If you have any free abelian group with an integral bilinear form embedded in the Lorentz space $\mathbb{R}^{n,1}$, you may consider the group of automorphisms generated by roots, i.e., reflections in vectors. The reflection hyperplanes will split the corresponding hyperbolic $n$-space into fundamental domains, and if you fix a chamber, you can choose simple roots corresponding to its walls. This setting includes all finite, affine, and hyperbolic Weyl groups. If there is a vector $\rho$ in the span of the roots satisfying $\Vert r - \rho \Vert = \Vert \rho \Vert$ for all simple roots $r$, then it is called a Weyl vector. This always exists in the finite and affine cases, and the other answers on this page give some description of the relevant geometry.

The existence of a Weyl vector gives a hyperbolic reflection group some arithmetic significance, and non-existence is generic - Lorentzian lattices of rank greater than 26 can't have Weyl vectors. From Barnard's thesis (and earlier work of Gristenko and Nikulin in small rank cases), if one has a lattice generated by roots with a Weyl vector, one may attach a vector-valued modular form, whose coefficients describe the root multiplicities of a Borcherds-Kac-Moody Lie algebra whose real simple roots are precisely those of the reflection group. The Lie algebra in turn has a Weyl denominator product that is a cusp expansion of an automorphic form on $O(n+1,2)$.

In the most extreme case, one may start with the 26-dimensional even unimodular Lorentzian lattice $I\! I_{25,1}$, and choose a chamber for its reflection group. The Dynkin diagram is naturally an affine space on the Leech lattice (by a theorem of Conway), and there is a norm zero Weyl vector $\rho$. There is an action of Leech, identified with the lattice quotient $\rho^\perp/\mathbb{Z}\rho$, on the fundamental domain by parabolic translation. Because there is a Weyl vector, one has a modular form whose coefficients control root multiplicities of a Lie algebra. In this case, we have the weight -12 form $1/\Delta$, and the Lie algebra is the fake monster Lie algebra, which apparently describes bosonic strings propagating in a 26-torus. The roots of norm $2n$ have multiplicity $p_{24}(1-n)$, i.e., the number of partitions in 24 colors.

In a different direction, there is a generalization of the Weyl character formula that holds for any Borcherds-Kac-Moody Lie algebra (not just hyperbolic), and $\rho$ appears here as any vector in the root space that satisfies the relation $\Vert r - \rho \Vert = \Vert \rho \Vert$ (equivalently, $r-2\rho \perp r$) for all simple roots $r$. In the BGG interpretation (worked out in detail in Jurisich's thesis), we find that $H_k(\mathfrak{n}_+, \mathbb{C})$ is spanned by the elements of $\bigwedge^k \mathfrak{n}_+$ whose weight $r$ satisfies $\Vert r - \rho \Vert = \Vert \rho \Vert$. In other words, when we throw away finiteness (and hence well-behaved flag varieties), $\rho$ still plays a role of selecting the part of the exterior power that contributes to the homology.

really greatexample of why MO is good. – Mariano Suárez-Alvarez♦ Nov 9 '11 at 2:32