Generative AI (GenAI) tools are increasingly being adopted in software development as productivity aids, since there is evidence that GenAI tools can improve individual aspects of productivity. However, productivity is multidimensional; accelerating one aspect of work may simply shift effort to another. In this paper, we investigate how GenAI adoption affects different dimensions of developer productivity. We surveyed 415 software practitioners to understand how they perceive productivity changes associated with AI adoption, using the SPACE framework (Satisfaction and well-being, Performance, Activity, Communication and collaboration, and Efficiency and flow). Our results reveal systematic redistribution of effort across SPACE dimensions. While frequent GenAI users reported faster task completion and higher output volume, these gains were offset by increased code review burden, persistent cognitive load from output verification, and unchanged collaboration patterns. We further provide an empirical mapping between the challenges perceived by developers and potential strategies to mitigate them. Overall, our findings suggest that, at the current stage of GenAI adoption, perceived productivity gains may be spurious -- surface-level acceleration, often accompanied by redistributed effort and hidden costs.