This paper describes the ReprGesture entry to the Generation and Evaluation of Non-verbal Behaviour for Embodied Agents (GENEA) challenge 2022. The GENEA challenge provides the processed datasets and performs crowdsourced evaluations to compare the performance of different gesture generation systems. In this paper, we explore an automatic gesture generation system based on multimodal representation learning. We use WavLM features for audio, FastText features for text and position and rotation matrix features for gesture. Each modality is projected to two distinct subspaces: modality-invariant and modality-specific. To learn inter-modality-invariant commonalities and capture the characters of modality-specific representations, gradient reversal layer based adversarial classifier and modality reconstruction decoders are used during training. The gesture decoder generates proper gestures using all representations and features related to the rhythm in the audio. Our code, pre-trained models and demo are available at https://github.com/YoungSeng/ReprGesture.