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In this paper, the problem of maximizing the sum of data rates of all users in an intelligent reflecting surface (IRS)-assisted millimeter wave multicast multiple-input multiple-output communication system is studied. In the considered model, one IRS is deployed to assist the communication from a multiantenna base station (BS) to the multi-antenna users that are clustered into several groups. Our goal is to maximize the sum rate of all users by jointly optimizing the transmit beamforming matrices of the BS, the receive beamforming matrices of the users, and the phase shifts of the IRS. To solve this non-convex problem, we first use a block diagonalization method to represent the beamforming matrices of the BS and the users by the phase shifts of the IRS. Then, substituting the expressions of the beamforming matrices of the BS and the users, the original sum-rate maximization problem can be transformed into a problem that only needs to optimize the phase shifts of the IRS. To solve the transformed problem, a manifold method is used. Simulation results show that the proposed scheme can achieve up to 28.6% gain in terms of the sum rate of all users compared to the algorithm that optimizes the hybrid beamforming matrices of the BS and the users using our proposed scheme and randomly determines the phase shifts of the IRS.