Adjoint Action
The adjoint action of an algebraic group \(G\) on its Lie algebra \(\mathfrak{g}\) is the differential of the conjugation action of \(G\) on itself. It is the basic geometric setting for nilpotent orbits, the Chevalley restriction theorem, and the Springer correspondence.
Definition
For an algebraic group \(G\) over a field \(k\) with Lie algebra \(\mathfrak{g} = \operatorname{Lie}(G)\), the adjoint action is the homomorphism of algebraic groups \[ \operatorname{Ad} : G \;\longrightarrow\; \mathrm{GL}(\mathfrak{g}), \] defined as the differential at the identity of the conjugation homomorphism \(c_g : G \to G\), \(h \mapsto g h g^{-1}\).
Concretely (for \(G \subseteq \mathrm{GL}_n\) and \(\mathfrak{g} \subseteq M_n\)): \[ \operatorname{Ad}(g)(X) \;=\; g X g^{-1}. \]
The induced map on Lie algebras is \(\mathrm{ad} : \mathfrak{g} \to \operatorname{End}(\mathfrak{g})\), \(X \mapsto [X, -]\).
Kernel and image
For \(G\) connected:
- \(\ker(\operatorname{Ad}) = Z(G)\), the centre of \(G\);
- \(\operatorname{Ad}(G) = G_{\mathrm{ad}} := G/Z(G)\), the adjoint group.
For \(G\) reductive, \(G_{\mathrm{ad}}\) is the reductive group with no centre (\(\pi_1\) is maximal, root datum is "adjoint" form).
Orbits
The adjoint action partitions \(\mathfrak{g}\) into adjoint orbits: \[ \mathfrak{g} \;=\; \bigsqcup_{[X]} G \cdot X. \]
Each orbit \(G \cdot X\) is a locally closed subvariety of \(\mathfrak{g}\), of dimension \(\dim G - \dim Z_G(X)\).
For \(\mathfrak{g}\) reductive, the orbits fall into three classes based on the Jordan decomposition \(X = X_s + X_n\):
- Semisimple orbits: \(X_n = 0\); parametrised by \(\mathfrak{t}/W\) for a fixed Cartan \(\mathfrak{t}\) (Chevalley).
- Nilpotent orbits: \(X_s = 0\); finite in number, classified by weighted Dynkin diagrams.
- Mixed orbits: both parts non-zero; an orbit through \(X\) is a "fibered" orbit \((G \cdot X_s) \times_{\bullet} (\text{nilpotent in } Z_G(X_s))\).
Chevalley restriction theorem
Theorem (Chevalley). Let \(\mathfrak{g}\) be a semisimple Lie algebra with Cartan \(\mathfrak{t}\) and Weyl group \(W = W(\mathfrak{g}, \mathfrak{t})\). The restriction of polynomial functions gives an isomorphism \[ k[\mathfrak{g}]^G \;\xrightarrow{\sim}\; k[\mathfrak{t}]^W. \]
In particular, the categorical quotient \(\mathfrak{g} /\!\!/ G = \operatorname{Spec}\, k[\mathfrak{g}]^G\) is isomorphic to \(\mathfrak{t}/W\), a polynomial variety \(\mathbb{A}^r\) (\(r = \mathrm{rank}\, \mathfrak{g}\)).
The fibres of the adjoint quotient \(\chi : \mathfrak{g} \to \mathfrak{t}/W\) are unions of adjoint orbits with the same semisimple part.
Examples
\(\mathfrak{gl}_n\)
Adjoint action: \(\operatorname{Ad}(g)(X) = g X g^{-1}\), conjugation. Orbits: similarity classes of \(n \times n\) matrices. Adjoint quotient: characteristic polynomial map \(\chi : \mathfrak{gl}_n \to \mathbb{A}^n\), \(X \mapsto (c_1(X), \ldots, c_n(X))\) (coefficients of \(\det(\lambda I - X)\)).
\(\mathfrak{sl}_2\)
Three orbits: \(\{0\}\); regular nilpotent (dim 2); semisimple orbit through any non-zero \(H \in \mathfrak{t}\) (a \(\mathbb{P}^1\)-family of size 1 modulo \(W\)).
Springer's resolution of orbit closures
For each nilpotent orbit \(\mathcal{O}_X = G \cdot X\), the closure \(\overline{\mathcal{O}_X}\) is the union of \(\mathcal{O}_X\) and strictly smaller orbits in the closure order. The Springer fibre \(\mathcal{B}_X\) is the preimage of \(X\) under \(\mu : T^*(G/B) \to \mathcal{N}\), and its cohomology carries a representation of the Weyl group (Springer).
Adjoint slice
The Kostant section of \(\mathfrak{g}\) along the adjoint quotient \(\chi\) is a slice \(\Sigma \subseteq \mathfrak{g}\) such that \(\chi : \Sigma \to \mathfrak{t}/W\) is an isomorphism. Concretely: \(\Sigma = e + Z_\mathfrak{g}(f)\) for any regular nilpotent \(\mathfrak{sl}_2\)-triple \((e, h, f)\).
The Kostant section makes the adjoint quotient a trivial bundle over its image and is essential in geometric Langlands.
Role in Langlands
- Springer correspondence is built on the adjoint orbit structure of \(\mathfrak{g}\).
- Lusztig's classification of irreducible representations of finite reductive groups is organised by the data of an adjoint orbit plus a local system on it.
- Wavefront sets of admissible representations are constrained to adjoint orbit closures in \(\mathfrak{g}^*\); they encode "asymptotic singularities."
- Geometric Langlands: the Hitchin moduli space is built from the adjoint quotient \(\mathfrak{g} \to \mathfrak{t}/W\) on a curve.
Mathlib
Mathlib.Algebra.Lie.Derivation provides general adjoint actions on
Lie algebras; specific algebraic-group cases are not formalised.
References
- Borel, Linear Algebraic Groups (1991), §3.16, §11.2.
- Chriss–Ginzburg, Representation Theory and Complex Geometry (1997), Ch. 3.
- Kostant, Lie group representations on polynomial rings (Amer.\ J.\ Math.\ 1963).