The affine Weyl group action on ${𝔥}_{ℂ}^{*}$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 27 July 2013

The affine Weyl group action on ${𝔥}_{ℂ}^{*}$

A reference for this section is [Kac1104219, §6.5].

Let $\stackrel{\circ }{R}$ be the set of roots of the finite dimensional Lie algebra $\stackrel{\circ }{𝔤}$ (see (1.10)). For $\alpha \in \stackrel{\circ }{R}$

$$\begin{array}{cc}\nu \left(h_{\alpha }\right)=\alpha \text{and}\nu \left(h_{{\alpha }^{\vee }}\right)={\alpha }^{\vee }=\frac{2\alpha }{⟨\alpha ,\alpha ⟩},& \text{(1.21)}\end{array}$$

where $[e_{\alpha },f_{\alpha }]=h_{{\alpha }^{\vee }},$ and $\lambda \left(h_{{\alpha }^{\vee }}\right)=⟨\lambda ,{\alpha }^{\vee }⟩\text{.}$

For $\alpha \in \stackrel{\circ }{R}$ and $k\in \mathbb{Z}$ let

$$h_{{(-\alpha +k\delta )}^{\vee }}=-h_{{\alpha }^{\vee }}+k{⟨e_{\alpha },f_{\alpha }⟩}_{0}K=-h_{{\alpha }^{\vee }}+k\frac{1}{2}{⟨h_{{\alpha }^{\vee }},h_{{\alpha }^{\vee }}⟩}_{0}K$$

and define

$$s_{-\alpha +k\delta }:{𝔥}_{ℂ}^{*}⟶{𝔥}_{ℂ}^{*}\text{by}{s}_{-\alpha +k\delta }\lambda =\lambda -\lambda \left(h_{{(-\alpha +k\delta )}^{\vee }}\right)(-\alpha +k\delta )\text{.}$$

Hence $[e_{\alpha },f_{\alpha }]=h_{{\alpha }^{\vee }}$ and

$$\bar{\lambda }\left(h_{{\alpha }^{\vee }}\right)=⟨\bar{\lambda },{\alpha }^{\vee }⟩$$

relates the evaluation pairing and the form on ${𝔥}^{*}\text{.}$ If $\lambda =\alpha \delta +\bar{\lambda }+m{\Lambda }_0$ and $s_{\alpha }\bar{\lambda }=\bar{\lambda }-⟨\bar{\lambda },{\alpha }^{\vee }⟩$ then

$$\begin{array}{ccc}{s}_{-\alpha +k\delta }\lambda & =& \lambda -\lambda \left(h_{{(-\alpha +k\delta )}^{\vee }}\right)(-\alpha +k\delta )\\ & =& \lambda -(a\delta +\bar{\lambda }+m{\Lambda }_0)(-h_{{\alpha }^{\vee }}+k\frac{1}{2}{⟨h_{{\alpha }^{\vee }},h_{{\alpha }^{\vee }}⟩}_{0}K)(-\alpha +k\delta )\\ & =& \lambda -a·0-⟨\bar{\lambda },-{\alpha }^{\vee }⟩(-\alpha +k\delta )-mk\frac{1}{2}⟨{\alpha }^{\vee },{\alpha }^{\vee }⟩(-\alpha +k\delta )\\ & =& \lambda -⟨\bar{\lambda },{\alpha }^{\vee }⟩\alpha +mk\frac{⟨\alpha ,{\alpha }^{\vee }⟩}{⟨\alpha ,\alpha ⟩}\alpha +(⟨\bar{\lambda },k{\alpha }^{\vee }⟩-\frac{1}{2}m⟨k{\alpha }^{\vee },k{\alpha }^{\vee }⟩)\delta \\ & =& \lambda -⟨\bar{\lambda },{\alpha }^{\vee }⟩\alpha +mk{\alpha }^{\vee }+(⟨\bar{\lambda },k{\alpha }^{\vee }⟩-\frac{1}{2}m⟨k{\alpha }^{\vee },k{\alpha }^{\vee }⟩)\delta \\ & =& (a+⟨\bar{\lambda },k{\alpha }^{\vee }⟩-\frac{1}{2}m⟨k{\alpha }^{\vee },k{\alpha }^{\vee }⟩)\delta +\bar{\lambda }-⟨\bar{\lambda },{\alpha }^{\vee }⟩\alpha +mk{\alpha }^{\vee }+m{\Lambda }_0\\ & =& (a+⟨\bar{\lambda },k{\alpha }^{\vee }⟩-\frac{1}{2}m⟨k{\alpha }^{\vee },k{\alpha }^{\vee }⟩)\delta +s_{\alpha }\bar{\lambda }\alpha +mk{\alpha }^{\vee }+m{\Lambda }_0\end{array}$$

