Topic Affine Weyl Groups
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Affine Weyl Groups

The affine Weyl group of a reductive group is the semidirect product of its finite Weyl group with the coroot lattice. It is the Coxeter group governing the affine Bruhat-Tits building, and the basic combinatorial structure on the affine flag variety and the Iwahori-Hecke algebra.

Definition

Let \((W, S)\) be the finite Weyl group of a root system \(\Phi \subseteq V\) with coroots \(\Phi^\vee\). The affine Weyl group is the semidirect product \[ W_{\mathrm{aff}} \;:=\; W \;\ltimes\; \mathbb{Z}\Phi^\vee, \] acting on \(V\) by: \[ (w, \lambda) \cdot v \;=\; w(v) + \lambda. \]

Equivalently, \(W_{\mathrm{aff}}\) is the group generated by the finite reflections \(s_\alpha\) (\(\alpha \in \Phi\)) together with the affine reflection \[ s_0 : v \mapsto v - (\langle v, \alpha_0^\vee \rangle - 1) \alpha_0, \] where \(\alpha_0\) is the highest root.

Extended affine Weyl group

For applications to non-simply-connected groups, the extended affine Weyl group is \[ \tilde W \;:=\; W \;\ltimes\; X_*, \] where \(X_*\) is the cocharacter lattice (containing \(\mathbb{Z}\Phi^\vee\) as a sublattice). The quotient \[ \Omega \;:=\; \tilde W / W_{\mathrm{aff}} \;=\; X_* / \mathbb{Z}\Phi^\vee \;=\; \pi_1(G) \] acts as diagram automorphisms on the affine Dynkin diagram.

For \(G\) simply connected: \(X_* = \mathbb{Z}\Phi^\vee\), so \(\Omega = 1\) and \(\tilde W = W_{\mathrm{aff}}\).

Coxeter structure

\(W_{\mathrm{aff}}\) is a Coxeter group with generators \(\{s_0\} \cup S\), where:

  • \(S = \{s_1, \ldots, s_r\}\) are the simple reflections of \(W\);
  • \(s_0\) is the affine reflection through the wall of the fundamental alcove opposite to the origin.

The braid relations include the new ones \((s_0 s_i)^{m_{0i}} = 1\) encoded by the affine Dynkin diagram \(\tilde \Phi\).

The extended affine Weyl group \(\tilde W\) is not a Coxeter group in general, but \(\Omega\) acts by automorphisms of the Coxeter system.

Alcoves

The fundamental domain for \(W_{\mathrm{aff}}\) acting on \(V\) is the fundamental alcove: \[ A_0 \;=\; \{v \in V : 0 \le \langle \alpha_i, v \rangle, \langle \alpha_0, v \rangle \le 1\}. \]

The \(W_{\mathrm{aff}}\)-translates of \(A_0\) tile \(V\) into alcoves, and \(W_{\mathrm{aff}}\) acts simply transitively on alcoves.

For \(\tilde W\), the action of \(\Omega\) permutes alcoves within an \(\Omega\)-orbit (= "stripe").

Examples

Type \(\tilde A_1\)

\(W = \mathbb{Z}/2\), \(\Phi^\vee = \mathbb{Z}\), so \(W_{\mathrm{aff}} = \mathbb{Z}/2 \ltimes \mathbb{Z} = D_\infty\) (infinite dihedral). Alcoves = intervals \([n, n+1] \subseteq \mathbb{R}\).

\(X_* = \frac{1}{2}\mathbb{Z}\) for \(\mathrm{PGL}_2\), so \(\Omega = \mathbb{Z}/2\) and \(\tilde W = (\mathbb{Z}/2) \ltimes \frac{1}{2}\mathbb{Z}\).

Type \(\tilde A_{n-1}\)

\(W = S_n\), \(\mathbb{Z}\Phi^\vee = \{(a_1, \ldots, a_n) : \sum a_i = 0\}\). The affine Dynkin diagram is a cycle of \(n\) nodes. Alcoves = top simplices in the triangulation of \(\{v \in \mathbb{R}^n : \sum v_i = 0\}\) by hyperplanes \(v_i - v_j \in \mathbb{Z}\).

For \(\mathrm{GL}_n\): \(X_* = \mathbb{Z}^n\), \(\Omega = \mathbb{Z}\) (the determinant lattice), generated by the "cyclic rotation" of the affine diagram.

Type \(\tilde G_2\)

\(W\) = dihedral of order 12. Affine Dynkin = "tail" attached to the finite \(G_2\) diagram. \(\Omega = 1\).

Length function

For \(\tilde w = w \cdot t_\lambda \in \tilde W\): \[ \ell(\tilde w) \;=\; \sum_{\alpha \in \Phi^+} \big| \langle \alpha, \lambda \rangle + \chi(w^{-1}\alpha < 0) \big|, \] where \(\chi\) is the indicator. This generalises the finite Weyl group length and is the foundation for Bruhat order on \(W_{\mathrm{aff}}\).

Bruhat decomposition for \(G(F)\)

For \(G\) reductive over a local field \(F\): \[ G(F) \;=\; \bigsqcup_{\tilde w \in \tilde W} I \cdot \tilde w \cdot I, \] where \(I\) is the Iwahori subgroup. This is the affine Bruhat decomposition and the source of the Iwahori-Hecke algebra.

Role in Langlands

  • Affine flag variety \(\mathcal{F}\ell_G = G(F)/I\) is stratified by \(\tilde W\); each stratum is an affine space.
  • Affine Grassmannian \(\mathrm{Gr}_G = G(F)/G(\mathcal{O})\) is stratified by \(\tilde W / W = X_*\); the closures of strata are affine Schubert varieties.
  • Iwahori-Hecke algebra \(\mathcal{H}(\tilde W) = \mathbb{Z}[v, v^{-1}][\tilde W]\) with quadratic relations; the affine Hecke algebra of \(G\).
  • Kazhdan-Lusztig: Kazhdan-Lusztig polynomials of \(\tilde W\) control local-Langlands branching and category O at non-integral weights.
  • Geometric Satake: equivalence of categories \(\mathrm{Perv}_{G(\mathcal{O})}(\mathrm{Gr}_G) \cong \mathrm{Rep}(\hat G)\) is parametrised by \(X_*^+ = X_*\) (dominant cocharacters).

Mathlib

Mathlib.LinearAlgebra.RootSystem.* has partial affine-root-system support; full affine Weyl group is not yet formalised.

References

  • Bourbaki, Groupes et Algèbres de Lie IV–VI (1968), §V.3.
  • Humphreys, Reflection Groups and Coxeter Groups (1990), Ch. 4.
  • Macdonald, Affine Hecke Algebras and Orthogonal Polynomials (2003).
  • Bruhat–Tits, Groupes réductifs sur un corps local I, II (1972, 1984).

Prerequisite Topics

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