Random walks in Weyl chambers and crystals

Abstract

We use Kashiwara crystal basis theory to associate a random walk 𝒲 to each irreducible representation V of a simple Lie algebra. This is achieved by endowing the crystal attached to V with a (possibly non-uniform) probability distribution compatible with its weight graduation. We then prove that the generalized Pitman transform defined for similar random walks with uniform distributions yields yet another Markov chain. When the representation is minuscule, and the associated random walk has a drift in the Weyl chamber, we establish that this Markov chain has the same law as 𝒲 conditioned to never exit the cone of dominant weights. For the defining representation V of 𝔤𝔩n, we notably recover the main result of O’Connell [Trans. Amer. Math. Soc. 355 (2003) 3669–3697]. At the heart of our proof is a quotient version of a renewal theorem which holds for a general random walk on a lattice. This theorem also has applications in representation theory since it permits the behaviour of some outer multiplicities for large dominant weights to be described.