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While increasing attention has been paid to co-speech gesture synthesis, most previous works neglect to investigate hand gestures with explicit and essential semantics. In this paper, we study co-speech gesture generation with an emphasis on specific hand gesture activation, which can deliver more instructional information than common body movements. To achieve this, we first build a high-quality dataset of 3D human body movements including a set of semantically explicit hand gestures that are commonly used by live streamers. Then we present a hybrid-modality gesture generation system GestureHYDRA built upon a hybrid-modality diffusion transformer architecture with novelly designed motion-style injective transformer layers, which enables advanced gesture modeling ability and versatile gesture operations. To guarantee these specific hand gestures can be activated, we introduce a cascaded retrieval-augmented generation strategy built upon a semantic gesture repository annotated for each subject and an adaptive audio-gesture synchronization mechanism, which substantially improves semantic gesture activation and production efficiency. Quantitative and qualitative experiments demonstrate that our proposed approach achieves superior performance over all the counterparts.