Exceptional Groups

The exceptional simple algebraic groups are the five families \(G_2, F_4, E_6, E_7, E_8\) — the simple groups not arising from classical bilinear forms. They are the most rigid objects in the classification and reveal the deepest structure of the Langlands programme.

The five Dynkin types

Type	Rank	Dim	Order of Weyl group	Special features
\(G_2\)	2	14	12	Octonion automorphisms; smallest exceptional
\(F_4\)	4	52	1152	Automorphisms of \(\mathfrak{e}_6\); both root lengths
\(E_6\)	6	78	51840	Diagram has \(\mathbb{Z}/2\) outer automorphism
\(E_7\)	7	133	2903040	Centre \(\mathbb{Z}/2\); contains \(E_6 \times \mathrm{GL}_1\)
\(E_8\)	8	248	696729600	Simply connected = adjoint; largest exceptional

Concrete constructions

\(G_2\): Automorphism group of the octonion algebra \(\mathbb{O}\). Equivalently, the stabiliser of a generic \(3\)-form on \(\mathbb{R}^7\). Compact form acts transitively on \(S^6\).
\(F_4\): Automorphism group of the exceptional Jordan algebra \(H_3(\mathbb{O})\) (\(3 \times 3\) Hermitian octonion matrices, 27-dimensional).
\(E_6\): Reduced norm automorphism group of \(H_3(\mathbb{O})\); acts on the Cayley plane \(\mathbb{OP}^2 = E_6/P\) of dim 16.
\(E_7\): Acts on the Freudenthal magic square at position \((4, 3)\); related to symmetries of \(H_3(\mathbb{O})\) at the level of Jordan triples.
\(E_8\): \(248\)-dimensional Lie algebra; the smallest faithful representation is the adjoint. \(E_8\) lattice is the unique even unimodular lattice in dimension \(8\).

Centres and fundamental groups

Type	\(Z\) (sc)	\(\pi_1\) (adj)	self-dual?
\(G_2\)	\(1\)	\(1\)	Yes (Langlands self-dual)
\(F_4\)	\(1\)	\(1\)	Yes
\(E_6\)	\(\mathbb{Z}/3\)	\(\mathbb{Z}/3\)	No: \(E_6^{\mathrm{sc}}\) ↔ \(E_6^{\mathrm{ad}}\) under duality
\(E_7\)	\(\mathbb{Z}/2\)	\(\mathbb{Z}/2\)	No: \(E_7^{\mathrm{sc}}\) ↔ \(E_7^{\mathrm{ad}}\)
\(E_8\)	\(1\)	\(1\)	Yes (sc = adj)

\(G_2\), \(F_4\), \(E_8\) are simply connected = adjoint; \(E_6\) and \(E_7\) have non-trivial isogenies between simply-connected and adjoint forms.

Forms over arbitrary fields

For each exceptional type, the non-split forms over a field \(k\) are classified by Galois cohomology of the split form, encoded by Tits indices (or Vogan diagrams for real forms):

\(G_2\): Two real forms (split, compact); no inner forms over \(p\)-adic fields except split.
\(F_4\): Two real forms (split, compact, and one intermediate); over local fields some non-split forms exist.
\(E_6\): Real forms include the split form, the compact form, and intermediate ones; outer forms include \({}^2 E_6\) (the Galois group acts non-trivially on the Dynkin diagram).
\(E_7\): Various real and \(p\)-adic forms; the "intermediate" real form \(E_{7(7)}\) plays a role in supergravity.
\(E_8\): Three real forms (split, compact, one intermediate); no outer forms (\(E_8\) Dynkin has no auto).

Number of nilpotent orbits

By Bala-Carter:

Type	Nilpotent orbits
\(G_2\)	5
\(F_4\)	16
\(E_6\)	21
\(E_7\)	45
\(E_8\)	70

These bound the complexity of unipotent representation theory.

Role in Langlands

\(G_2\) and \(F_4\): serve as test cases for the conjectural unipotent local Langlands; the theta correspondence for \(G_2 \subset \mathrm{SO}_7\) is well-studied.
\(E_6\): Shimura varieties of \(E_6\)-type provide explicit realisations of Galois representations.
\(E_7\): Cohomology of \(E_7\)-Shimura varieties; the \(\mathrm{GL}_{56}\)-functoriality conjecture.
\(E_8\): A "universal" target for Langlands functoriality; conjecturally all Galois representations factor through some \({}^L G \subseteq E_8\).
Magic square and Freudenthal triple systems: exceptional groups appear in unified constructions involving composition algebras.

References

Jacobson, Exceptional Lie Algebras (1971).
Adams, Lectures on Exceptional Lie Groups (1996).
Carter, Finite Groups of Lie Type (1985), Ch. 7.
Tits, Classification of algebraic semisimple groups (Proc.\ AMS Symp.\ 1966).
Springer–Steinberg, Conjugacy classes (Lecture Notes 1970).