Central Isogenies

A central isogeny is a surjective homomorphism of connected algebraic groups with finite central kernel. Central isogenies relate the various forms of a semisimple group within an isogeny class — the chain \(G^{\mathrm{sc}} \to G_{\mathrm{der}} \to G_{\mathrm{ad}}\) is the prototype.

Definition

Let \(G\) and \(G'\) be connected algebraic groups over a field \(k\). A homomorphism \(\phi : G \to G'\) is an isogeny if it is surjective with finite kernel.

\(\phi\) is central if \(\ker \phi \subseteq Z(G)\) — equivalently, \(\phi\) is étale-locally a quotient by a finite subgroup of multiplicative type in the centre.

For semisimple \(G\): every isogeny is automatically central (since finite normal subgroups of a connected semisimple group are central).

Isogeny classes

Two connected semisimple groups \(G\) and \(G'\) are isogenous if there are isogenies \(G \to G'\) (or via an intermediate group). Within an isogeny class:

All groups have the same root system \(\Phi\) (not just root datum);
The cocharacter lattices range from \(Q^\vee = \mathbb{Z}\Phi^\vee\) (adjoint extreme) to \(P^\vee\) (simply-connected extreme);
Each intermediate lattice gives a different group, related to its neighbours by central isogenies.

For type \(A_{n-1}\): the isogeny class contains \(\mathrm{SL}_n\) and \(\mathrm{PGL}_n\), with intermediate groups \(\mathrm{SL}_n / \mu_d\) for each \(d | n\), all connected by central isogenies.

Universal cover and adjoint quotient

For a connected semisimple group \(G\) over \(k\) (good characteristic):

Simply connected cover \(G^{\mathrm{sc}}\): the unique (up to isomorphism) simply-connected group with an isogeny \(G^{\mathrm{sc}} \to G\); kernel \(= \pi_1(G)\), a finite group of multiplicative type.
Adjoint quotient \(G^{\mathrm{ad}} = G/Z(G)\): the quotient by the centre; isogeny \(G \to G^{\mathrm{ad}}\) with kernel \(Z(G)\).

The composite \(G^{\mathrm{sc}} \to G^{\mathrm{ad}}\) is the "longest" isogeny, with kernel \(Z(G^{\mathrm{sc}})\).

Examples

\(A_{n-1}\) isogeny class

\[ \mathrm{SL}_n \;\xrightarrow{}\; \mathrm{SL}_n/\mu_d \;\xrightarrow{}\; \cdots \;\xrightarrow{}\; \mathrm{PGL}_n, \] with isogeny kernels \(\mu_d\) for divisors \(d | n\).

\(D_n\) isogeny class (\(n \ge 2\))

\(\mathrm{Spin}_{2n}\) (sc), \(\mathrm{SO}_{2n}\), \(\mathrm{HSpin}_{2n}\) (half-spin), and \(\mathrm{PSO}_{2n}\) (adjoint). Kernels are subgroups of \(Z(\mathrm{Spin}_{2n}) = \mathbb{Z}/2 \times \mathbb{Z}/2\) (for \(n\) even) or \(\mathbb{Z}/4\) (for \(n\) odd).

\(E_6\) isogeny class

\(E_6^{\mathrm{sc}}\) and \(E_6^{\mathrm{ad}}\), related by isogeny with kernel \(\mu_3\).

\(E_8\), \(F_4\), \(G_2\)

Trivial isogeny class (sc = adjoint).

Functoriality on root data

Under root-datum classification: a central isogeny \(G \to G'\) corresponds to an inclusion of cocharacter lattices \(X_*(T') \subseteq X_*(T)\) (with the same coroots), or equivalently to a surjection of character lattices \(X^*(T) \twoheadrightarrow X^*(T')\).

Specifically:

\(X_*(T) / X_*(T')\) is the kernel \(\ker(G \to G')\) (as a finite group of multiplicative type).

Categorical interpretation

The category of connected semisimple groups in an isogeny class with fixed root system \(\Phi\) is equivalent to the partially ordered set of intermediate lattices \(\mathbb{Z}\Phi^\vee \subseteq L \subseteq P^\vee\). Morphisms are central isogenies; the maximal lattice \(P^\vee\) gives the simply-connected form; the minimal \(\mathbb{Z}\Phi^\vee\) gives the adjoint form.