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Finding chamber weights for LSL2

By trdunlap2

Lets \tilde LSL_2 be the double extension of the loop group: taking a universal central extension and an etension by loop rotation. Take a maximal torus \tilde T_{LSL_2} = \mathbb{C}^\times_{central}\times T_{SL_2}\times\mathbb{C}^\times_{rotation} where T_{SL_2}=\mathbb{C}^\times is the torus of constant loops in SL_2. The characters of T_{LSL_2}, denoted \hat T_{LSL_2}, are the weights of \tilde LSL_2 and form a three dimensional lattice.

The roots, i.e. those weights which appear as eigenvalues of the adjoint action of T_{LSL_2} on \tilde L\mathfrak{sl}_2 (minus the zero weight), divide into two groups. The real roots are the elements of the form (0,\alpha,n) where \alpha\in\hat R_{SL_2} is a root of SL_2 and by n we mean \theta\mapsto\theta^n. The imaginary roots are of the form (0,0,n\neq 0). (Note/Recall that root lattice for SL_2 has index two in the weight lattice.)

The coweights of \tilde LSL_2 are \check T_{LSL_2} also a 3 dimensional lattice.
We take the eigenvectors \tilde e_{\tilde \alpha} for each root \tilde\alpha=(0,\alpha,n) in our case these are the matrices z^ne_\alpha where e_\alpha (recall from SL_2) is either \left(\begin{matrix} 0 && 1 \\ 0 && 0\end{matrix}\right) or \left(\begin{matrix} 0 && 0 \\ 1 && 0\end{matrix}\right). We do this to calculate the coroots which will have dirivative given by \tilde h_{\tilde \alpha}=[\tilde e_{\tilde \alpha},\tilde e_{-\tilde \alpha}]=(n,h_\alpha,0) That is in the case of the real roots — the complex roots seem to give the points (2n,0,0) but there is some debate about scaling right now so we will restrict ourselves for the most part in dealing with real roots for the remainder of this post.

Our choice for positive real roots will be \{(0,2,n\geq 1),(0,-2,n\geq 0)\} then the positive real coroots will be \{(n\geq1,1,0),(n\geq0,-1,0)\}. The real span of these coweights will be a fan generated by (1,1,0) and (0,-1,0). Its dual fan (using the obvious pairing) will be spanned by the weights (1,-1,0), (1,0,0), and (0,0,1). These will be called the fundamental weights, but the last will be disregarded for now as imaginary.

Side note: I have my suspicions that this is not the proper duality between \hat T_{LSL_2}\otimes \mathbb R and \check T_{LSL_2}\otimes \mathbb R but maybe I can post on that later.

Now that we have the fundamental weights we take their orbits under the affine Weyl group to get what we call chamber weights. The Weyl group is generated by the elements:

s_1 = s =\left(\begin{matrix} 0 && -1 \\ 1 && 0\end{matrix}\right)
s_2 = i \tau s = \left(\begin{matrix} 0 && -iz \\ iz^{-1} && 0\end{matrix}\right)

so chosen, we will see, because they are reflections and each one fixes one of the fundamental weights.

The action of W_{aff} on the weights we work with for now is taken in modified form from Loop Groups

s\cdot (l,m,n) =(l,-m,n) fixes (1,0,0)
\tau\cdot(l,m,n) =(l,m+2l,n+m+l)
\tau s \cdot (l,m,n) = (l,2l-m,n+l-m) fixes (1,1,0)

The two fundamental weights will have parabola shaped orbits.

This entry was posted on October 9, 2008 at 5:50 pm and is filed under Uncategorized. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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