Nilpotent Orbits
A nilpotent orbit in a reductive Lie algebra \(\mathfrak{g}\) is an adjoint orbit of nilpotent elements. The set of nilpotent orbits is finite, partially ordered, and stratifies the nilpotent cone — making it one of the basic combinatorial invariants of \(\mathfrak{g}\).
Definition
Let \(G\) be a connected reductive algebraic group over an algebraically closed field \(k\) of characteristic zero (or "good" characteristic for the type of \(G\)), with Lie algebra \(\mathfrak{g} = \operatorname{Lie}(G)\). An element \(X \in \mathfrak{g}\) is nilpotent if \(\operatorname{ad}(X)\) is nilpotent, equivalently if \(X\) lies in the nilpotent cone \(\mathcal{N} \subseteq \mathfrak{g}\).
A nilpotent orbit is an orbit \(\mathcal{O} = G \cdot X \subseteq \mathfrak{g}\) for the adjoint action, where \(X \in \mathcal{N}\).
Finiteness
Theorem. There are only finitely many nilpotent orbits in \(\mathfrak{g}\).
The proof uses the Jacobson–Morozov theorem: every nilpotent element \(X \ne 0\) extends to an \(\mathfrak{sl}_2\)-triple \((X, H, Y)\), and \(\mathfrak{sl}_2\)-triples are classified up to \(G\)-conjugacy by a finite combinatorial datum (a weighted Dynkin diagram).
Examples
\(\mathfrak{gl}_n\) and \(\mathfrak{sl}_n\)
Nilpotent orbits in \(\mathfrak{gl}_n\) are in bijection with partitions of \(n\), via Jordan canonical form. A nilpotent \(n \times n\) matrix is conjugate to a sum of Jordan blocks; the partition records the block sizes.
For \(\mathfrak{sl}_n\) the bijection is the same (the trace-zero condition is automatic for nilpotent matrices).
Classical types
For \(\mathfrak{so}_{2n+1}\), \(\mathfrak{sp}_{2n}\), \(\mathfrak{so}_{2n}\): nilpotent orbits are classified by partitions satisfying parity constraints:
| Type | Partitions of \(n\) (or \(2n+1\)) with constraint |
|---|---|
| \(\mathfrak{sp}_{2n}\) | odd parts have even multiplicity |
| \(\mathfrak{so}_{2n+1}\) | even parts have even multiplicity |
| \(\mathfrak{so}_{2n}\) | even parts have even multiplicity (very even ones split into two) |
Exceptional types
For \(E_8, E_7, E_6, F_4, G_2\) the classification is via the Bala-Carter labels; there is no partition shortcut.
The closure order
Nilpotent orbits are partially ordered by \(\mathcal{O}_1 \le \mathcal{O}_2 \iff \mathcal{O}_1 \subseteq \overline{\mathcal{O}_2}\) (closure in the Zariski topology). The poset has:
- a minimum: the zero orbit \(\{0\}\);
- a maximum: the principal (= regular) nilpotent orbit, of dimension \(\dim \mathfrak{g} - \operatorname{rank} \mathfrak{g}\).
For \(\mathfrak{gl}_n\), the partition order on orbits is the dominance order on partitions.
Springer theory
The Springer resolution \(\mu : \tilde{\mathcal{N}} \to \mathcal{N}\) (with \(\tilde{\mathcal{N}}\) the cotangent bundle of the flag variety \(G/B\)) realises the nilpotent cone as the image of a smooth projective variety. The fibres are the Springer fibres \(\mathcal{B}_X = \mu^{-1}(X)\), and their cohomology carries a \(W\)-action (the Springer representation), giving a fundamental bijection:
\[ \{\text{nilpotent orbits in } \mathfrak{g}\} \;\hookrightarrow\; \widehat{W} \]
(the Springer correspondence, when refined by the component group of the stabiliser \(Z_G(X)\)).
Role in Langlands
- Local Langlands at the unipotent block (Lusztig, Kazhdan-Lusztig) is organised by nilpotent orbits in the dual Lie algebra \(\operatorname{Lie}(\hat G)\).
- Wavefront sets and Whittaker support of admissible representations are tracked by their associated nilpotent orbit.
- Endoscopy transfers nilpotent orbits between groups via the Langlands–Shelstad transfer map.
- Bala-Carter classification is the type-independent way to label orbits in the dual Lie algebra appearing in L-parameters.
References
- Collingwood–McGovern, Nilpotent Orbits in Semisimple Lie Algebras (1993).
- Carter, Finite Groups of Lie Type (1985), Ch. 5.
- Jantzen, Nilpotent Orbits in Representation Theory (2004).
- Springer, Trigonometric sums, Green functions of finite groups of Lie type and representations of Weyl groups (Inventiones 1976).