Extended Groups for Real Forms
The extended group \(G^\Gamma\) (in the ABV formalism) is a semi- direct product of the complex reductive group with the absolute Galois group of \(\mathbb{C}/\mathbb{R}\). It is the natural setting for parametrising all real forms (and their L-packets) simultaneously.
Definition
For a connected reductive complex group \(G_\mathbb{C}\), let \(\Gamma = \mathrm{Gal}(\mathbb{C}/\mathbb{R}) = \mathbb{Z}/2\), and let \(\theta : G_\mathbb{C} \to G_\mathbb{C}\) be a fixed Cartan involution (an inner automorphism realising the compact form).
The extended group is \[ G^\Gamma \;:=\; G_\mathbb{C} \;\rtimes\; \mathbb{Z}/2, \] where the non-trivial element of \(\mathbb{Z}/2\) acts by an anti-holomorphic involution \(\delta\) realising one of the real forms.
The "double coset" structure of \(G^\Gamma\) encodes all the real forms simultaneously: each real form corresponds to a conjugacy class of involutions in \(G^\Gamma\) lifting the non-trivial element of \(\mathbb{Z}/2\).
Strong real forms via \(G^\Gamma\)
A strong real form corresponds to a specific conjugacy class of order-2 elements in \(G^\Gamma\) projecting non-trivially. Two strong real forms differing by an inner automorphism correspond to elements in the same \(G_\mathbb{C}\)-conjugacy class within \(G^\Gamma\).
The set \[ \mathcal{X}(G^\Gamma) \;:=\; \{g \in G^\Gamma : g^2 = 1, \;g \notin G_\mathbb{C}\} / G_\mathbb{C}\text{-conj} \] classifies strong real forms.
Why the extended group?
- Uniform framework: all real forms appear as conjugacy classes in a single object.
- Canonical Cartan: the choice of base Cartan involution \(\theta\) makes the construction work coherently across forms.
- Cohomology: \(H^1(\mathbb{R}, G_\mathbb{C}) = H^1(\Gamma, G_\mathbb{C}(\mathbb{C}))\) is computable via standard group-cohomology techniques on the semidirect product.
ABV book setting
In the Adams-Barbasch-Vogan book (1992), the extended group is the fundamental object for stating and proving the local Langlands correspondence at \(\mathbb{R}\). The L-parameter \(\varphi : W_\mathbb{R} \to {}^L G\) and the strong real form \((G_\mathbb{R}, \delta)\) are paired via the extended group:
- \(W_\mathbb{R}\) has a specific structure with non-trivial Frobenius \(j\) (with \(j^2 = -1\)) — a non-trivial central extension of \(\mathbb{Z}/2\) by \(\mathbb{C}^\times\).
- The map \(\varphi\) lands in \({}^L G = \hat G \rtimes \Gamma\), and the "Vogan duality" matches it with strong-form data via the extended group.
Examples
\(G = \mathrm{SL}_2\)
\(G^\Gamma = \mathrm{SL}_2(\mathbb{C}) \rtimes \mathbb{Z}/2\) (with the non-trivial Galois acting by complex conjugation followed by transpose-inverse). Two real forms:
- Trivial conjugacy class of involutions → \(\mathrm{SU}(2)\) (compact).
- Non-trivial conjugacy class → \(\mathrm{SL}_2(\mathbb{R})\) (split).
\(G = \mathrm{GL}_n\)
Multiple real forms (\(\mathrm{GL}_n(\mathbb{R})\), \(\mathrm{U}(p, q)\), \(\mathrm{GL}_n(\mathbb{H})\) for \(n\) even) all appear as conjugacy classes in \(G^\Gamma\).
Generalisations to \(p\)-adic and global
For \(p\)-adic fields: the analogous construction uses a Galois gerbe (Kaletha), not just a semidirect product, because \(H^1(F, G)\) is generally more complex.
For global fields: the global ABV / Kaletha framework uses a global Galois gerbe and the "Hasse-principle gerbe" connecting it to local data at all places.
Role in Langlands
- Real L-packets: the ABV parametrisation indexes packets by representations of component groups of L-parameters within \(G^\Gamma\).
- Kazhdan-Lusztig polynomials: real KL polynomials are computed on the extended group, controlling composition multiplicities of cohomological induction.
- Vogan duality: an involution on representations realised at the level of the extended group.
References
- Adams–Barbasch–Vogan, The Langlands Classification and Irreducible Characters for Real Reductive Groups (1992).
- Vogan, Representations of Real Reductive Lie Groups (1981).
- Borel, Automorphic L-functions (Corvallis 1979), §10.