Kottwitz Homomorphism

The Kottwitz homomorphism \(\kappa_G : B(G) \to \pi_1(G)_\Gamma\) is the canonical map from \(\sigma\)-conjugacy classes in \(G(\breve F)\) to the Galois-coinvariants of the algebraic fundamental group. Together with the Newton map it makes \(B(G)\) into a tractable combinatorial set.

Setup

Let \(F\) be a non-archimedean local field with residue field \(\mathbb{F}_q\), completion of the maximal unramified extension \(\breve F\), Frobenius \(\sigma\), absolute Galois group \(\Gamma = \operatorname{Gal}(\bar F/F)\). Let \(G\) be a connected reductive group over \(F\) and \(\pi_1(G)\) its algebraic fundamental group (see node).

Construction

Case 1: \(G = T\) a torus

For a torus \(T\) over \(F\), \(\pi_1(T) = X_*(T)\) and the Kottwitz map is given by the valuation: \[ \kappa_T : B(T) \;\longrightarrow\; X_*(T)_\Gamma, \qquad [t \in T(\breve F)] \;\longmapsto\; v(t), \] where \(v : T(\breve F) \to X_*(T)\) extracts the valuation \(\langle \chi, t \rangle \mapsto v(\chi(t))\) coordinate-wise.

This map is a bijection \(\kappa_T : B(T) \xrightarrow{\sim} X_*(T)_\Gamma\).

Case 2: \(G\) connected reductive

For general \(G\), Kottwitz constructs \(\kappa_G\) by combining functoriality and the torus case. There is a canonical surjection \(G \twoheadrightarrow G/G_{\mathrm{der}}\) onto the abelianisation (a torus), inducing \[ \kappa_G : B(G) \;\longrightarrow\; B(G/G_{\mathrm{der}}) \;=\; X_*(G/G_{\mathrm{der}})_\Gamma \;=\; \pi_1(G)_\Gamma. \]

(More precisely: Kottwitz uses a \(z\)-extension or the Borovoi-Kottwitz crossed-module formalism to handle non-simply-connected derived groups uniformly.)

Properties

Functoriality. For \(f : G \to G'\), the diagram \[ \begin{array}{ccc} B(G) & \to & B(G') \\ \downarrow & & \downarrow \\ \pi_1(G)_\Gamma & \to & \pi_1(G')_\Gamma \end{array} \] commutes.
Surjectivity. \(\kappa_G : B(G) \to \pi_1(G)_\Gamma\) is surjective.
Fibres. The fibre over \(\bar\mu \in \pi_1(G)_\Gamma\) is parametrised by the Newton map restricted to that fibre.

The Kottwitz theorem

Theorem (Kottwitz 1997). The product map \[ (\kappa_G, \nu_G) : B(G) \;\longrightarrow\; \pi_1(G)_\Gamma \times (X_*(T)_\mathbb{Q})^\Gamma/W \] is injective, with image characterised by a compatibility condition (image of \(\kappa_G\) under the natural map equals the projection of \(\nu_G\)).

This gives a complete combinatorial description of \(B(G)\).

Examples

\(G = \mathrm{GL}_n\)

\(\pi_1(\mathrm{GL}_n) = \mathbb{Z}\) via the determinant; Frobenius acts trivially, so \(\pi_1(\mathrm{GL}_n)_\Gamma = \mathbb{Z}\).

The Kottwitz map \(\kappa : B(\mathrm{GL}_n) \to \mathbb{Z}\) sends an isocrystal of slope sum \(s\) (i.e.\ Newton polygon ending at height \(s\)) to \(s\). Equivalently, \(\kappa(b) = v(\det b)\) for any representative \(b \in \mathrm{GL}_n(\breve F)\).

\(G = \mathrm{SL}_n\)

\(\pi_1(\mathrm{SL}_n) = 0\), so \(\kappa\) is trivial. Hence \(B(\mathrm{SL}_n)\) is classified by the Newton point alone (subject to the constraint that the slope sum is \(0\)).

\(G = \mathrm{PGL}_n\)

\(\pi_1(\mathrm{PGL}_n) = \mathbb{Z}/n\); \(\kappa\) detects the isocrystal modulo \(n\), i.e.\ the Brauer class in \(\mathrm{Br}(F)[n]\).

Local Langlands and inner forms

\(\pi_0(\kappa_G) := \ker(\kappa_G \to \pi_1(\mathrm{ad}\, G)_\Gamma)\) classifies the inner forms of \(G\):

\[ H^1(F, G_{\mathrm{ad}}) \;\cong\; \pi_1(G_{\mathrm{ad}})_\Gamma \;\cong\; \text{(inner forms of } G\text{)}. \]

This is the bridge to local Langlands: each inner form gives an L-packet, and the union of L-packets over inner forms is the "extended" L-packet whose internal structure is governed by the Kottwitz map.

References

Kottwitz, Isocrystals with additional structure II (Compositio Math. 1997).
Rapoport–Richartz, On the classification and specialization of F-isocrystals with additional structure (Compositio 1996).
Borovoi, Abelian Galois cohomology of reductive groups (Mem. AMS 1998).