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We study some path transformations related to Pitman's theorem on Brownian motion and the three dimensional Bessel process. We relate these to Littelmann path model, and give applications to representation theory and to Brownian motion in a Weyl chamber.1991 Mathematics Subject Classification. Primary ; Secondary . Braid relations for the Pitman transformsDefinition 2.1. The Pitman transform P α is defined on the set of continuous paths π : [0, T ] → V , satisfying π(0) = 0, by the formula: P α π(t) = π(t) − inf t≥s≥0 α ∨ (π(s))α, T ≥ t ≥ 0.

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