Affine Lie algebras and a conditioned space-time Brownian motion in an affine Weyl chamber

We construct a sequence of Markov processes on the set of dominant weights of an affine Lie algebra $mathfrak{g}$ considering tensor product of irreducible highest weight modules of $mathfrak{g}$ and specializations of the characters involving the Weyl vector $rho$. We show that it converges towards a space-time Brownian motion with a drift, conditioned to remain in a Weyl chamber associated to the root system of $mathfrak{g}$.