Topic Nilpotent Cone
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Nilpotent Cone

The nilpotent cone \(\mathcal{N} \subseteq \mathfrak{g}\) is the closed subvariety of nilpotent elements in a Lie algebra. It is the geometric scaffolding for the entire theory of nilpotent orbits, the target of the Springer resolution, and a source of remarkable varieties in representation theory.

Definition

Let \(\mathfrak{g}\) be the Lie algebra of a connected reductive algebraic group \(G\) over an algebraically closed field \(k\) of characteristic 0 (or good characteristic). The nilpotent cone is \[ \mathcal{N} \;:=\; \{X \in \mathfrak{g} : X \text{ is nilpotent}\}. \]

Equivalently: \[ \mathcal{N} \;=\; \{X \in \mathfrak{g} : \operatorname{ad}(X) \text{ is nilpotent}\} \;=\; \{X : 0 \in \overline{G \cdot X}\}. \]

Basic properties

  • \(\mathcal{N}\) is a closed conical subvariety (closed under scalar multiplication).
  • Irreducible for \(\mathfrak{g}\) semisimple.
  • Dimension: \(\dim \mathcal{N} = \dim \mathfrak{g} - \mathrm{rank}\, \mathfrak{g}\).
  • Normal as an algebraic variety.
  • Singular at every non-regular nilpotent.

Kostant's theorem

Theorem (Kostant 1963). Let \(\mathfrak{g}\) be semisimple. The adjoint quotient \(\chi : \mathfrak{g} \to \mathfrak{g} /\!\!/ G\) is flat, and \(\mathcal{N} = \chi^{-1}(0)\) is a complete intersection in \(\mathfrak{g}\), cut out by the \(r\) basic \(G\)-invariants \(p_1, \ldots, p_r\) (\(r = \mathrm{rank}\,\mathfrak{g}\)).

For \(\mathfrak{gl}_n\): \(p_i = \mathrm{tr}(X^i)\), and \(\mathcal{N}\) is cut out by \(\mathrm{tr}(X) = \mathrm{tr}(X^2) = \cdots = \mathrm{tr}(X^n) = 0\), equivalently by the vanishing of the non-constant coefficients of the characteristic polynomial.

Springer resolution

The Springer resolution \[ \mu : \tilde{\mathcal{N}} \;\longrightarrow\; \mathcal{N} \] is a proper birational morphism, where \(\tilde{\mathcal{N}} = T^*(G/B) = \{(X, B') : X \in \operatorname{Lie}(R_u(B'))\}\). It is an isomorphism over the open regular nilpotent orbit and contracts other fibres (Springer fibres \(\mathcal{B}_X = \mu^{-1}(X)\)) to lower-dimensional varieties.

Examples

\(\mathfrak{gl}_n\) and \(\mathfrak{sl}_n\)

\(\mathcal{N}\) = nilpotent matrices; \(\dim \mathcal{N} = n^2 - n\). Orbits indexed by partitions of \(n\); closure order = dominance.

\(\mathfrak{sl}_2\)

\(\mathcal{N}\) is the 2-dimensional quadric cone \(\{X \in \mathfrak{sl}_2 : X^2 = 0\}\), with three orbits: 0, regular nilpotent (open), the rays connecting them collapsed.

Exceptional types

\(E_8\): \(\dim \mathcal{N} = 240\); 70 orbits stratifying it. \(G_2\): \(\dim \mathcal{N} = 12\); 5 orbits.

Stratification by orbits

\(\mathcal{N}\) is the disjoint union of finitely many \(G\)-orbits, each locally closed and even-dimensional. The closure relations give a partial order with:

  • minimum: \(\{0\}\) (zero orbit);
  • maximum: regular nilpotent orbit (= principal), open dense;
  • minimum non-trivial: minimal nilpotent orbit, of dimension \(2h^\vee - 2\) (\(h^\vee\) = dual Coxeter number).

Connection to flag variety

The Springer resolution is the cotangent bundle \(\tilde{\mathcal{N}} = T^*(G/B)\), so \(\mathcal{N}\) is "transverse" to the flag variety. Concretely:

  • Steinberg variety \(Z = \tilde{\mathcal{N}} \times_{\mathcal{N}} \tilde{\mathcal{N}}\) carries a convolution algebra structure;
  • \(H_{\bullet}^{BM}(Z, \mathbb{C}) \cong \mathbb{C}[W]\) (Borel-Moore homology = group algebra of Weyl group) — this is the Springer algebra.

Geometric Satake

Under Mirković-Vilonen / geometric Satake, the nilpotent cone of \(\hat G\) corresponds to the affine Grassmannian of \(G\), with strata bijection: \(\hat G\)-orbits in \(\hat{\mathcal{N}}\) ↔ \(G(\mathcal{O})\)-orbits in \(\mathrm{Gr}_G\).

Role in Langlands

  • Springer correspondence: nilpotent orbits ↔ irreps of \(W\) (with extra component-group data), via top-cohomology of Springer fibres.
  • A-parameters: the \(\mathrm{SL}_2\) factor in Arthur parameters corresponds (via Jacobson-Morozov) to a nilpotent orbit in \(\mathrm{Lie}(\hat G)\); refines L-packets to A-packets.
  • Wavefront sets: invariants of admissible representations parametrised by nilpotent orbits in \(\mathfrak{g}\).

Mathlib

Not formalised.

References

  • Collingwood–McGovern, Nilpotent Orbits in Semisimple Lie Algebras (1993).
  • Chriss–Ginzburg, Representation Theory and Complex Geometry (1997).
  • Kostant, Lie group representations on polynomial rings (Amer.\ J.\ Math.\ 1963).

Prerequisite Topics

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