Probability Theory and Related Fields

, Volume 165, Issue 3–4, pp 649–665 | Cite as

Affine Lie algebras and conditioned space-time Brownian motions in affine Weyl chambers

  • Manon Defosseux


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 \(\mathfrak {g}\). This extends in particular the results of Defosseux (arXiv:1401.3115, 2014) to any affine Lie algebras, in the case with a drift.

Mathematics Subject Classification

17B67 35R37 60J65 


  1. 1.
    Crank, J.: Free and Moving Boundary Problems. Clarendon Press, Oxford (1984)zbMATHGoogle Scholar
  2. 2.
    Defosseux, M.: The affine Lie algebra \(\hat{\mathfrak{sl}_2}\) and a conditioned space-time Brownian motion, arXiv:1401.3115 (2014)
  3. 3.
    Kac, V.G.: Infinite dimensional Lie algebras, 3rd edn. Cambridge University Press, Cambridge (1990)CrossRefzbMATHGoogle Scholar
  4. 4.
    Lecouvey, C., Lesigne, E., Peigné, M.: Conditioned random walks from Kac-Moody root systems, arXiv:1306.3082 [math.CO] (2013)
  5. 5.
    Stembridge, J.R.: Multiplicity-free products and restrictions of Weyl characters. Represent. Theory 7, 404–439 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Appliquées à Paris 5, Université Paris 5ParisFrance

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