Unipotent Elements and the Unipotent Variety

Definition

Let \(G\) be a linear algebraic group. An element \(g \in G(k)\) is unipotent if its semisimple part in the Jordan decomposition is trivial: \(g = g_u\), i.e.\ \(g_s = 1\).

Equivalently, for any faithful representation \(\rho : G \to \mathrm{GL}(V)\), \(\rho(g)\) is a unipotent matrix: all eigenvalues equal \(1\), i.e.\((\rho(g) - I)^N = 0\) for some \(N \le \dim V\).

The unipotent variety

The unipotent variety (or unipotent cone) is \[ \mathcal{U} = \{g \in G : g \text{ is unipotent}\}. \] This is a closed, \(G\)-stable (under conjugation) subvariety of \(G\).

Dimension

\[ \dim \mathcal{U} = \dim G - \mathrm{rank}(G). \]

Irreducibility

For \(G\) connected reductive, \(\mathcal{U}\) is irreducible: every unipotent element is conjugate to an element of the unipotent radical \(U_B\) of a Borel subgroup \(B\), and \(G/B \times U_B \to \mathcal{U}\) is a resolution (the Springer resolution).

Relation to the nilpotent cone

In characteristic \(0\), the exponential map gives a \(G\)-equivariant isomorphism \[ \exp : \mathcal{N} \xrightarrow{\;\sim\;} \mathcal{U}, \] where \(\mathcal{N} \subset \mathrm{Lie}(G)\) is the nilpotent cone. This identifies unipotent orbits in \(G\) with nilpotent orbits in \(\mathrm{Lie}(G)\).

In characteristic \(p > 0\), the exponential map does not converge globally. Instead, the identification goes through the Springer isomorphism (a \(G\)-equivariant isomorphism \(\mathcal{N} \xrightarrow{\sim} \mathcal{U}\) that exists under mild hypotheses on \(p\), e.g.\ \(p\) good for \(G\)).

Unipotent subgroups

The unipotent elements in a connected unipotent algebraic group \(U\) fill all of \(U\). For a reductive \(G\), unipotent elements appear in:

• Unipotent radicals \(R_u(P)\) of parabolic subgroups \(P\).
• Root subgroups \(U_\alpha \cong \mathbb{G}_a\).
• The big cell \(B^- \cdot B\) (open dense in \(G\)).