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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 $ρ$. 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 $\mathfrak{g}$.