Affine \SL_2

Let V be the geometric representation of the \widetilde{A}_2 root system, with basis the two simple roots \set{\alpha_1, \alpha_2} with \alpha_2 the affine root. A convenient alternative basis is \set{\alpha_1, \delta} with \delta = \alpha_1 + \alpha_2 the “null root” (not actually a root, just a member of the root lattice). \delta is null in the sense that \innprod{\alpha^\vee, \delta} = 0 for all coroots \alpha^\vee.

The W-orbit of the simple roots is the root system \Phi \subseteq V, shown below on the right. The horizontal axis is the hyperplane where the finite roots \set{\pm \alpha_1} live, and the vertical axis is along the null root \delta. Whenver a root \beta is selected, its reflecting hyperplane H_\beta = \set{\mu \in V^* \mid \innprod{\mu, \beta} = 0} is shown on the left in red.

In the dual space V^*, the coroots \alpha_1^\vee and \alpha_2^\vee are linearly dependent (\alpha_1^\vee + \alpha_2^\vee = 0) and so the W-action does not generate a root system, but instead of roots we have chamber geometry. Define the fundamental chamber C = \set{\mu \in V^* \mid \innprod{\mu, \alpha_i} > 0 \text{ for } i=1,2}, then the W-orbit of C is a convex cone \cC called the Tits cone, which in this case is the upper half-plane. The chamber C is a fundamental domain for the action of W on \cC. By intersecting each wC with the hyperplane \innprod{-, \delta} = 1 (written as the equation ”\delta = 1” for short), we obtain the alternative picture of W as a group of affine reflections acting on the line \delta = 1.

The dual space V^* is shown on the left, in the basis dual to (\alpha_1, \delta). Whenever a chamber wC is selected, the root inversion set \Phi^+(w) = \set{\beta \in \Phi^+ \mid w(\beta) < 0} is shown circled on the right on the right. The element w \alpha_1 is shown on the right with a light green background. Since every element of W fixes \bbR \delta pointwise, knowing w \alpha_1 is enough to determine the action of w on V.

δ=1\delta = 1
α1\alpha_1^\vee
α2\alpha_2^\vee
Λ1\Lambda_1^\vee
Λ2\Lambda_2^\vee
δ\delta^*
ϖ1\varpi_1^\vee
-12,13 -13,12 -11,12 -12,11 -10,11 -11,10 -9,10 -10,9 -8,9 -9,8 -7,8 -8,7 -6,7 -7,6 -5,6 -6,5 -4,5 -5,4 -3,4 -4,3 -2,3 -3,2 -1,2 -2,1 0,-1 -1,0 1,0 0,1 2,1 1,2 3,2 2,3 4,3 3,4 5,4 4,5 6,5 5,6 7,6 6,7 8,7 7,8 9,8 8,9 10,9 9,10 11,10 10,11 12,11 11,12
α1\alpha_1
α2\alpha_2
δ\delta
Word:

Coordinates

Declare V to be the vector space with basis (\alpha_1, \alpha_2) of simple roots (so the simple roots are linearly independent by definition), then the simple coroots (\alpha_1^\vee, \alpha_2^\vee) are the unique vectors in V^* so that their pairing with the simple roots gives the Cartan matrix of affine \SL_2:

\alpha_1^\vee, \alpha_2^\vee \in V^* \text{ satisfy } \innprod{\alpha_i^\vee, \alpha_j^\vee} = a_{ij}, \text{ where } [a_{ij}] = \begin{pmatrix}2 & -2 \\ -2 & 2 \end{pmatrix}.

The simple coroots will be linearly independent if and only if the Cartan matrix [a_{ij}] has full rank, so in this case the simple coroots are dependent. We also get the uniquely determined fundamental coweights \Lambda_1^\vee, \Lambda_2^\vee \in V^* by requiring the condition \innprod{\Lambda_i^\vee, \alpha_j} = \delta_{ij}. The fundamental coweights are linearly independent, because the matrix [\delta_{ij}] has full rank. Finally note that in this realisation, we cannot define the fundamental weights, since the simple coroots are linearly dependent which is incompatible with the condition \innprod{\alpha_i^\vee, \Lambda_j} = \delta_{ij}.

What we have constructed is called the adjoint realisation of affine \SL_2: if we were to define a Kac-Moody group using V = \mathfrak{h}_\bbR as the Cartan and V^* = \mathfrak{h}^*_\bbR as the dual Cartan, we would end up with a group of adjoint type. By switching the roles of V and V^*, we would have the simply-connected realisation.

Aside: In the standard construction of Kac-Moody algebras, neither the adjoint nor simply-connected realisations are used. Instead, (in the case of an affine algebra) the dimension of V is enlarged by 1, which gives enough “room” so that the simple roots and simple coroots can be embedded so that both are linearly independent. This means that both the simple coweights and simple weights are defined, and one gets a proper root system and chamber geometry on both sides of the realisation, rather than just on one. If one is only wanting to study the underlying Coxeter group (rather than constructing Kac-Moody groups), then looking at the adjoint realisation is enough, and all the standard root system results apply in this smaller case.

After construction, it is helpful to change basis to (\alpha_1, \delta) where \delta = \alpha_1 + \alpha_2 is the null root, so the affine root \alpha_2 = \delta - \alpha_1. In general, the coefficients of \delta are the coefficients of a minimal integral linear dependence in the columns of the Cartan matrix A, so \delta is a canonical coordinate for any affine algebra. Both simple reflections fix \bbR \delta pointwise, and on \alpha_1 we have s_1(\alpha_1) = - \alpha_1 and s_2(\alpha_1) = 2\delta - \alpha_1. In terms of matrices in the (\alpha_1, \delta)-basis,

[s_1] = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}, \quad [s_2] = \begin{pmatrix} -1 & 0 \\ 2 & 1 \end{pmatrix}.

To calculate inversion sets, we are using the fact that if l(sw) > l(w) then \Phi^+(sw) = \Phi^+(w) \cup \set{w^{-1} \alpha_s}.