Topic Maximal Tori
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Maximal Tori

A maximal torus of a reductive group \(G\) is a torus \(T \subseteq G\) of maximal dimension. Maximal tori are the "Cartan-like" subgroups of \(G\): they encode the rank, give rise to the root datum, and (over non-algebraically-closed fields) capture the rational structure of \(G\).

Definition

A torus in an algebraic group \(G\) is a closed subgroup isomorphic to \(\mathbb{G}_m^r\) over the algebraic closure (or more generally, an algebraic torus). A maximal torus is a torus not properly contained in any larger torus.

Existence and conjugacy (algebraic closure)

Theorem. Let \(G\) be a connected linear algebraic group over an algebraically closed field \(k\). Then:

  1. \(G\) contains a maximal torus.
  2. All maximal tori of \(G\) are conjugate by elements of \(G(k)\).
  3. The dimension of any maximal torus is the same; this common dimension is the rank of \(G\).

For \(G\) reductive: the maximal tori all have the same character lattice and root datum, giving the well-defined notions of root datum, Weyl group, etc.

Over arbitrary fields

For \(G\) reductive over a non-closed field \(k\):

  • Maximal tori always exist (by Galois descent of a geometric maximal torus).
  • Different conjugacy classes of maximal tori over \(k\) exist; they are parametrised by \(H^1(k, N_G(T_0))\) for a fixed reference torus.
  • The stable conjugacy of maximal tori is governed by the geometric Weyl group action.

Split, non-split, and anisotropic maximal tori

A maximal torus \(T \subseteq G\) is:

  • Split if \(T \cong \mathbb{G}_m^r\) over \(k\);
  • Anisotropic if \(T\) has no \(k\)-rational characters (\(X^*(T)^\Gamma = 0\));
  • General \(T\) is a "twisted" form combining split and anisotropic parts.

The maximal split subtorus \(A \subseteq G\) is the largest torus that is split over \(k\); its dimension is the \(k\)-split rank of \(G\).

Examples

\(\operatorname{GL}_n\)

Standard maximal torus: \(T = \) diagonal invertible matrices, \(T \cong \mathbb{G}_m^n\). Always split.

Other maximal tori: stabilisers of decompositions \(k^n = V_1 \oplus \cdots \oplus V_r\) where \(V_i\) are absolutely irreducible \(\mathrm{Gal}(\bar k/k)\)-modules. Over \(\mathbb{Q}\): includes anisotropic tori coming from extension fields \(k(\alpha)\) with degree dividing \(n\).

\(\operatorname{SO}_n\) (real)

Maximal split torus has rank \(\lfloor n/2 \rfloor\). The compact form \(\mathrm{SO}_n^{\mathrm{cpt}}\) has an anisotropic maximal torus (the standard compact maximal torus from symmetric-space theory).

Anisotropic example: \(R^{(1)}_{K/k}\mathbb{G}_m\)

The norm-one torus is anisotropic over \(k\) (the fixed lattice \(X^*(T)^\Gamma\) is trivial); maximal anisotropic torus in \(\mathrm{SL}_1(D)\) for a division algebra \(D/k\).

Normaliser and Weyl group

For a maximal torus \(T \subseteq G\): \[ N_G(T) / T \;\cong\; W(G, T), \] the Weyl group of \(G\) relative to \(T\). Over \(\bar k\), this agrees with the abstract Weyl group of the root datum.

For non-split \(G\): the rational Weyl group \(N_G(T)(k)/T(k)\) is the Galois fixed points of \(W\), often a proper subgroup of the absolute one.

Cartan subalgebras

The Lie algebra of a maximal torus, \(\mathfrak{t} = \operatorname{Lie}(T)\), is a Cartan subalgebra of \(\mathfrak{g}\): an abelian self-normalising subalgebra of maximal dimension.

In characteristic 0 (or good positive characteristic), Cartan subalgebras and maximal tori are in bijection (under \(\mathfrak{t} \mapsto Z_G(\mathfrak{t})\)).

Steinberg's theorem on tori

Theorem (Steinberg). Let \(G\) be a simply connected semisimple algebraic group over a perfect field \(k\). The conjugacy classes of maximal tori of \(G\) over \(k\) are in bijection with the \(F\)-conjugacy classes in \(W\) (the absolute Weyl group), where \(F\) is the Frobenius automorphism for \(k\) over its prime field.

This is the foundational result for the Deligne-Lusztig theory of representations of finite reductive groups: each maximal torus contributes a "twisted" Deligne-Lusztig variety.

Role in Langlands

  • Cuspidal Levis: every Levi subgroup is the centraliser of a rational maximal torus's split component.
  • Deligne-Lusztig representations: for finite reductive groups, characters are parametrised by pairs \((T, \theta)\) with \(T\) a rational maximal torus and \(\theta\) a character of \(T(\mathbb{F}_q)\).
  • Endoscopy: endoscopic data are organised by stable conjugacy classes of maximal tori (or their dual cocenters).
  • Local Langlands at the Iwahori block: indexed by characters of maximal tori restricted to maximal compact subgroups.

Mathlib

Mathlib.LinearAlgebra.RootSystem.* handles Cartan-like data abstractly; specific algebraic-group maximal tori are not formalised.

References

  • Borel, Linear Algebraic Groups (1991), §11, §13.
  • Springer, Linear Algebraic Groups (1998), §6, §13.
  • Steinberg, Endomorphisms of linear algebraic groups (Mem.\ AMS 1968).
  • Conrad, Reductive group schemes (Notes, 2014), §3.

Prerequisite Topics

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