  We supply two different descriptions of the pushing process driving the
reflected Brownian motion in Weyl chambers, when the latter domains are
simplexes. The first one shows that a simple root lies in one and only one
orbit if and only if the pushing process in the direction of that simple root
increases as the sum of all the Brownian local times in the directions of the
orbit's positive elements. The last one shows that the pushing process may be
written as the sum of an inward normal vector at the chamber's boundary and an
inward normal vector at the origin, yielding a kind of a multivoque stochastic
differential equation for the reflected process. We finally give a particles
system interpretation of the reflected process and construct a multidimensional
skew Brownian motion.
