Let $\frak{g}$ be a finite-dimensional complex simple Lie algebra together with a choice of Cartan subalgebra and associated root system $(\Delta, (-,-))$. Also we denote the half-sum of positive roots by
$$
\rho := \frac{1}{2}\sum\_{\alpha \in \Delta\_+} \alpha.
$$
What can one say about the scalar
$$
(\alpha,\rho)?
$$
I feel this is an easy question (perhaps too easy for M.O.) but I also feel I am missing a simple trick to produce the answer.



Pairing a root with the half-sum of positive roots
I'm sorry, my original answer was about the expansion of the *highest root* $\theta$ into the fundamental weights $\omega\_1,\ldots,\omega\_n$. For the Weyl vector $\rho$, something much simpler is true: we have $\rho=\sum\_{i=1}^{n} \omega\_i$. It is easy to see this since the simple reflection $s\_{\alpha\_i}$ sends $\alpha\_i$ to $-\alpha\_i$, while permuting the other positive roots.


A precise reference for this fact is Proposition 29 of $\S$1.10 of Chapter VI (Root Systems) of Bourbaki's book on Lie Groups and Lie Algebras.


**EDIT**: As requested by LSpice in the comments, let me restore (and correct) my comments about the expansion of the highest root $\theta$ into fundamental weights. We have $\theta=\sum\_{i}c\_i\omega\_i$ where $c\_i$ is the number of edges from the special "affine" node $0$ in the extended Dynkin diagram of $\mathfrak{g}$ to the node $i$. (For the simply laced case there is no ambiguity regarding what is meant by extended Dynkin diagram; for non-simply laced cases there are two different versions of the "extended Dynkin diagram" and they give the fundamental weight coefficients for the highest root and highest short root, respectively.)


I'm fairly certain this statement about $\theta$ appears somewhere in Bourbaki as well, but I'm having trouble locating precisely where.


