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Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak{g}$. The group has a maximal torus $T$ with Lie algebra $\mathfrak{t}$. Let $\rho = \frac{1}{2}\sum_{\alpha > 0} \alpha$ be the half-sum of positive roots, the so-called "Weyl vector."

When one considers the classical compact groups $U(n)$,$Sp(n)$,$SO(2n)$,$SO(2n+1)$ one gets the following Weyl vectors $$\left(\frac{n-1}{2}, \frac{n-3}{2}, \dots, -\frac{n-1}{2} \right)$$ $$\left(n, n-1, \dots, 1 \right)$$ $$\left(n-1, n-2, \dots, 0 \right)$$ $$\left(n-\frac{1}{2}, n-\frac{3}{2}, \dots, \frac{1}{2} \right)$$ I have two questions: firstly I note that the difference between any two components is an integer, $\rho_i - \rho_j \in \mathbb{Z}$. Is there a general way of proving this?

Of course I am implicitly choosing a basis of $\mathfrak{t}$ and $\mathfrak{t}^\ast$ when I write the above vectors. And this is my second question. I am taking a basis $e_1, \dots, e_n \in \mathfrak{t}$, $e_1^\ast, \dots, e_n^\ast \in \mathfrak{t}^\ast$, such that $\exp(2\pi e_j) = \mathrm{Id}$ and such that $\langle e_i , e_j \rangle = \delta_{ij}$, where $\langle \cdot, \cdot \rangle$ is an Ad-invariant inner product. In the groups considered I don't see any systematic reason why such a basis (with these properties) should exist, though in each case a computation shows that it does. Is there some property that all these groups share that makes this possible?

UtilityMaximiser
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1 Answers1

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The basis you have chosen to write $\rho$ in is the basis of fundamental weights, so for $U(n)$ for example you are saying that $$\rho = \frac{n - 1}{2} \varpi_1 + \frac{n - 3}{2} \varpi_2 + \cdots - \frac{n - 1}{2} \varpi_{n - 1}.$$ We can then rewrite your question: since the fundamental weights are dual to the simple coroots, why is it true for any two simple coroots $\alpha^\vee, \beta^\vee$ that $\langle \rho, \alpha^\vee - \beta^\vee \rangle$ is an integer? In fact, more is true: each of $\langle \rho, \alpha^\vee \rangle$ and $\langle \rho, \beta^\vee \rangle$ will be an integer.

Let $S \subseteq \Delta^+ \subseteq \Delta \subseteq \mathfrak{t}^*$ denote the simple roots, positive roots, and roots. For every simple root $\alpha \in S$, the simple reflection $s_\alpha$ permutes the other positive roots $s_\alpha(\Delta^+ - \{\alpha\}) = \Delta^+ - \{\alpha\}$ and of course acts on $\alpha$ as $s_\alpha(\alpha) = -\alpha$, hence we get that $s_\alpha(\rho) = \rho - \alpha$. By definition the reflection $s_\alpha(\rho) = \rho - \langle \rho, \alpha^\vee \rangle \alpha$ (where $\alpha^\vee \in \mathfrak{t}$ is the coroot corresponding to $\alpha$) and therefore we see that for $\alpha \in S$, $\langle \rho, \alpha^\vee \rangle = 1$.

If the Lie algebra $\mathfrak{g}$ is semisimple, then we can replace the fact "$\langle \rho, \alpha^\vee \rangle = 1$ for all simple coroots $\alpha^\vee$" with the equivalent (and easier to remember fact) $\rho = \varpi_1 + \cdots + \varpi_n$.

Joppy
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