Fock Spaces for the Complex Dunkl Operator and Deformed (1, 1) Algebra

In this paper, we study a class of generalized Fock spaces for the complex Dunkl operator associated with the cyclic group, generalizing the Bargmann representations for Lie algebra s u (1,1)-generators. We present an explicit realization of C r -extended oscillator in terms of this operator. Furthermore, we construct the wave functions and we determine the coherent states. We also determine an explicit realization of a polynomial deformation s u d (1,1) of the classical Lie algebra s u (1,1). Finally, we give a deforming functional which maps s u d (1,1) to s u (1,1).