot \tau\) where \(\tau\) is the compact involution. The choice of \(\delta\) (modulo inner) gives the Cartan involution (modulo inner).
So: \[ \{\text{strong real forms}\}/\sim \;\stackrel{\sim}{\longleftrightarrow}\; \{\text{Cartan involutions of } G_\mathbb{R}\}/\mathrm{inner}. \]
Local Langlands at \(\mathbb{R}\)
For an L-parameter \(\varphi : W_\mathbb{R} \to {}^L G\), the Adams-Barbasch-Vogan L-packet at a strong real form \((G_\mathbb{R}, \delta)\) is: \[ \Pi_\varphi(G_\mathbb{R}, \delta) \;\stackrel{\sim}{\longleftrightarrow}\; \mathrm{Irr}(S_\varphi^+), \] the irreducible representations of the (extended) component group \(S_\varphi^+ = \pi_0(Z_{\hat G}(\varphi))\).
The choice of strong-form data is essential to make the bijection canonical; without it, the bijection depends on arbitrary choices.
Role in Langlands
- ABV book: the foundational work for real local Langlands; packets indexed by strong real forms.
- Vogan duality: pairs of L-packets across strong real forms related by an involutive duality.
- Kaletha extension: strong real forms are the prototype that generalises to rigid inner twists in the \(p\)-adic case.
References
- Adams–Barbasch–Vogan, The Langlands Classification and Irreducible Characters for Real Reductive Groups (Progress in Math.\ 1992).
- Vogan, Representations of Real Reductive Lie Groups (Progress in Math.\ 1981).
- Knapp, Lie Groups Beyond an Introduction (2002).