  The isotropic Dunkl oscillator model in the plane is investigated. The model
is defined by a Hamiltonian constructed from the combination of two independent
parabosonic oscillators. The system is superintegrable and its symmetry
generators are obtained by the Schwinger construction using parabosonic
creation/annihilation operators. The algebra generated by the constants of
motion, which we term the Schwinger-Dunkl algebra, is an extension of the Lie
algebra u(2) with involutions. The system admits separation of variables in
both Cartesian and polar coordinates. The separated wavefunctions are
respectively expressed in terms of generalized Hermite polynomials and products
of Jacobi and Laguerre polynomials. Moreover, the so-called Jacobi-Dunkl
polynomials appear as eigenfunctions of the symmetry operator responsible for
the separation of variables in polar coordinates. The expansion coefficients
between the Cartesian and polar bases (overlap coefficients) are given as
linear combinations of dual -1 Hahn polynomials. The connection with the
Clebsch-Gordan problem of the sl_{-1}(2) algebra is explained.
