Derived Group, Center, Adjoint Group, and Simply Connected Cover
A connected reductive group \(G\) carries four canonical subgroups / quotients that organise its isogeny structure:
| Object | Notation | Role |
|---|---|---|
| Derived subgroup | \(G_{\mathrm{der}} = [G, G]\) | maximal semisimple subgroup |
| Centre | \(Z(G)\) | maximal subgroup of multiplicative type |
| Adjoint quotient | \(G_{\mathrm{ad}} = G / Z(G)\) | trivial-centre form |
| Simply connected cover | \(G^{\mathrm{sc}}\) of \(G_{\mathrm{der}}\) | trivial-π_1 form |
The basic exact sequences
For any connected reductive \(G\):
\[ 1 \;\longrightarrow\; Z(G)^\circ \;\longrightarrow\; G \;\longrightarrow\; G/Z(G)^\circ \;\longrightarrow\; 1 \]
\[ G \;=\; Z(G)^\circ \cdot G_{\mathrm{der}}, \qquad Z(G)^\circ \cap G_{\mathrm{der}} \subseteq Z(G_{\mathrm{der}}) \]
with \(Z(G)^\circ\) a connected torus.
For the derived (semisimple) part: \[ 1 \;\longrightarrow\; Z(G^{\mathrm{sc}}) \;\longrightarrow\; G^{\mathrm{sc}} \;\stackrel{\pi}{\longrightarrow}\; G_{\mathrm{der}} \;\longrightarrow\; 1, \] with \(Z(G^{\mathrm{sc}})\) finite of multiplicative type.
And: \[ 1 \;\longrightarrow\; Z(G_{\mathrm{der}}) \;\longrightarrow\; G_{\mathrm{der}} \;\longrightarrow\; G_{\mathrm{ad}} \;\longrightarrow\; 1. \]
Derived subgroup \(G_{\mathrm{der}}\)
\(G_{\mathrm{der}} = [G, G]\) is the closed normal subgroup generated by commutators. For connected reductive \(G\): \(G_{\mathrm{der}}\) is connected semisimple, with the same root datum as \(G\) modulo the abelianisation lattice.
Examples:
- \(\mathrm{GL}_n\): \(G_{\mathrm{der}} = \mathrm{SL}_n\).
- \(\mathrm{Sp}_{2n}\): \(G_{\mathrm{der}} = \mathrm{Sp}_{2n}\) (already semisimple, perfect).
- \(\mathrm{GSp}_{2n}\) (general symplectic): \(G_{\mathrm{der}} = \mathrm{Sp}_{2n}\).
Centre \(Z(G)\)
The centre \(Z(G)\) is the kernel of the adjoint representation \(\mathrm{Ad} : G \to \mathrm{GL}(\mathfrak{g})\). It is a group of multiplicative type (a closed subgroup of an algebraic torus).
The connected centre \(Z(G)^\circ\) is a torus (the "central torus"). The component group \(Z(G)/Z(G)^\circ\) is finite.
Examples:
- \(Z(\mathrm{GL}_n) = \mathbb{G}_m\) (scalar matrices); connected.
- \(Z(\mathrm{SL}_n) = \mu_n\); finite, not connected unless \(n = 1\).
- \(Z(\mathrm{Sp}_{2n}) = \mu_2\); finite.
- \(Z(\mathrm{Spin}_{2n+1}) = \mu_2\); finite.
- \(Z(E_8) = 1\); trivial.
Adjoint group \(G_{\mathrm{ad}}\)
\(G_{\mathrm{ad}} := G / Z(G)\), the adjoint form:
- \(Z(G_{\mathrm{ad}}) = 1\) (trivial centre);
- \(\pi_1(G_{\mathrm{ad}})\) is the dual of \(Z(G^{\mathrm{sc}})\);
- The root datum of \(G_{\mathrm{ad}}\) has \(X^* = \mathbb{Z}\Phi\) (root lattice).
Examples:
- \(\mathrm{PGL}_n = \mathrm{GL}_n / \mathbb{G}_m = \mathrm{SL}_n / \mu_n\).
- \(\mathrm{PSp}_{2n} = \mathrm{Sp}_{2n} / \mu_2\).
- \(\mathrm{PSO}_n = \mathrm{SO}_n / Z(\mathrm{SO}_n)\).
Simply connected cover \(G^{\mathrm{sc}}\)
For a semisimple \(G\), the simply connected cover is the unique isogeny \(G^{\mathrm{sc}} \to G\) with \(G^{\mathrm{sc}}\) simply connected (\(\pi_1 = 0\)). The kernel is a finite central subgroup, isomorphic to \(\pi_1(G)\) (under Pontryagin duality with \(Z(G^{\mathrm{sc}})\)).
For a reductive group: define \(G^{\mathrm{sc}}\) as the simply connected cover of \(G_{\mathrm{der}}\); but \(G^{\mathrm{sc}}\) does NOT cover \(G\) directly — only \(G_{\mathrm{der}}\). There's a canonical homomorphism \(G^{\mathrm{sc}} \to G\), but it need not be surjective onto \(G\).
Examples:
- \((\mathrm{PGL}_n)^{\mathrm{sc}} = \mathrm{SL}_n\).
- \((\mathrm{SO}_n)^{\mathrm{sc}} = \mathrm{Spin}_n\).
- \((\mathrm{PSp}_{2n})^{\mathrm{sc}} = \mathrm{Sp}_{2n}\).
Z-extensions and pullbacks
For working with non-simply-connected \(G\), one often uses a \(z\)-extension: \[ 1 \;\to\; Z \;\to\; G' \;\to\; G \;\to\; 1 \] with \(Z\) an induced torus and \(G'^{\mathrm{der}}\) simply connected. The existence is a theorem of Langlands and is essential for the Kottwitz formalism of stable conjugacy and rigid inner forms.
Lattice description
For a split reductive \(G\) with root datum \((X, \Phi, X^\vee, \Phi^\vee)\):
| Object | \(X\) | \(X^\vee\) |
|---|---|---|
| \(G\) | \(X\) | \(X^\vee\) |
| \(G_{\mathrm{der}}\) | \(X / X^{W \cdot \Phi^\perp}\) | \(\mathbb{Z}\Phi^\vee \subseteq X^\vee\) |
| \(G_{\mathrm{ad}}\) | \(\mathbb{Z}\Phi\) | \(P^\vee\) (weight lattice) |
| \(G^{\mathrm{sc}}\) | \(P\) (weight lattice) | \(\mathbb{Z}\Phi^\vee\) |
The isogeny chain \(G^{\mathrm{sc}} \to G_{\mathrm{der}} \to G_{\mathrm{ad}}\) corresponds dually to the lattice inclusions \(\mathbb{Z}\Phi^\vee \subseteq X^\vee_{\mathrm{der}} \subseteq P^\vee\).
Role in Langlands
- L-group construction: the dual group \(\hat G\) swaps simply- connected and adjoint properties; \(\hat{G^{\mathrm{sc}}} = \hat G^{\mathrm{ad}}\).
- Rigid inner forms (Kaletha): defined using \(z\)-extensions to rigidify non-simply-connected cohomological subtleties.
- Hasse principle: simply connected groups satisfy the Hasse principle (Chernousov, Kneser); adjoint groups generally don't.
- Stable conjugacy: defined relative to a \(z\)-extension; lifts cleanly to the simply-connected cover.
References
- Borel, Linear Algebraic Groups (1991), §14.
- Conrad, Reductive group schemes (Notes, 2014), §6.
- Milne, Reductive Groups (notes), §19–21.