Real Forms

A real form of a complex reductive group \(G_\mathbb{C}\) is an \(\mathbb{R}\)-form: a real algebraic group \(G_\mathbb{R}\) with \(G_\mathbb{R} \otimes_\mathbb{R} \mathbb{C} \cong G_\mathbb{C}\). Real forms are classified by Cartan involutions, and they give the real Lie groups of "classical Lie theory."

Definition

For a connected reductive algebraic group \(G_\mathbb{C}\) over \(\mathbb{C}\), a real form is an algebraic group \(G_\mathbb{R}\) over \(\mathbb{R}\) such that \[ G_\mathbb{R} \otimes_\mathbb{R} \mathbb{C} \;\cong\; G_\mathbb{C}. \]

Equivalently: a \(\mathbb{C}\)-anti-linear involution \(\sigma : G_\mathbb{C} \to G_\mathbb{C}\) (with \(G_\mathbb{R} = G_\mathbb{C}^\sigma\)) commuting with the group structure.

Classification via Galois cohomology

\(\mathrm{Gal}(\mathbb{C}/\mathbb{R}) = \mathbb{Z}/2\), so: \[ \{\text{real forms of } G_\mathbb{C}\} \;\cong\; H^1(\mathbb{R}, \mathrm{Aut}(G_\mathbb{C})). \]

The Galois cohomology of \(\mathrm{Aut}(G_\mathbb{C})\) decomposes into inner and outer parts via the exact sequence: \[ 1 \to G_{\mathbb{C}, \mathrm{ad}} \to \mathrm{Aut}(G_\mathbb{C}) \to \mathrm{Out}(G_\mathbb{C}) \to 1. \]

Cartan involutions

A Cartan involution of a real Lie group \(G_\mathbb{R}\) is an involution \(\theta : G_\mathbb{R} \to G_\mathbb{R}\) such that \(K = G_\mathbb{R}^\theta\) is a maximal compact subgroup.

Theorem (Cartan). Every connected reductive real Lie group has a Cartan involution, unique up to inner automorphism by an element of \(K\). The bijection: \[ \{\text{real forms of } G_\mathbb{C}\}/\mathrm{conj} \;\stackrel{\sim}{\longleftrightarrow}\; \{\text{Cartan involutions of complex form}\}/\mathrm{conj}. \]

Examples

Real forms of \(\mathrm{SL}_n(\mathbb{C})\)

Form	Description
\(\mathrm{SL}_n(\mathbb{R})\)	split form
\(\mathrm{SU}(n)\)	compact form
\(\mathrm{SU}(p, q)\), \(p + q = n\)	mixed signature; inner forms of split
\(\mathrm{SL}_n(\mathbb{H}) = \mathrm{SU}^*(2n)\), \(n\) even	quaternionic; outer form

Real forms of \(\mathrm{Spin}_{2n+1}(\mathbb{C})\)

\(\mathrm{Spin}(p, q)\) for \(p + q = 2n+1\); the compact form is \(\mathrm{Spin}(2n+1)\).

Real forms of exceptional groups

Type	Compact	Split	Intermediate
\(G_2\)	\(G_{2(-14)}\)	\(G_{2(2)}\)	—
\(F_4\)	\(F_{4(-52)}\)	\(F_{4(4)}\)	\(F_{4(-20)}\)
\(E_6\)	\(E_{6(-78)}\)	\(E_{6(6)}\)	\(E_{6(-26)}, E_{6(-14)}, E_{6(2)}\) (and \({}^2 E_6\) forms)
\(E_7\)	compact	split \(E_{7(7)}\)	various intermediates
\(E_8\)	compact	split \(E_{8(8)}\)	\(E_{8(-24)}\)

The notation \(E_{6(s)}\) records the signature \(s\) of the Killing form on the Lie algebra (= dim non-compact minus dim compact).

Compact and split forms

Two distinguished real forms:

Compact form: \(K \le G_\mathbb{R}\) is a maximal compact subgroup; for the compact form, \(K = G_\mathbb{R}\) (so \(G_\mathbb{R}\) is itself compact).
Split form: maximal \(\mathbb{R}\)-split torus has dimension = rank of \(G_\mathbb{C}\); equivalently, \(G_\mathbb{R}\) has the largest possible Cartan subgroup defined over \(\mathbb{R}\).

Every semisimple complex group has a unique compact form (up to isomorphism) and a unique split form.

Vogan diagrams

A Vogan diagram is a Dynkin diagram with additional decorations encoding a real form:

Vertices coloured "compact" or "non-compact" (black/white circles);
Optional diagram automorphism (for outer real forms).

Theorem (Borel-de Siebenthal, Vogan). Vogan diagrams classify real forms of complex semisimple Lie algebras up to inner automorphism.

Cartan decomposition

For a real form \(G_\mathbb{R}\) with Cartan involution \(\theta\): \[ \mathfrak{g}_\mathbb{R} \;=\; \mathfrak{k} \;\oplus\; \mathfrak{p}, \] where \(\mathfrak{k} = \mathfrak{g}_\mathbb{R}^\theta\) (Lie algebra of \(K\)) and \(\mathfrak{p} = \mathfrak{g}_\mathbb{R}^{-\theta}\) (the "non-compact" direction).

The Killing form is negative-definite on \(\mathfrak{k}\) and positive-definite on \(\mathfrak{p}\); this is the foundation of symmetric-space theory.

Restricted root system

The action of \(\mathfrak{a} \subseteq \mathfrak{p}\) (a maximal abelian subalgebra) on \(\mathfrak{g}_\mathbb{R}\) decomposes \(\mathfrak{g}_\mathbb{R}\) into restricted root spaces, with the non-zero weights forming the restricted root system.

The restricted root system can be smaller than the absolute root system (and need not be reduced — type \(BC_n\) occurs).

Role in Langlands

Real Langlands: parameters \(W_\mathbb{R} \to {}^L G\) where \(W_\mathbb{R}\) is the Weil group of \(\mathbb{R}\) (an extension of \(\mathbb{Z}/2\) by \(\mathbb{C}^\times\)).
Discrete series: real reductive groups with equal-rank maximal compact subgroup carry discrete series representations (Harish-Chandra). The condition is real-form-specific.
Vogan-Zuckerman duality: relates real-form L-packets to cohomological packets via Vogan-Zuckerman cohomological induction.