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Summary form only given. We study the interaction of a Bose-Einstein condensate with the light field in a driven high-quality optical cavity. The light field acts as an optical potential for the condensate while simultaneously the condensate changes the optical path length of the cavity according to its refractive index. We thus obtain a set of two coupled nonlinear equations of motion for the light and the condensate wave function. We solve these equations numerically and discuss the properties of the obtained stationary ground state of the compound system formed by the condensate and the cavity. Furthermore, we calculate the spectrum of collective excitations which, in contrast to the case of fixed external potentials, involves oscillation of the condensate wave function as well as of the intracavity light intensity. For appropriate system parameters, the dissipative cavity dynamics via the cavity decay leads to damping of the collective condensate excitations.