# Properties of the Weyl vector $\rho = \frac{1}{2} \sum_{\alpha > 0} \alpha$

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?

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) $$\rho = \varpi_1 + \cdots + \varpi_n$$.