Affine Root Systems

An affine root system is the set of affine roots \(\Psi = \{\alpha + n\}\) on a Euclidean space, arising from a reductive group over a local field. They classify the hyperplane arrangement of a Bruhat-Tits apartment and give the affine Coxeter group of the affine Weyl group.

Definition

An affine root system in a Euclidean space \(\mathbb{A} = V\) is a set \(\Psi\) of affine linear functionals \(\psi : \mathbb{A} \to \mathbb{R}\) of the form \(\psi(x) = \alpha(x) + c\) for \(\alpha\) in the "vector part" \(\Phi\) (an ordinary root system on \(V^*\)) and \(c \in \mathbb{R}\), satisfying:

(AR1) \(\Psi\) is closed under affine reflections \(s_\psi\) (which preserve hyperplanes \(\psi^{-1}(0)\)).

(AR2) The set of constants \(\{c : \alpha + c \in \Psi\}\) for each \(\alpha\) forms a discrete sub-additive group of \(\mathbb{R}\).

(AR3) The vector part \(\Phi = \{\alpha : \exists c, \alpha + c \in \Psi\}\) is a finite root system.

Construction from reductive groups

For \(G\) a connected reductive group over a non-archimedean local field \(F\) with maximal \(F\)-split torus \(S\) and apartment \(\mathbb{A} = X_*(S) \otimes \mathbb{R}\):

The affine root system \(\Psi(G, S, F)\) consists of the affine roots \(\alpha + n\) (or fractional translates, in ramified cases) coming from:

The root subgroups \(U_\alpha \subseteq G(F)\) for relative roots \(\alpha \in \Phi(G, S)\);
The Moy-Prasad filtrations of \(U_\alpha\) by depth.

Reduced vs.\ non-reduced

Reduced: each "ray" \(\{\alpha + c : c \in \mathbb{R}\}\) gives one affine root (up to translation); the affine root system is \(\tilde X_n\) for some classical type.
Non-reduced: some rays carry both \(\alpha\) and \(2\alpha\) contributions (e.g.\ for ramified \(\mathrm{SU}_n\)); the affine root system is of "twisted" or \(BC\)-type.

Non-reduced affine root systems classify ramified non-split groups, which is essential for general local Langlands.

Classification

Affine Dynkin diagrams (Kac labels):

Type	Affine Dynkin	Description
\(\tilde A_n\)	cycle of \(n+1\) vertices	from \(\mathrm{SL}_{n+1}\)
\(\tilde B_n, \tilde C_n, \tilde D_n\)	classical extended diagrams	from \(\mathrm{Spin}, \mathrm{Sp}, \mathrm{SO}\)
\(\tilde E_6, \tilde E_7, \tilde E_8\)	exceptional extended	from \(E_6, E_7, E_8\)
\(\tilde F_4, \tilde G_2\)	exceptional extended	from \(F_4, G_2\)
\(\tilde{BC}_n\), \(\tilde{C}_n^\vee\), etc.\	twisted affine	from ramified non-split groups

Affine Weyl group

The reflections \(s_\psi\) for \(\psi \in \Psi\) generate the affine Weyl group \(W_{\mathrm{aff}}\): \[ W_{\mathrm{aff}} \;=\; W \;\ltimes\; \mathbb{Z}\Phi^\vee, \] which is the Coxeter group of the affine Dynkin diagram.

Alcoves and walls

For a reduced affine root system: the alcoves (= chambers of \(W_{\mathrm{aff}}\) acting on \(\mathbb{A}\)) are the connected components of the complement of \(\bigcup_\psi \psi^{-1}(0)\). They form the top simplices of the Bruhat-Tits apartment.

Examples

\(\tilde A_1\)

\(\Psi = \{\pm\alpha + n : n \in \mathbb{Z}\}\); the affine roots are "single rays at integer translates." Walls at \(\frac{1}{2}\mathbb{Z}\); alcoves = \([n, n+\frac{1}{2}]\). Affine Weyl group = \(D_\infty\).

\(\tilde A_n\)

\(\Psi\) consists of the affine roots \(e_i - e_j + n\) for \(i \ne j\), \(n \in \mathbb{Z}\). Alcoves are simplices in the triangular lattice.

\(\tilde{BC}_n\)

For ramified non-split groups: includes "half-integer" affine roots \(\frac{1}{2}\alpha + n\) in addition to the integer ones.

Role in Langlands

Affine Hecke algebra: defined relative to the affine Weyl group; controls the representation theory of \(G(F)\) at the Iwahori block.
Macdonald polynomials: defined via affine root systems; generalise Hall-Littlewood and Jack polynomials.
Cherednik DAHA: double affine Hecke algebra, with quantum affine Weyl group; foundational for geometric Langlands at the affine level.

Mathlib

Partial infrastructure via Mathlib.LinearAlgebra.RootSystem.*; affine root systems specifically not formalised.