Algebraic Fundamental Group

The algebraic fundamental group \(\pi_1(G)\) of a connected reductive group \(G\) is the cokernel of the coroot lattice in the cocharacter lattice of a maximal torus. It plays the role for reductive groups that \(\pi_1\) of a Lie group plays in topology, and it is the target of the Kottwitz map.

Definition

Let \(G\) be a connected reductive group over a field \(k\), with maximal torus \(T\), cocharacter lattice \(X_*(T)\), and coroots \(\Phi^\vee \subseteq X_*(T)\). Then \[ \pi_1(G) \;:=\; X_*(T) \,/\, \mathbb{Z}\Phi^\vee, \] the cokernel of the inclusion of the coroot lattice \(Q^\vee := \mathbb{Z}\Phi^\vee\) into \(X_*(T)\).

This depends only on \(G\), not on the choice of \(T\) (since maximal tori are conjugate). For \(G\) over \(k\) (not necessarily split), the Galois group \(\Gamma = \operatorname{Gal}(\bar k/k)\) acts on \(\pi_1(G)\).

Equivalent description via short exact sequences

Let \(G_{\mathrm{sc}}\) be the simply connected cover of the derived group \(G_{\mathrm{der}}\), and let \(D(G) := G/G_{\mathrm{der}}\) be the cocharacter quotient (a torus). Then there is a short exact sequence \[ 1 \;\longrightarrow\; G_{\mathrm{sc}} \;\longrightarrow\; G \;\longrightarrow\; D(G) \;\longrightarrow\; 1. \] \(\pi_1(G)\) fits into a long exact sequence relating it to \(\pi_1(G_{\mathrm{der}})\) (the fundamental group of the root system) and \(X_*(D(G))\): \[ 0 \;\to\; \pi_1(G_{\mathrm{der}}) \;\to\; \pi_1(G) \;\to\; X_*(D(G)) \;\to\; 0. \]

Examples

Topological analogy

For a complex reductive group \(G(\mathbb{C})\), regarded as a topological group, the algebraic fundamental group \(\pi_1(G)\) defined above agrees with the topological fundamental group of the underlying real Lie group of \(G(\mathbb{C})\). For example, \(\pi_1^{\mathrm{top}}(\mathrm{PGL}_n(\mathbb{C})) = \mathbb{Z}/n\), \(\pi_1^{\mathrm{top}}(\mathrm{SL}_n(\mathbb{C})) = 0\), etc.

This is why the same symbol \(\pi_1\) is used.

Galois action

For \(G/k\) and an algebraic closure \(\bar k\), the Galois group \(\Gamma\) acts on \(X_*(T_{\bar k})\) and on the set of coroots, hence on \(\pi_1(G_{\bar k})\). The resulting \(\Gamma\)-module \(\pi_1(G)\) is the basic Galois-theoretic invariant of \(G\).

For local fields \(F\), the Frobenius coinvariants \(\pi_1(G)_\Gamma\) appear as the target of the Kottwitz homomorphism.

Role in Galois cohomology and Langlands

• Borovoi's abelianisation. For a connected reductive group \(G\) over a field \(k\), Borovoi defines an "abelianised Galois cohomology" \[ H^i_{\mathrm{ab}}(k, G) \;:=\; H^i(\Gamma, \pi_1(G_{\bar k}) \to X_*(T_{\bar k})), \] and shows \[ H^1(k, G) \;\twoheadrightarrow\; H^1_{\mathrm{ab}}(k, G) \] with fibres governed by \(H^1(k, G_{\mathrm{sc}})\).
• Kottwitz map. \(\kappa_G : B(G) \to \pi_1(G)_\Gamma\) detects the "discrete part" of \(\sigma\)-conjugacy classes.
• Dual group. Under Langlands duality, \(\pi_1(G)\) becomes \(Z(\hat G)\), the centre of the dual group: \[ \pi_1(G) \;=\; X^*(Z(\hat G)), \qquad \pi_1(G)_\Gamma \;=\; X^*(Z(\hat G))^\Gamma. \] So Kottwitz's map factors through characters of \(Z(\hat G)^\Gamma\), which directly correspond to L-packets via Langlands parametrisation.

References

• Borovoi, Abelian Galois cohomology of reductive groups (Mem. AMS 1998).
• Kottwitz, Stable trace formula: cuspidal tempered terms (Duke 1984) and Isocrystals II (Compositio 1997).
• Milne, Reductive Groups (notes), §17.

Prerequisite Topics

• Reductive Groups
• Root Data
• Fundamental Group of a Root Datum