We introduce a deformation of the affine Hecke algebra of type GL with four parameters.
Making use of its representation on the space of polynomials,
we can construct a discrete analogue of integration operators satisfying the braid relations.
It determines a difference operator which can be regarded as a discretization
of the Hamiltonian of the one-dimensional delta Bose gas.
By specializing the parameters of the discrete Hamiltonian,
we get the transition rate matrix of an integrable stochastic particle system
called (a continuous time limit of ) the q-Hahn system.