Fundamental Group of a Root Datum

The fundamental group \(\pi_1(\Psi)\) of a root datum \(\Psi = (X, \Phi, X^\vee, \Phi^\vee)\) is the quotient of the cocharacter lattice by the coroot lattice. Combinatorially, it classifies the "isogeny gap" between simply-connected and adjoint forms; geometrically, it agrees with the algebraic fundamental group of the associated reductive group.

Definition

For a root datum \(\Psi = (X, \Phi, X^\vee, \Phi^\vee)\) with coroot lattice \(Q^\vee = \mathbb{Z}\Phi^\vee \subseteq X^\vee\):

\[ \pi_1(\Psi) \;:=\; X^\vee \big/ Q^\vee. \]

Dually, the weight lattice is \[ P^\vee \;:=\; \{x \in X^\vee \otimes \mathbb{Q} : \langle \alpha, x \rangle \in \mathbb{Z} \text{ for all } \alpha \in \Phi\}, \] and one has the inclusions \[ Q^\vee \;\subseteq\; X^\vee \;\subseteq\; P^\vee \] (when \(\Phi\) spans \(X \otimes \mathbb{Q}\), i.e.\ in the semisimple case). The full fundamental group of the root system is \(P^\vee / Q^\vee\); the "lattice gap" \(P^\vee / X^\vee\) is the centre of the associated group.

Isogeny structure

For an irreducible root system \(\Phi\), the lattice gap \(P^\vee / Q^\vee\) is a finite abelian group, and any intermediate lattice \(Q^\vee \subseteq X^\vee \subseteq P^\vee\) gives a different isogeny form of the same group:

Type	\(P^\vee / Q^\vee\)	Forms
\(A_{n-1}\)	\(\mathbb{Z}/n\)	\(\mathrm{SL}_n\) (sc, \(X^\vee = Q^\vee\)) and \(\mathrm{PGL}_n\) (ad, \(X^\vee = P^\vee\)), plus intermediate quotients
\(B_n\)	\(\mathbb{Z}/2\)	\(\mathrm{Spin}_{2n+1}\) and \(\mathrm{SO}_{2n+1}\)
\(C_n\)	\(\mathbb{Z}/2\)	\(\mathrm{Sp}_{2n}\) and \(\mathrm{PSp}_{2n}\)
\(D_n\), \(n\) odd	\(\mathbb{Z}/4\)	\(\mathrm{Spin}_{2n}\), \(\mathrm{SO}_{2n}\), \(\mathrm{HSpin}_{2n}\), adjoint
\(D_n\), \(n\) even	\(\mathbb{Z}/2 \times \mathbb{Z}/2\)	similar
\(E_6\)	\(\mathbb{Z}/3\)	sc and adjoint
\(E_7\)	\(\mathbb{Z}/2\)	sc and adjoint
\(E_8\), \(F_4\), \(G_2\)	\(1\)	sc = ad (unique form)

Relation to the algebraic fundamental group

For a connected reductive group \(G\) with root datum \(\Psi\): \[ \pi_1(G) \;=\; \pi_1(\Psi) \;=\; X_*(T) / \mathbb{Z}\Phi^\vee. \] The combinatorial \(\pi_1\) agrees with the algebraic fundamental group (see node), which for complex groups also agrees with the topological fundamental group of \(G(\mathbb{C})\).

Examples

\(\mathrm{GL}_n\)

Root datum has \(X^\vee = \mathbb{Z}^n\), \(Q^\vee = \{(a_1, \dots, a_n) : \sum a_i = 0\}\), so \(\pi_1 = \mathbb{Z}\) (via the sum). Topologically, \(\mathrm{GL}_n(\mathbb{C})\) deformation-retracts to \(\mathrm{U}(n)\), which has \(\pi_1 = \mathbb{Z}\).

\(\mathrm{SL}_n\)

\(X^\vee = Q^\vee\), so \(\pi_1 = 0\). \(\mathrm{SL}_n(\mathbb{C})\) is simply connected.

\(\mathrm{PGL}_n\)

\(X^\vee = P^\vee\) (maximally large), and \(P^\vee / Q^\vee = \mathbb{Z}/n\). \(\pi_1^{\mathrm{top}}(\mathrm{PGL}_n(\mathbb{C})) = \mathbb{Z}/n\).

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

\(\pi_1 = \mathbb{Z}/2\); the universal cover is the spin group \(\mathrm{Spin}_{2n+1}\).

Duality

Under root-datum duality, the lattice gap swaps: \[ \pi_1(\Psi) \;=\; X^\vee / Q^\vee \;\xleftrightarrow{\text{dual}}\; X / Q \;=\; Z(\hat\Psi). \] This is the Kottwitz duality: \[ \pi_1(G) \;=\; X^*(Z(\hat G)), \] which makes the centre of the dual group into the Pontryagin dual of the fundamental group.

Mathlib

Mathlib.LinearAlgebra.RootSystem.* partially formalises the lattice chains \(Q \subseteq X \subseteq P\); the explicit quotients giving \(\pi_1\) for classical types are accessible.