Unipotent Orbits
Definition
Let \(G\) be a connected reductive group over an algebraically closed field \(k\). The unipotent orbits of \(G\) are the orbits of \(G\) acting on the unipotent variety \(\mathcal{U}\) by conjugation: \[ g \cdot u = g u g^{-1}, \quad g \in G(k),\; u \in \mathcal{U}(k). \]
There are finitely many unipotent orbits; their number depends only on the root datum (root system) of \(G\).
Classification
The unipotent orbits of \(G\) are classified by the same combinatorial data as nilpotent orbits in \(\mathrm{Lie}(G)\):
| Method | Description |
|---|---|
| Bala-Carter | Distinguished parabolic subgroups \(P\) with Levi \(L\); pairs \((L, \mathcal{O}_L^{\mathrm{reg}})\) |
| Weighted Dynkin diagrams | Labellings of the Dynkin diagram by \(\{0, 1, 2\}\) |
| Partitions (type A) | Partitions of \(n\) parametrise unipotent orbits of \(\mathrm{GL}_n\) |
| Partitions (types B,C,D) | B-partitions / D-partitions for \(\mathrm{SO}_n\), \(\mathrm{Sp}_{2n}\) |
The regular and subregular orbits
- Regular orbit: The unique dense orbit; consists of elements \(u \in \mathcal{U}\) with \(\dim C_G(u) = \mathrm{rank}(G)\). Exists in all characteristics.
- Subregular orbit: The second largest orbit (by closure order); its closure is \(\overline{\mathcal{O}_{\mathrm{sub}}} = \mathcal{U} \setminus \mathcal{O}_{\mathrm{reg}}\).
- Trivial orbit: The single-point orbit \(\{1\}\).
Closure ordering
The set of unipotent orbits carries the closure order: \(\mathcal{O}_1 \le \mathcal{O}_2\) if \(\mathcal{O}_1 \subseteq \overline{\mathcal{O}_2}\). This gives a partial order whose Hasse diagram is the same as for nilpotent orbits.
Correspondence with nilpotent orbits
In characteristic \(0\) (or good characteristic \(p\) via the Springer isomorphism), the identification \(\mathcal{U} \cong \mathcal{N}\) is \(G\)-equivariant, giving a bijection: \[ \{\text{unipotent orbits in }G\} \xleftrightarrow{\;\sim\;} \{\text{nilpotent orbits in }\mathrm{Lie}(G)\}. \]