Topic Basic Sigma-Conjugacy Classes
topic admitted

Basic Sigma-Conjugacy Classes

A \(\sigma\)-conjugacy class \([b] \in B(G)\) is basic if its Newton point is central: \(\nu([b]) \in X_*(Z(G))_\mathbb{Q}\). Basic classes are the "constant-slope" classes — the analogues of "supersingular" isocrystals — and they classify the inner forms of \(G\) and the Rapoport-Zink spaces.

Definition

Let \(b \in G(\breve F)\) and let \(\nu_b \in X_*(T)_\mathbb{Q}^\Gamma / W\) be its Newton point. The class \([b] \in B(G)\) is basic if \(\nu_b\) is invariant under the Weyl group action on \(X_*(T)\) — equivalently, \(\nu_b\) is central in \(G\): \[ \nu_b \in X_*(Z(G))_\mathbb{Q}^\Gamma. \]

The set of basic classes is denoted \(B(G)_{\mathrm{basic}}\).

Characterisation

Equivalent conditions for \([b]\) to be basic:

  1. The Newton point \(\nu_b\) is central (i.e.\ a cocharacter of \(Z(G)\)).
  2. The slope decomposition is "all one slope" in the \(G\)-isocrystal picture.
  3. The Jacobson-Morozov \(\mathrm{SL}_2\)-triple attached to \(b\) is trivial (the unipotent factor of the Mazur-Kottwitz Hodge decomposition is trivial).
  4. \([b]\) is minimal in its connected component of \(B(G)\) under the dominance order on Newton points.

Bijection with inner forms

Theorem (Kottwitz). \[ B(G)_{\mathrm{basic}} \;\stackrel{\sim}{\longleftrightarrow}\; H^1(F, G_{\mathrm{ad}}) \;=\; \{\text{inner forms of } G \text{ over } F\}. \]

The bijection is via the Kottwitz map: a basic class \([b]\) has \(\kappa_G([b]) \in \pi_1(G)_\Gamma\), which (via the dual identification \(\pi_1(G_{\mathrm{ad}}) = H^1(F, G_{\mathrm{ad}})\) for basic elements) determines an inner form.

Each basic class \([b]\) determines:

  • An inner twist \(G_b\) of \(G\) (the centraliser of \(\nu_b\) in \(G\), an inner form of \(G\) itself in the basic case).
  • A basic Rapoport-Zink space \(\mathcal{M}_{G, b, \mu}\) whose geometry encodes part of local Langlands.

Examples

\(\mathrm{GL}_n\)

\(B(\mathrm{GL}_n)_{\mathrm{basic}}\) = constant-slope Newton polygons, i.e.\ \(\nu = (s, s, \ldots, s)\) for \(s \in \mathbb{Q}\) with denominator dividing \(n\).

The basic class with slope \(s = k/n\) (\(\gcd(k, n) = d\)) corresponds to the central simple algebra \(D_{k/n} = D\) of degree \(n/d\) and invariant \(k\); the inner form is \(\mathrm{GL}_{d}(D)\).

In particular:

  • \(s = 0\): split form \(\mathrm{GL}_n\) (trivial Brauer class).
  • \(s = k/n\) with \(\gcd(k, n) = 1\): "supersingular" class, inner form is \(\mathrm{SL}_1(D)\) for \(D\) a division algebra of degree \(n\) and invariant \(k\).

\(\mathrm{SL}_n\)

\(\pi_1(\mathrm{SL}_n) = 0\), so \(B(\mathrm{SL}_n)_{\mathrm{basic}}\) is a single point: the unique inner form is \(\mathrm{SL}_n\) itself. (Actually \(H^1(F, \mathrm{PGL}_n) \ne 0\), but inner forms of \(\mathrm{SL}_n\) are \(\mathrm{SL}_1(D)\)'s for division algebras \(D\) of degree \(n\) — which are the basic classes in \(B(\mathrm{PGL}_n)\) — but as inner forms of the simply-connected \(\mathrm{SL}_n\) they don't appear; the relation goes via \(z\)-extensions.)

Tori

For a torus \(T\): every \(\sigma\)-conjugacy class is basic (\(\nu = \kappa\)), and \(B(T) = X_*(T)^\Gamma = \pi_1(T)_\Gamma\) is in trivial bijection with itself.

Connectedness of \(B(G)\)

\(B(G)\) has finitely many connected components in the closure order, each labelled by an element of \(\pi_1(G)_\Gamma\) via the Kottwitz map. Each component has a unique basic class at its minimum (the Kottwitz image), and a unique generic class at its maximum (the "ordinary" or top stratum).

Role in Shimura varieties

For a Shimura datum \((G, X)\) with associated minuscule cocharacter \(\{\mu\}\): the Newton stratification of \(\mathrm{Sh}_K \pmod p\) has strata indexed by the admissible set \(B(G, \mu) \subseteq B(G)\), i.e.\ classes \([b]\) with Newton point \(\nu_b\) lying in the convex hull of \(W\mu\).

The basic stratum corresponds to the unique basic class in \(B(G, \mu)\); it is the "supersingular" stratum, and its geometry is governed by Rapoport-Zink spaces and \(p\)-adic uniformisation.

Role in Langlands

  • Bijection with inner forms: basic classes ↔ inner forms is the source of the isocrystal version of local Langlands.
  • Basic Rapoport-Zink spaces: deformation spaces of basic isocrystals; their cohomology realises local Langlands at the basic block (Harris-Taylor for \(\mathrm{GL}_n\)).
  • Fargues-Fontaine curve: provides a more flexible framework unifying all classes, with basic classes as the simplest case.

References

  • Kottwitz, Isocrystals with additional structure II (Compositio 1997).
  • Rapoport–Richartz, On the classification and specialization of F-isocrystals with additional structure (Compositio 1996).
  • Rapoport–Zink, Period spaces for p-divisible groups (Annals of Math.\ Studies 1996).

Prerequisite Topics