Topic Vogan Diagrams
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Vogan Diagrams

A Vogan diagram is a Dynkin diagram with two additional decorations encoding a real form: vertices coloured "compact" or "non-compact," and an optional diagram automorphism. Vogan diagrams classify real forms of complex semisimple Lie algebras combinatorially.

Definition

A Vogan diagram of a simple Lie algebra of type \(X_n\) is a Dynkin diagram of type \(X_n\) together with:

  1. A choice of base \(\Delta\) (simple roots);
  2. A 2-colouring of \(\Delta\): each \(\alpha \in \Delta\) marked white (compact) or black (non-compact);
  3. Optionally: a diagram automorphism \(\sigma\) compatible with the colouring.

Constraints:

  • If \(\sigma = \mathrm{id}\) (trivial diagram automorphism): the Vogan diagram corresponds to an inner real form.
  • If \(\sigma \ne \mathrm{id}\): the Vogan diagram corresponds to an outer real form.

From Vogan diagram to real form

A Vogan diagram determines a real form \(G_\mathbb{R}\) of the complex form \(G_\mathbb{C}\) of type \(X_n\) via the Cartan involution:

  • \(\theta\) acts on the Cartan as \(\sigma\) (diagram automorphism);
  • For each simple root \(\alpha\):
  • White: \(\theta\) is trivial on \(\mathfrak{g}_\alpha\) (compact direction);
  • Black: \(\theta\) is \(-1\) on \(\mathfrak{g}_\alpha\) (non-compact direction).

The fixed points of \(\theta\) give the Cartan involution / maximal compact subgroup of \(G_\mathbb{R}\).

Equivalence classes

Two Vogan diagrams give the same real form iff they differ by a sequence of Cayley transforms through non-compact simple roots.

For each Dynkin type, there is a standard "normal form" Vogan diagram for each real form; classification is done by tabulating these.

Examples

\(\mathfrak{sl}_n\) (type \(A_{n-1}\))

Vogan diagram Real form
all white, \(\sigma = \mathrm{id}\) \(\mathfrak{su}(n)\) (compact)
all black, \(\sigma = \mathrm{id}\) \(\mathfrak{sl}_n(\mathbb{R})\) (split)
pattern of black/white \(\mathfrak{su}(p, q)\) (mixed signature)
with \(\sigma\) = reverse \(\mathfrak{sl}_n(\mathbb{H})\) (\(n\) even)

\(\mathfrak{so}_{2n+1}\) (type \(B_n\))

\(\mathrm{Out} = 1\), so only inner real forms:

  • \(\mathfrak{so}(2n+1)\) (compact, all white);
  • \(\mathfrak{so}(p, q)\) with \(p + q = 2n+1\) (mixed black/white).

\(\mathfrak{e}_8\)

Three real forms: compact \(\mathfrak{e}_{8(-248)}\), split \(\mathfrak{e}_{8(8)}\), and the intermediate \(\mathfrak{e}_{8(-24)}\). Vogan diagrams: respectively all white, all black, and a specific colouring of 8 nodes.

Borel-de Siebenthal theorem

Theorem (Borel-de Siebenthal). Real forms of a complex simple Lie algebra \(\mathfrak{g}_\mathbb{C}\) of type \(X_n\) are classified by:

  • Vogan diagrams modulo equivalence (Cayley transforms);
  • Equivalently, conjugacy classes of order-2 elements in \(\mathrm{Aut}(\mathfrak{g}_\mathbb{C})\).

The conjugacy classes of involutions are computable via extended Dynkin diagrams (Kac labels): for each node of the extended Dynkin diagram, there is a corresponding involution.

Real-form invariants

From a Vogan diagram \((X_n, \text{colouring}, \sigma)\) one can read:

  • Compact rank = number of white vertices + dim of automorphism fixed space (for outer forms).
  • Non-compact rank = number of black vertices.
  • Signature of the Killing form = (compact dim) - (non-compact dim).

Role in Langlands

  • Real Langlands: Vogan diagrams give the input for the ABV / Vogan parametrisation of L-packets.
  • Discrete series: existence of discrete series for a real form is read off from the Vogan diagram (compact-modulo-centre maximal torus condition).
  • Cohomological induction: Vogan-Zuckerman cohomological induction is parametrised by sub-diagrams of the Vogan diagram.

References

  • Vogan, Representations of Real Reductive Lie Groups (1981), §6.
  • Knapp, Lie Groups Beyond an Introduction (2002), Ch. VI.
  • Onishchik–Vinberg, Lie Groups and Algebraic Groups (1990).
  • Adams–Barbasch–Vogan (1992), §3.

Prerequisite Topics