Let $\frak{g}$ be a finitedimensional complex simple Lie algebra together with a choice of Cartan subalgebra and associated root system $(\Delta, (,))$. Also we denote the halfsum 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.
2 Answers
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 nonsimply 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.

1$\begingroup$ Your original answer for the highest root was also interesting, and I didn't know it. Would you consider restoring it, say after the correct answer, with a note that it is an answer to a different question? $\endgroup$– LSpiceCommented Apr 29, 2023 at 15:09

1$\begingroup$ @DidierdeMontblazon This is the definition of $s_i$ (and in fact the formula for a reflection in any vector space): The reflection through $v$ sends $w$ to $w  2( w,v) v / (v,v)$. $\endgroup$ Commented Apr 29, 2023 at 15:32

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1$\begingroup$ @LSpice: Okay, I restored the comments about the highest root $\theta$ and extended Dynkin diagram. I also found a reference for the basic facts about $\rho$ (but unfortunately not what I wrote about $\theta$). $\endgroup$ Commented Apr 29, 2023 at 15:52

1$\begingroup$ Yes, I suppose that's right, you could simply view this as the definition of extended Dynkin diagram (and yes, to get the two versions in the nonsimply laced cases you take either the highest root or highest short root: the point is these are the only roots in the fundamental chamber). $\endgroup$ Commented Apr 29, 2023 at 16:07
Sorry but I have to quibble: $(\alpha,\rho)$ isn't canonical, it depends on the choice of $W$invariant inner product. What is canonical is $\langle \alpha^\vee,\rho\rangle$ where $\alpha^\vee$ is a coroot and $\langle,\rangle$ is the canonical pairing between roots and coroots. (IMHO it is always better to pair roots with coroots, as in Bourbaki, not roots with roots.)
Anyway as per the preceding discussion, $\langle\alpha^\vee,\rho\rangle=1$ for $\alpha$ a simple root; therefore $\langle\alpha^\vee,\rho\rangle=\operatorname{height}(\alpha)$ for any root; and this is equal to the Coxeter number minus 1 (the sum of the labels on the Dynkin diagram) for the highest root.