TITLE: What is significant about the half-sum of positive roots?
QUESTION [57 upvotes]: I apologize for the somewhat vague question: there may be multiple answers but I think this is phrased in such a way that precise answers are possible.
Let $\mathfrak{g}$ be a semisimple Lie algebra (say over $\mathbb{C}$) and $\mathfrak{h} \subset \mathfrak{g}$ a Cartan subalgebra. All the references I have seen which study the representation theory of $\mathfrak{g}$ in detail make use of the half-sum of positive roots, which is an element of $\mathfrak{h}^\ast$: e.g. Gaitsgory's notes on the category O introduce the "dotted action" of the Weyl group on $\mathfrak{h}^\ast$, the definition of which involves this half-sum.
Is there a good general explanation of why this element of $\mathfrak{h}^\ast$ is important? The alternative, I suppose, is that it is simply convenient in various situations, but this is rather unsatisfying.

REPLY [3 votes]: Parts of what I will explain here have been said in other answers, but not quite in this way.
One thing about the weight $\rho$ is that the places it tends to show up, it is really the weight $-\rho$ that is the important one. Note that the "dot" action is just a shift to make $-\rho$ the new "$0$".
So why would we like to have $-\rho$ be our new "$0$"? Because it turns out to be the weight that behaves like a $0$ should when we consider cohomology.
This becomes a bit clearer when we go to the closely related setting of a semisimple simply connected algebraic group $G$ with Borel subgroup $B$. Here we consider the functor $\operatorname{Ind}_B^G$ inducing from $B$-modules to $G$-modules (for full details of definitions and proofs, one can consult for example Jantzen's book).
To make things simpler, for a weight $\lambda$ we denote also by $\lambda$ the $1$-dimensional $B$-module where the maximal torus in $B$ acts as $\lambda$, and $B$ acts via the projection to this torus.
For $i\geq 0$ write $H^i(\lambda) = R^i\operatorname{Ind}_B^G(\lambda)$ for the $i$'th right derived functor of induction applied to the $B$-module $\lambda$. Note that this is the $i$'th cohomology of the flag variety $G/B$ with coefficients in the line bundle defined by $\lambda$.
Now the reason for wanting $-\rho$ to be our "$0$" starts to be clear, as $H^i(-\rho) = 0$ for all $i\geq 0$. But there are other weights satisfying this, so what makes $-\rho$ extra special?
What makes $-\rho$ so special is that the vanishing of the above cohomology groups in some sense happens "as early as possible". More precisely, if $P = P(\alpha)$ is a parabolic subgroup corresponding to a simple root $\alpha$, then already $R^i\operatorname{Ind}_B^P(-\rho) = 0$, and we can factor $\operatorname{Ind}_B^G$ as $\operatorname{Ind}_P^G\operatorname{Ind}_B^P$.
Moreover, if $\lambda\neq -\rho$ then there is some simple root $\alpha$ such that either $\operatorname{Ind}_B^{P(\alpha)}(\lambda)\neq 0$ or $R^1\operatorname{Ind}_B^{P(\alpha)}(\lambda)\neq 0$.