Jordan Decomposition in Algebraic Groups

Statement

Let \(G\) be a linear algebraic group over an algebraically closed field \(k\). For every element \(g \in G(k)\), there exists a unique decomposition \[ g = g_s \, g_u = g_u \, g_s \] where \(g_s\) is semisimple (diagonalisable over \(k\)) and \(g_u\) is unipotent (\((g_u - 1)^N = 0\) for some \(N\), equivalently all eigenvalues of \(\rho(g_u)\) equal \(1\) for any faithful representation \(\rho\)).

This is the abstract Jordan-Chevalley decomposition in \(G\).

Functoriality

The decomposition is functorial: for any morphism of algebraic groups \(\phi : G \to H\), \[ \phi(g)_s = \phi(g_s), \quad \phi(g)_u = \phi(g_u). \] In particular, the decomposition is intrinsic to \(G\) and does not depend on a choice of embedding \(G \hookrightarrow \mathrm{GL}_n\).

Construction

Embed \(G \hookrightarrow \mathrm{GL}_n\). In \(\mathrm{GL}_n(k)\), the classical Jordan decomposition (of matrices over \(k\)) gives \(g = g_s g_u\) with \(g_s\) semisimple and \(g_u\) unipotent. One verifies:

1. \(g_s, g_u \in G(k)\) (they lie in every closed subgroup containing \(g\)).
2. The decomposition is independent of the embedding.

The key lemma: the semisimple and unipotent parts of \(g \in \mathrm{GL}_n(k)\) lie in the Zariski closure of the cyclic group \(\langle g \rangle\), hence in any closed subgroup containing \(g\).

Conjugacy classes and Jordan decomposition

The Jordan decomposition partitions \(G(k)\) into:

• Semisimple elements: elements with \(g = g_s\) (unipotent part trivial). These are \(G\)-conjugate iff they are conjugate in some maximal torus.
• Unipotent elements: elements with \(g = g_u\) (semisimple part \(= 1\)). These form a closed \(G\)-stable subvariety, the unipotent variety.
• Mixed elements: general \(g\) with both \(g_s \ne 1\) and \(g_u \ne 1\). The study of mixed conjugacy classes reduces to unipotent elements in the centraliser \(C_G(g_s)^\circ\).

Relation to the Lie algebra

The differential of the Jordan decomposition at \(1\) gives the Jordan decomposition in \(\mathrm{Lie}(G)\): \[ X = X_s + X_n \in \mathrm{Lie}(G), \quad [X_s, X_n] = 0, \] where \(X_s\) is semisimple and \(X_n\) is nilpotent (as endomorphisms via the adjoint representation). The exponential map (when it exists, e.g.\ in characteristic \(0\)) interchanges \(\exp(X_n) \leftrightarrow g_u\).