Bala-Carter Classification
The Bala-Carter classification is the type-independent classification of nilpotent conjugacy classes (orbits) in a complex (or, over an algebraically closed field of good characteristic) simple Lie algebra \(\mathfrak{g}\). It assigns to each nilpotent orbit a labeled subdiagram of the Dynkin diagram.
The classification
Theorem (Bala–Carter). Nilpotent \(G\)-orbits in \(\mathfrak{g}\) are in canonical bijection with pairs \((\mathfrak{l}, \mathcal{O}_{\mathfrak{l}})\) where:
- \(\mathfrak{l} \subseteq \mathfrak{g}\) is a Levi subalgebra of some parabolic of \(\mathfrak{g}\), taken up to \(G\)-conjugacy;
- \(\mathcal{O}_{\mathfrak{l}}\) is a distinguished nilpotent orbit in the derived subalgebra \([\mathfrak{l}, \mathfrak{l}]\) (one whose centraliser in the corresponding group contains no non-trivial torus).
A distinguished orbit, in turn, is labelled by a weighted Dynkin diagram of \(\mathfrak{l}\) with weights in \(\{0, 2\}\).
Examples
\(\mathfrak{gl}_n\) and \(\mathfrak{sl}_n\)
For \(\operatorname{GL}_n\) (and \(\operatorname{SL}_n\)), nilpotent orbits are classified by partitions of \(n\) (the Jordan canonical form). The Bala-Carter Levi is the standard Levi corresponding to the partition, and the distinguished orbit is the principal nilpotent (regular Jordan block) in that Levi.
| Partition | Levi | Distinguished orbit |
|---|---|---|
| \((n)\) | \(\mathfrak{gl}_n\) itself | principal |
| \((n-1, 1)\) | \(\mathfrak{gl}_{n-1}\) | principal in \(\mathfrak{gl}_{n-1}\) |
| \((1^n)\) | \(\mathfrak{t} = \mathfrak{gl}_1^n\) | zero |
Exceptional groups
For \(E_8\) there are 70 nilpotent orbits; for \(E_7\) there are 45; for \(E_6\) there are 21; for \(F_4\) there are 16; for \(G_2\) there are 5. The Bala–Carter labels are the standard way to refer to them.
Distinguished orbits and weighted Dynkin diagrams
A nilpotent element \(e \in \mathfrak{g}\) is distinguished if the identity component of the centraliser \(Z_G(e)^\circ\) is unipotent (equivalently, contains no non-trivial torus). For each nilpotent orbit, the weighted Dynkin diagram (see node) is obtained from a Jacobson–Morozov \(\mathfrak{sl}_2\)-triple \((e, h, f)\) by recording the values \(\alpha_i(h)\) for each simple root \(\alpha_i\). The weights are in \(\{0, 1, 2\}\), and for distinguished orbits they are in \(\{0, 2\}\).
Why this matters for Langlands
- Jacobson-Morozov theorem sets up the correspondence between nilpotent orbits and \(\mathfrak{sl}_2\)-triples.
- Springer correspondence relates nilpotent orbits to representations of the Weyl group, with components of the Springer fibre over each orbit indexed by \(W\)-representations.
- Local Langlands: the local Langlands correspondence at the unipotent block involves the parametrisation of nilpotent orbits in the dual Lie algebra and the Springer correspondence on each side.
- Lusztig's classification of unipotent representations of finite groups of Lie type is organised by Bala–Carter labels.
References
- Bala–Carter, Classes of unipotent elements in simple algebraic groups I, II (Math. Proc. Cambridge Philos. Soc., 1976).
- Collingwood–McGovern, Nilpotent Orbits in Semisimple Lie Algebras (1993).
- Carter, Finite Groups of Lie Type (1985).