This paper presents a method to mitigate traffic fluctuations in mixed traffic flows of autonomous vehicles (CAVs) and human-driven vehicles (HDVs). To address the computational feasibility issues of existing predictive control frameworks and the challenges of modeling the nonlinear and heterogeneous behavior of HDVs, we propose an adaptive deep Koopman predictive control framework (AdapKoopPC). AdapKoopPC features a novel deep Koopman network (AdapKoopnet) that adaptively learns from naturalistic data to represent the complex HDV vehicle-following dynamics as a linear system in a high-dimensional space. This learned linear representation is integrated into a model predictive control (MPC) technique, enabling real-time, scalable, and optimal CAV control. Validated through a HighD dataset and extensive numerical simulations, AdapKoopnet achieves superior trajectory prediction accuracy compared to baseline models, while the AdapKoopPC controller performs robustly even at low AV ratios and significantly reduces traffic fluctuations with minimal computational overhead. This study provides a scalable, data-driven solution for improving safety in realistic mixed traffic environments, and the code is publicly available.