Based Root Data
A based root datum is the combinatorial invariant that classifies split connected reductive groups. The classification theorem (Demazure / Chevalley) states:
\[ \{\text{split connected reductive } G/k\} \;\leftrightarrow\; \{\text{based root data}\}. \]
This is the algebraic-geometric analogue of "compact Lie groups are classified by their root data" in classical Lie theory.
Definition
A root datum is a quadruple \((X, \Phi, X^\vee, \Phi^\vee)\) where:
- \(X, X^\vee\) are finite-rank free \(\mathbb{Z}\)-modules paired perfectly \(\langle -, - \rangle : X \times X^\vee \to \mathbb{Z}\);
- \(\Phi \subseteq X\) is a finite set (roots);
- \(\Phi^\vee \subseteq X^\vee\) is a finite set (coroots);
- there is a bijection \(\Phi \leftrightarrow \Phi^\vee\), \(\alpha \leftrightarrow \alpha^\vee\) satisfying:
- \(\langle \alpha, \alpha^\vee \rangle = 2\);
- reflections \(s_\alpha(x) = x - \langle x, \alpha^\vee \rangle \alpha\) preserve \(\Phi\) (and dually \(\Phi^\vee\)).
A based root datum adds a choice of base (= simple roots) \(\Delta \subseteq \Phi\): a subset such that every root is a non-negative or non-positive integer combination of \(\Delta\).
Interpretation for a reductive group
For a split connected reductive group \(G\) over \(k\) with split maximal torus \(T\):
- \(X = X^*(T) = \operatorname{Hom}(T, \mathbb{G}_m)\) is the character lattice (a free \(\mathbb{Z}\)-module of rank \(= \dim T\));
- \(X^\vee = X_*(T) = \operatorname{Hom}(\mathbb{G}_m, T)\) is the cocharacter lattice;
- \(\Phi \subseteq X\) is the set of nonzero weights of \(T\) acting on \(\operatorname{Lie}(G)\) (roots);
- For each \(\alpha \in \Phi\), there is a canonical "SL_2-triple" giving \(\alpha^\vee : \mathbb{G}_m \to T\) (coroot).
A choice of Borel subgroup \(B \supseteq T\) gives the base: the simple roots are the weights of \(T\) on \(\operatorname{Lie}(B/T)\) that are not sums of two positive roots.
Classification
Theorem (Chevalley / Demazure–Grothendieck). Over any field \(k\), the assignment \(G \mapsto (\text{based root datum})\) is a bijection between isomorphism classes of split connected reductive \(k\)-groups and isomorphism classes of based root data.
Moreover, this works over \(\mathbb{Z}\): every based root datum lifts canonically to a Chevalley group scheme \(G_{\mathbb{Z}}\) over \(\mathbb{Z}\), and the split form over any base is obtained by base change.
Connection to Dynkin diagrams
A based root datum determines a Dynkin diagram:
- vertices = simple roots;
- edges record the angles between simple roots via the Cartan integers \(c_{ij} = \langle \alpha_i, \alpha_j^\vee \rangle\).
The connected Dynkin diagrams classify simple root systems (and hence simply connected simple algebraic groups):
\[ A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2. \]
For semisimple groups, the based root datum decomposes as a product of "isogeny classes" indexed by the simply connected and adjoint forms in each Dynkin type.
Duality
The dual root datum \((X^\vee, \Phi^\vee, X, \Phi)\) swaps roots and coroots. Under the classification theorem, this corresponds to the Langlands dual group \({}^L G\) of \(G\):
\[ G = \mathrm{SL}_n \leftrightarrow {}^L G = \mathrm{PGL}_n, \qquad G = \mathrm{SO}_{2n+1} \leftrightarrow {}^L G = \mathrm{Sp}_{2n}, \]
and so on. Langlands duality is the cornerstone of the Langlands correspondence.
Outer automorphisms
\(\operatorname{Out}(G) \cong \operatorname{Aut}(\text{based root datum})\) for connected reductive \(G\). This is the rigidity that makes the classification a classification: there is no continuous moduli of reductive groups, only the discrete combinatorial data.