I am trying to understand lowest representations of loop groups as developed in Pressley and Segal's book. Specifically I want to be able to compute the weight spaces that appear in a lowest weight representation. I realize there is a formula for this -my question is along the lines of how to apply the formula correctly.


I tried to do a small example with $LSL\_3$ (actually $\mathbb{C}^\times\_{\theta} \ltimes \tilde LSL\_3$) and something fishy happened so I was hoping someone could point out my mistake. The maximal torus is $\mathbb{C}^\times\_\theta \times T \times \mathbb{C}^\times$ where $\mathbb{C}^\times\_\theta$ is the loop rotations and the other $\mathbb{C}^\times$ is central. 


The fundamental weights are $w\_0 = (0,0,1)$, $w\_1 = (0,-\omega\_1,1)$, $w\_2 = (0, -\omega\_2,1)$. The positive roots are $(0,\alpha\_1,0), (0,\alpha\_2,0),(1,-\alpha\_3,0)$. Where $\omega\_i,\alpha\_i$ are fundamental weights and positive roots of $SL\_3$. Pressley and Segal normalize the Killing form so $\langle H\_{\alpha\_i},H\_{\alpha\_i}\rangle = 2$. Choosing coordinates $H\_{\alpha\_1} = [1\ \ 0]^T$, $H\_{\alpha\_2} = [0\ \ 1]^T$, $\alpha\_1 = [2 \ \ -1]$, $\alpha\_2 = [-1 \ \ 2]$, $\omega\_1 = [1\ \ 0]$, $\omega\_2 = [0\ \ 1]$ and the restriction of the Killing form to the torus is just the Cartan matrix $(B\_{11} = B\_{22} = 2, B\_{12} = B\_{21} = -1)$.


I'm interested in the representation $V\_{\tilde \lambda}$ of lowest weight $\tilde\lambda = (0, - \alpha\_3,3) = w\_0 + w\_1 + w\_2$. Let $\tilde \mu = (m,\mu, 3)$ be a weight of $V\_{\tilde \lambda}$. According to *Loop Groups* (11.1.1) it is the case that $\tilde \mu - \tilde \lambda = (m,\mu +\alpha\_3, 0)$ is a sum of positive roots. Viewing $B$ as a map from co-characters to characters and noting that $\alpha\_i = BH\_{\alpha\_i}$ it follows that we can write $\mu = B[a\ \ b]^T$ for some $a,b$.


According to (9.3.7) on pg 180 of *Loop Groups* the $\tilde\mu =(m, \mu,3)$ which satisfy


$3\langle \mu,\mu\rangle - 6m = 6 = 3\langle -\alpha\_3,-\alpha\_3\rangle$


appear among the weight of $V\_{\tilde \lambda}$. This says 


$m = {1 \over 2}\langle\mu,\mu\rangle-1 = {1\over 2}[a\ \ b]B[a\ \ b]^T - 1 = a^2 + b^2- ab -1$. 


Taking $a,b = 0$ produces the weight $\tilde \mu = (-1, 0, 3)$ but then $\tilde \mu - \tilde \lambda = -(1,-\alpha\_3,0)$. Which is certainly not a *sum* of positive roots. So what gives?



lowest weight representation of loop groups
The formula for the invariant bilinear form is given in $(4.9.3)$ on
page 64 $$\langle (x\_1,\xi\_1, y\_1),(x\_2,\xi\_2,y\_2) \rangle=\langle
\xi\_1, \xi\_2 \rangle - x\_1 y\_2-y\_1x\_2$$ As I mentioned in the
comments, $(9.3.7)$ becomes then $||\mu||^2-6m=2$. So your last
equation would be $m=\frac{1}{3}(a^2-ab+b^2)-\frac{1}{3}$. As you
said, there's no more worries about negative $m$ and as a consistency
check, for $m=0$, the solutions for $[a \ \ b]$ are $[\pm 1 \ \ 0]$,
$[0 \ \ \pm 1]$, $[1 \ \ 1 ]$, and $[-1 \ \ -1]$. Applying, $B$
gives you the six weights in the Weyl orbit of $-\alpha\_3$ (the
roots).


