Weighted Dynkin Diagrams
A weighted Dynkin diagram is a labelling of the simple roots of a reductive Lie algebra by weights in \(\{0, 1, 2\}\), serving as the canonical combinatorial invariant of a nilpotent orbit (or more precisely, of an \(\mathfrak{sl}_2\)-triple) up to \(G\)-conjugacy.
Construction
Let \(\mathfrak{g}\) be a complex semisimple Lie algebra (or reductive over a field of good characteristic) with fixed Cartan subalgebra \(\mathfrak{h}\) and simple roots \(\Delta = \{\alpha_1, \dots, \alpha_r\}\). Given a nonzero nilpotent \(X \in \mathfrak{g}\), the Jacobson–Morozov theorem produces an \(\mathfrak{sl}_2\)-triple \((X, H, Y)\) with \(H \in \mathfrak{g}\) semisimple.
\(H\) can be \(G\)-conjugated into the dominant chamber of \(\mathfrak{h}\). The weighted Dynkin diagram of \(X\) is the labelling \[ \alpha_i \;\longmapsto\; d_i := \alpha_i(H) \in \{0, 1, 2\}. \]
Theorem (Dynkin, Kostant). The map \[ \{\text{nilpotent } G\text{-orbits in } \mathfrak{g}\} \;\longrightarrow\; \{\text{weighted Dynkin diagrams on } \Delta\} \] is injective, with image characterised by combinatorial conditions on each Dynkin type. In particular, weighted Dynkin diagrams classify nilpotent orbits.
Why the weights are in {0, 1, 2}
For an \(\mathfrak{sl}_2\)-triple \((X, H, Y)\) inside a complex semisimple Lie algebra, all eigenvalues of \(\operatorname{ad}(H)\) on \(\mathfrak{g}\) are integers (\(\mathfrak{sl}_2\)-representation theory). The fact that the simple-root values are in \(\{0, 1, 2\}\) specifically uses that the dominant representative of \(H\) acts on the simple-root spaces \(\mathfrak{g}_{\alpha_i}\) by an eigenvalue \(\alpha_i(H) \ge 0\), and the height-grading bound for nilpotent elements gives the upper bound \(2\).
Distinguished orbits
A weighted Dynkin diagram with all weights in \(\{0, 2\}\) corresponds to a distinguished nilpotent orbit (an orbit whose stabiliser \(Z_G(X)^\circ\) is unipotent).
Theorem (Bala-Carter). Every nilpotent orbit \(\mathcal{O}\) in \(\mathfrak{g}\) is induced from a distinguished orbit in a Levi subalgebra \(\mathfrak{l}\). The weighted Dynkin diagram of \(\mathcal{O}\) restricts on \(\mathfrak{l}\) to the (\(\{0,2\}\)) diagram of the distinguished orbit, and is zero on simple roots outside \(\mathfrak{l}\).
This reduces the classification to listing distinguished orbits in each Levi.
Examples
\(\mathfrak{sl}_3\) (type \(A_2\))
Three nilpotent orbits:
| Orbit | Partition | Weighted Dynkin |
|---|---|---|
| \(\{0\}\) | \((1, 1, 1)\) | \(0 - 0\) |
| subregular | \((2, 1)\) | \(1 - 1\) |
| principal | \((3)\) | \(2 - 2\) |
\(G_2\)
Five nilpotent orbits with weighted Dynkin diagrams \(0 - 0\), \(0 - 1\), \(1 - 0\), \(0 - 2\), \(2 - 2\) (labels on short root then long root).
The diagram \(0 - 1\) corresponds to the minimal nilpotent orbit; \(2 - 2\) is the principal.
The Dynkin-Kostant labelling vs. Bala-Carter
Both are bijections from nilpotent orbits to combinatorial data, but the encodings differ:
- Dynkin-Kostant = weighted diagram on the full Dynkin diagram of \(\mathfrak{g}\).
- Bala-Carter = pair (Levi Dynkin subdiagram, distinguished orbit in that Levi).
A Bala-Carter label of the form "Levi type, distinguished decoration" (e.g.\ \(A_2 + A_1\), \(D_4(a_1)\), \(E_6(a_3)\)) corresponds to a unique weighted Dynkin diagram and vice versa.
Role for Langlands
- Arthur parameters include an \(\mathfrak{sl}_2\)-factor in \(\hat G \times W_F \times \mathrm{SL}_2(\mathbb{C})\), whose associated nilpotent orbit determines the Arthur SL_2 type; weighted Dynkin diagrams are the natural labels.
- Endoscopic transfer of nilpotent orbits is described in terms of weighted Dynkin diagrams.
References
- Collingwood–McGovern, Nilpotent Orbits in Semisimple Lie Algebras (1993), Ch. 3.
- Carter, Finite Groups of Lie Type (1985), §5.7.
- Bala-Carter, Classes of unipotent elements in simple algebraic groups (Math. Proc. Camb. Phil. Soc. 1976).