Centers and Dual Centers

Under root-datum duality, the centre of \(G\) and the fundamental group of the dual side swap roles. This Kottwitz duality of finite abelian groups is the foundation of L-parameter central characters and of the parametrisation of L-packets.

Centres

For a connected reductive group \(G\) with root datum \((X, \Phi, X^\vee, \Phi^\vee)\): \[ Z(G) \;=\; \ker\Big(\prod_{\alpha \in \Phi} \alpha : T \to \mathbb{G}_m^{\#\Phi}\Big) \;=\; \mathrm{D}(X / \mathbb{Z}\Phi), \]

where \(\mathrm{D}\) is the Cartier dual (a finite group of multiplicative type from the abelian group \(X/\mathbb{Z}\Phi\)).

Equivalently: \(Z(G) = T \cap \ker(\mathrm{ad})\), with character group \[ X^*(Z(G)) \;=\; X / \mathbb{Z}\Phi. \]

This is the abelianisation lattice of \(X\) by the root lattice.

Fundamental group

The algebraic fundamental group of \(G\) (= the fundamental group of the root datum \(\Psi\)) is \[ \pi_1(G) \;=\; X^\vee / \mathbb{Z}\Phi^\vee. \]

(See fundamental group of a root datum.)

Duality identity

Theorem (Kottwitz). For any connected reductive group \(G\): \[ Z(\hat G) \;=\; \mathrm{Hom}(\pi_1(G), \mathbb{C}^\times), \] the Pontryagin dual of the finite abelian group \(\pi_1(G)\).

In terms of lattices: \[ X^*(Z(\hat G)) \;=\; X_*(G) / \mathbb{Z}\Phi^\vee \;=\; \pi_1(G). \]

Dually: \[ X^*(Z(G)) \;=\; X^*(G) / \mathbb{Z}\Phi \;=\; \pi_1(\hat G). \]

Examples

The swap pattern is consistent: the column \(Z(G)\) matches the column \(Z(\hat G)\) of the dual entry (under the natural Pontryagin duality).

Central characters of representations

A representation of \(G\) has a central character valued in the character group \(X^*(Z(G))\). Under local Langlands, this central character corresponds to an L-parameter component in: \[ \mathrm{Hom}(Z(\hat G), \mathbb{C}^\times) \;=\; \pi_1(G)^{\mathrm{dual}}, \] giving a fundamental compatibility between representation-side and parameter-side data.

For the local Langlands correspondence: \[ \Pi(G(F)) \;\to\; \Phi(W_F \to {}^L G), \qquad \pi \mapsto \varphi_\pi \] sends the central character \(\omega_\pi : Z(G)(F) \to \mathbb{C}^\times\) to the "central character" of \(\varphi_\pi\) (its value on \(W_F^{\mathrm{ab}}\) restricted via \(Z(\hat G) \subseteq \hat G\)).

Twisted version with Galois action

For non-split \(G\) with Galois action on the root datum: the centre becomes a Galois module, and: \[ H^1(F, Z(G)) \;\cong\; \pi_0(Z(\hat G)^{W_F}) \;\cong\; \mathrm{Irr}(S_\varphi), \] where \(S_\varphi\) is the component group of an L-parameter \(\varphi\). This is the Kottwitz-Shelstad component group, which labels L-packet members.

Mathlib

Mathlib.LinearAlgebra.RootSystem.* provides infrastructure for the lattice gap \(\mathbb{Z}\Phi \subseteq X \subseteq P\); the explicit identification with centres/fundamental groups requires algebraic group infrastructure.

References

• Kottwitz, Stable trace formula: cuspidal tempered terms (Duke 1984), §1.
• Borel, Automorphic L-functions (Corvallis 1979), §2.
• Springer, Linear Algebraic Groups (1998), §7.

Prerequisite Topics

• Fundamental Group of a Root Datum
• Dual Root Data