Rational Nilpotent Orbits

When the base field \(k\) is not algebraically closed, a geometric nilpotent orbit \(\mathcal{O} \subseteq \mathfrak{g}_{\bar k}\) may break up into multiple rational (= \(G(k)\)) orbits. The classification of rational orbits is a finer invariant requiring Galois cohomology.

Setup

Let \(G\) be a connected reductive group over a field \(k\), \(\mathfrak{g} = \operatorname{Lie}(G)\), and consider the action of \(G(k)\) on the nilpotent cone \(\mathcal{N}(k) = \mathcal{N} \cap \mathfrak{g}(k)\).

A geometric orbit is a \(G(\bar k)\)-orbit in \(\mathcal{N}(\bar k)\), defined over \(k\) iff fixed by Galois.

A rational orbit is a \(G(k)\)-orbit in \(\mathcal{N}(k)\).

A single geometric orbit defined over \(k\) may contain multiple rational orbits.

Classification

Theorem. For \(X \in \mathcal{N}(k)\) and geometric orbit \(\mathcal{O} = G(\bar k) \cdot X\), the set of rational orbits inside \(\mathcal{O}(k) = \mathcal{O} \cap \mathfrak{g}(k)\) is in bijection with \[ \ker\Big(H^1(k, Z_G(X)) \to H^1(k, G)\Big). \]

In particular, if \(H^1(k, G) = 1\) (e.g.\ \(k\) finite, by Lang's theorem), rational orbits inside \(\mathcal{O}\) are parametrised by \(H^1(k, Z_G(X))\).

Examples

\(k = \mathbb{F}_q\) finite

Lang's theorem: \(H^1(\mathbb{F}_q, G) = 1\). So: \[ \#\{\text{rational orbits inside } \mathcal{O}\} \;=\; \#H^1(\mathbb{F}_q, Z_G(X)). \]

For nilpotent \(X\) in a finite reductive Lie algebra, the centraliser \(Z_G(X)\) is typically a unipotent extension of a smaller reductive group; explicit computations give a count of "Springer points" in the fibre.

\(k = \mathbb{R}\)

Real nilpotent orbits decompose as follows: a complex orbit \(\mathcal{O}_\mathbb{C} \subseteq \mathfrak{g}_\mathbb{C}\) defined over \(\mathbb{R}\) (\(= \mathfrak{g}\otimes\mathbb{C}\)) splits into real orbits parametrised by \(H^1(\mathbb{R}, Z_G(X))\); since \(\mathrm{Gal}(\mathbb{C}/\mathbb{R}) = \mathbb{Z}/2\), this is a sum of \((\mathbb{Z}/2)^k\) for some \(k\).

For \(\mathfrak{sl}_n(\mathbb{R})\): a partition \(\lambda\) gives a geometric orbit, but the rational orbits are labelled by signed Young diagrams (real signature data).

For \(\mathfrak{u}(p, q)\): rational orbits classified by signed Young diagrams compatible with the signature \((p, q)\).

\(k\) a \(p\)-adic field

\(H^1(F, T)\) for \(T\) the centraliser of a regular nilpotent is computable via local Tate duality; the result is the abelianised Galois cohomology of the centraliser, computable from the root datum.

For \(\mathrm{SL}_2(\mathbb{Q}_p)\): the regular nilpotent orbit splits into two rational orbits, parametrised by \(\mathbb{Q}_p^\times / (\mathbb{Q}_p^\times)^2\) (modulo squares — a group of order 4 for \(p\) odd).

Connection to Galois cohomology

The bijection rests on the fundamental fact: \[ G(\bar k) \cdot X \;\cong\; G/Z_G(X) \] (as varieties), so rational points are computed by: \[ (G/Z_G(X))(k) \;\to\; H^1(k, Z_G(X)) \;\to\; H^1(k, G), \] the exact sequence of pointed sets associated to \(1 \to Z_G(X) \to G \to G/Z_G(X) \to 1\).

Role in Langlands

Local Langlands: nilpotent orbits in the dual Lie algebra \(\mathrm{Lie}(\hat G)\), but the relevant orbits are typically rational orbits over an appropriate field (\(\mathbb{C}\) or \(\bar{\mathbb{Q}}_\ell\) in arithmetic context).
Endoscopy: transfer of nilpotent orbits between inner forms involves identifying rational orbits with the same Newton/Bala-Carter data but in different forms.
Wavefront set: a \(p\)-adic representation's wavefront set is a union of rational orbits in \(\mathfrak{g}(\bar F)\).

References

Collingwood–McGovern, Nilpotent Orbits in Semisimple Lie Algebras (1993), §9 (real and \(p\)-adic).
Bhargava–Gross, The arithmetic invariant theory of certain nilpotent orbits (Galois lifts paper, 2014).
Reeder–Yu, Epipelagic representations and invariant theory (J.\ Amer.\ Math.\ Soc.\ 2014).