Central Isogenies and Duality

A central isogeny between reductive groups dualises under Langlands duality to an isogeny in the opposite direction. The kernels are Pontryagin-dual finite abelian groups. This swap is the source of many "adjoint vs.\ simply-connected" inversions in the Langlands programme.

Statement

For a central isogeny \(\phi : G \to G'\) of connected reductive groups with finite central kernel \(K = \ker \phi\), the dual isogeny is \[ \hat \phi : \hat{G'} \;\longrightarrow\; \hat G, \] with kernel \(\hat K = \mathrm{Hom}(K, \mathbb{C}^\times)\) (the Pontryagin dual of \(K\)).

The direction of the arrow reverses: \(G\) being a "covering" of \(G'\) corresponds to \(\hat G\) being "covered" by \(\hat{G'}\).

Examples

\(\mathrm{SL}_n \to \mathrm{PGL}_n\)

Central isogeny with kernel \(K = \mu_n\). Dualises to: \[ \widehat{\mathrm{PGL}_n} = \mathrm{SL}_n \;\to\; \mathrm{PGL}_n = \widehat{\mathrm{SL}_n}, \] again an isogeny with kernel \(\mu_n\) (= dual of \(\mu_n\)).

So \(\mathrm{SL}_n \to \mathrm{PGL}_n\) is self-dual as an isogeny (modulo swapping which side is "covering").

\(\mathrm{Spin}_{2n+1} \to \mathrm{SO}_{2n+1}\)

Kernel \(\mu_2\). Dualises to: \[ \widehat{\mathrm{SO}_{2n+1}} = \mathrm{Sp}_{2n} \;\to\; \mathrm{PSp}_{2n} = \widehat{\mathrm{Spin}_{2n+1}}, \] with kernel \(\mu_2\).

So spin double-cover dualises to symplectic-to-projective-symplectic double cover.

\(\mathrm{Spin}_{2n} \to \mathrm{SO}_{2n}\)

For \(n\) odd: kernel \(\mu_4\), dualises to a similar size-4 isogeny.

For \(n\) even: kernel \(\mathbb{Z}/2 \times \mathbb{Z}/2\), with three "intermediate" forms — the kernel is its own Pontryagin dual.

Lattice description

In terms of root data: a central isogeny \(\phi : (X, \Phi, X^\vee, \Phi^\vee) \to (X', \Phi, X'^\vee, \Phi^\vee)\) corresponds to:

An inclusion of character lattices \(X' \subseteq X\) with the same roots;
Dually, an inclusion of cocharacter lattices \(X^\vee \subseteq X'^\vee\) with the same coroots;
Kernel \(K = D(X/X')\) = Cartier dual of the quotient lattice.

Under root-datum duality, \((X, X^\vee) \leftrightarrow (X^\vee, X)\), so the lattice inclusions swap direction:

Original: \(X' \subseteq X\) (the \(G \to G'\) direction);
Dual: \(\hat{X'^\vee} \subseteq \hat X^\vee\), i.e.\ \(X^\vee \subseteq X'^\vee\) (the \(\hat{G'} \to \hat G\) direction).

Composition

The composition of central isogenies dualises as expected (with arrows reversed): \[ G_1 \to G_2 \to G_3 \quad \rightsquigarrow \quad \hat G_3 \to \hat G_2 \to \hat G_1. \]

The kernel of the composition is an extension of the dual kernels.

Isogeny class duality

Within an isogeny class of semisimple groups with fixed root system \(\Phi\):

\[ G^{\mathrm{sc}} \;\longrightarrow\; \cdots \;\longrightarrow\; G^{\mathrm{ad}} \]

dualises to:

\[ \widehat{G^{\mathrm{ad}}} = (\hat G)^{\mathrm{sc}} \;\longrightarrow\; \cdots \;\longrightarrow\; (\hat G)^{\mathrm{ad}} = \widehat{G^{\mathrm{sc}}}. \]

So the sc / adjoint extremes swap. The intermediate forms (= lattices between \(\mathbb{Z}\Phi^\vee\) and \(P^\vee\)) are in self-dual correspondence with the intermediate dual forms.

Role in Langlands

Central isogeny invariance of L-packets: Functoriality is compatible with central isogenies: composing with an isogeny \(G^{\mathrm{sc}} \to G\) refines the L-packet structure.
Strong vs.\ weak functoriality: defined relative to the isogeny class of the source/target groups.
Kottwitz formalism: uses \(z\)-extensions to handle the difference between \(G\) and \(G^{\mathrm{sc}}\) cleanly.