For every q, there is a Markov process on reverse plane partitions whose projection to any skew shape is again a Markov process, giving a q-deformed analogue of O'Connell's discrete Whittaker processes.
Whittaker functions on quantum groups and q-deformed Toda operators
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In this paper we q-deform a construction of Kazhdan and Kostant from 1970's which produces quantum Toda Hamiltonians by considering the action of Casimirs of a simple Lie algebra on Whittaker functions on the corresponding Lie group. We also give the affine analog of this generalization. This is done by extending the notion of a Whittaker function to quantum groups and quantum affine algebras. We compute the q-deformed Toda Hamiltonians for Lie algebras of type A and show that they coincide with those known in the theory of integrable systems.
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$q$-Deformed Discrete Whittaker Processes
For every q, there is a Markov process on reverse plane partitions whose projection to any skew shape is again a Markov process, giving a q-deformed analogue of O'Connell's discrete Whittaker processes.