Topic Roots, Coroots, and Root Subgroups
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Roots, Coroots, and Root Subgroups

To a reductive group \(G\) with maximal torus \(T\) one associates:

  • a finite set \(\Phi \subseteq X^*(T)\) of roots;
  • a corresponding set \(\Phi^\vee \subseteq X_*(T)\) of coroots;
  • for each \(\alpha \in \Phi\), a root subgroup \(U_\alpha \subseteq G\) isomorphic to \(\mathbb{G}_a\).

These are the building blocks of the entire structure theory of \(G\).

Roots

Let \(G\) be a connected reductive group over an algebraically closed field \(k\) with maximal torus \(T\). The torus acts on the Lie algebra \(\mathfrak{g} = \operatorname{Lie}(G)\) by the adjoint representation, decomposing it into weight spaces: \[ \mathfrak{g} \;=\; \mathfrak{t} \;\oplus\; \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha, \] where \(\mathfrak{t} = \operatorname{Lie}(T)\) and \(\mathfrak{g}_\alpha = \{X \in \mathfrak{g} : \mathrm{Ad}(t) X = \alpha(t) X \text{ for all } t \in T\}\) for non-zero weight \(\alpha \in X^*(T)\).

The roots are \[ \Phi \;=\; \Phi(G, T) \;:=\; \{\alpha \in X^*(T) \setminus \{0\} : \mathfrak{g}_\alpha \ne 0\}. \]

Properties:

  • \(\Phi\) is finite;
  • \(\dim \mathfrak{g}_\alpha = 1\) for each \(\alpha \in \Phi\);
  • \(\Phi = -\Phi\) (roots come in pairs \(\pm \alpha\));
  • \(\Phi\) generates a sublattice of \(X^*(T) \otimes \mathbb{Q}\) — its rank is the semisimple rank of \(G\).

Root subgroups

For each \(\alpha \in \Phi\), there is a unique connected closed subgroup \(U_\alpha \subseteq G\) with:

  • \(U_\alpha\) is isomorphic to \(\mathbb{G}_a\) (as algebraic group);
  • \(\operatorname{Lie}(U_\alpha) = \mathfrak{g}_\alpha\);
  • \(T\) normalises \(U_\alpha\), acting through \(\alpha\): \(t \cdot u_\alpha(x) \cdot t^{-1} = u_\alpha(\alpha(t) \cdot x)\).

The \(U_\alpha\) are the root subgroups of \(G\) relative to \(T\).

Coroots

For each \(\alpha \in \Phi\), the subgroup \(\langle U_\alpha, U_{-\alpha}, T \rangle\) generates a subgroup \(G_\alpha\) of \(G\) (the rank-1 subgroup) that is isogenous to \(\mathrm{SL}_2\). The coroot \[ \alpha^\vee : \mathbb{G}_m \;\longrightarrow\; T \] is defined as the unique cocharacter such that: \[ \alpha^\vee(\mathbb{G}_m) \;=\; G_\alpha \cap T, \qquad \langle \alpha, \alpha^\vee \rangle = 2. \]

This packages the \(\mathrm{SL}_2\)-data near the \(\alpha\)-axis into a 1-parameter subgroup of \(T\).

Commutation relations

For non-proportional roots \(\alpha, \beta \in \Phi\), the Steinberg relation holds: \[ [u_\alpha(x), u_\beta(y)] \;=\; \prod_{i\alpha + j\beta \in \Phi, \; i, j > 0} u_{i\alpha + j\beta}(c_{i,j}^{\alpha, \beta} x^i y^j), \] where the integers \(c_{i,j}^{\alpha, \beta}\) are the Chevalley structure constants (an explicit description in each Dynkin type).

For \(\alpha, -\alpha\): \[ \langle U_\alpha, U_{-\alpha} \rangle \;\subseteq\; \mathrm{SL}_2\text{-type subgroup}. \]

Examples

\(\mathrm{GL}_n\)

\(T\) = diagonal matrices; \(X^*(T) = \mathbb{Z}\langle \varepsilon_1, \ldots, \varepsilon_n \rangle\); \(\Phi = \{\varepsilon_i - \varepsilon_j : i \ne j\}\).

For \(\alpha = \varepsilon_i - \varepsilon_j\):

  • \(U_\alpha\) = matrices with single non-zero off-diagonal entry at position \((i, j)\), i.e.\ \(I + x E_{ij}\) for \(x \in k\).
  • \(\alpha^\vee\) = the cocharacter \(t \mapsto \mathrm{diag}(1, \ldots, t, \ldots, t^{-1}, \ldots, 1)\) with \(t\) in position \(i\) and \(t^{-1}\) in position \(j\).

\(\mathrm{Sp}_{4}\) (type \(C_2\))

8 roots: \(\pm \varepsilon_1 \pm \varepsilon_2\), \(\pm 2\varepsilon_1\), \(\pm 2\varepsilon_2\). Long roots \(\pm 2 \varepsilon_i\) have coroots \(\alpha^\vee = \varepsilon_i^\vee\) (themselves a basis of \(X_*\)).

Functoriality

For a morphism \(f : (G, T) \to (G', T')\) of reductive groups matching maximal tori:

  • \(f\) induces a map of character lattices \(X^*(T') \to X^*(T)\);
  • \(f\) maps \(U_\alpha\) to \(U'_{f^*(\alpha)}\) (compatibly with Steinberg relations).

This makes the root subgroup structure functorial in pairs \((G, T)\).

Borel subgroups and the choice of positive roots

A choice of Borel subgroup \(B \supseteq T\) determines a system of positive roots \(\Phi^+ \subseteq \Phi\) (those occurring in \(\operatorname{Lie}(B)\)). The Borel decomposes as: \[ B \;=\; T \cdot \prod_{\alpha \in \Phi^+} U_\alpha, \] with the product taken in any order (the resulting variety is well- defined, independent of ordering).

The corresponding opposite Borel \(B^-\) has the opposite positivity, and \(U \cdot U^- \cdot T\) is the big cell, open dense in \(G\).

Role in Langlands

  • Local fields: parahoric subgroups and Iwahori subgroups are defined via the root subgroups (each contributes either trivially or as a "level" in the parahoric filtration).
  • Bruhat decomposition \(G = \sqcup_{w \in W} BwB\): each cell involves explicit root subgroup products via the Steinberg relations.
  • Galois cohomology of root subgroups: \(H^1(k, U_\alpha) = 0\) (since \(U_\alpha = \mathbb{G}_a\) is "additively trivial"), contributing to vanishing in the calculation of \(H^1(k, G)\).

Mathlib

Root subgroups specifically not formalised. Abstract roots and coroots via Mathlib.LinearAlgebra.RootSystem.*.

References

  • Borel, Linear Algebraic Groups (1991), §13.
  • Springer, Linear Algebraic Groups (1998), §7.
  • Steinberg, Lectures on Chevalley Groups (1968).
  • Carter, Simple Groups of Lie Type (1972).

Prerequisite Topics

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