so that, in matrix form with respect to a basis $\delta ,{\hat{h}}_1,\dots ,{\hat{h}}_{\ell },{\Lambda }_0$ of $𝔥,$ where ${\hat{h}}_1,\dots ,{\hat{h}}_{\ell }$ is an orthonormal basis of $\stackrel{\circ }{𝔥},$

$$s_{-\alpha +k\delta }=\left(\begin{array}{ccc}1& k{\alpha }^{\vee }& -\frac{1}{2}⟨k{\alpha }^{\vee },k{\alpha }^{\vee }⟩\\ \begin{array}{c}⋮\\ 0\\ ⋮\end{array}& \begin{array}{ccc}\phantom{0}& \phantom{0}& \phantom{0}\\ \phantom{0}& s_{\alpha }& \phantom{0}\\ \phantom{0}& \phantom{0}& \phantom{0}\end{array}& \begin{array}{ccc}\phantom{0}& \phantom{0}& \phantom{0}\\ \phantom{0}& k{\alpha }^{\vee }& \phantom{0}\\ \phantom{0}& \phantom{0}& \phantom{0}\end{array}\\ 0& \begin{array}{ccc}\dots & 0& \dots \end{array}& \begin{array}{ccc}\phantom{\dots }& 1& \phantom{\dots }\end{array}\end{array}\right)$$

Then

$$\begin{array}{ccc}{t}_{k{\alpha }^{\vee }}=s_{-\alpha +k\delta }{s}_{\alpha }& =& \left(\begin{array}{ccc}1& k{\alpha }^{\vee }& -\frac{1}{2}⟨k{\alpha }^{\vee },k{\alpha }^{\vee }⟩\\ \begin{array}{c}⋮\\ 0\\ ⋮\end{array}& \begin{array}{ccc}\phantom{0}& \phantom{0}& \phantom{0}\\ \phantom{0}& s_{\alpha }& \phantom{0}\\ \phantom{0}& \phantom{0}& \phantom{0}\end{array}& \begin{array}{ccc}\phantom{0}& \phantom{0}& \phantom{0}\\ \phantom{0}& k{\alpha }^{\vee }& \phantom{0}\\ \phantom{0}& \phantom{0}& \phantom{0}\end{array}\\ 0& \begin{array}{ccc}\dots & 0& \dots \end{array}& \begin{array}{ccc}\phantom{\dots }& 1& \phantom{\dots }\end{array}\end{array}\right)\left(\begin{array}{ccc}1& \begin{array}{ccc}\phantom{\dots }& 0& \phantom{\dots }\end{array}& 0\\ \begin{array}{c}⋮\\ 0\\ ⋮\end{array}& \begin{array}{ccc}\phantom{0}& \phantom{0}& \phantom{0}\\ \phantom{0}& s_{\alpha }& \phantom{0}\\ \phantom{0}& \phantom{0}& \phantom{0}\end{array}& \begin{array}{c}\phantom{⋮}\\ 0\\ \phantom{⋮}\end{array}\\ 0& \begin{array}{ccc}\dots & 0& \dots \end{array}& 1\end{array}\right)\\ & =& \left(\begin{array}{ccc}1& -k{\alpha }^{\vee }& -\frac{1}{2}⟨k{\alpha }^{\vee },k{\alpha }^{\vee }⟩\\ \begin{array}{c}⋮\\ 0\\ ⋮\end{array}& \begin{array}{ccc}\phantom{0}& \phantom{0}& \phantom{0}\\ \phantom{0}& 1& \phantom{0}\\ \phantom{0}& \phantom{0}& \phantom{0}\end{array}& \begin{array}{ccc}\phantom{0}& \phantom{0}& \phantom{0}\\ \phantom{0}& k{\alpha }^{\vee }& \phantom{0}\\ \phantom{0}& \phantom{0}& \phantom{0}\end{array}\\ 0& \begin{array}{ccc}\dots & 0& \dots \end{array}& \begin{array}{ccc}\phantom{\dots }& 1& \phantom{\dots }\end{array}\end{array}\right)\text{.}\end{array}$$

Thus the group generated by the $\left\{s_{\alpha +k\delta }\hspace{0.17em}\right|\hspace{0.17em}\alpha \in \stackrel{\circ }{R},k\in \mathbb{Z}\}$ is

$$W=W_0⋊Q^{\vee }=\left\{w{t}_{{\mu }^{\vee }}\hspace{0.17em}\right|\hspace{0.17em}w\in W_0,{\mu }^{\vee }\in Q^{\vee }\},\text{where}{Q}^{\vee }=\sum _{\alpha \in \stackrel{\circ }{R}}\mathbb{Z}{\alpha }^{\vee }\text{.}$$

and $W_0$ is the group generated by the $\left\{s_{\alpha }\hspace{0.17em}\right|\hspace{0.17em}\alpha \in \stackrel{\circ }{R}\}\text{.}$

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