## 1 Introduction  
While cellular systems are basically designed for access points (APs) to deliver separate messages for individual users, multicast (MC) transmission techniques supplement the basic mode by allowing the AP to distribute common messages to groups of users [1, 2, 3]. Since the weakest channel among the user channels served by the same message determines the throughput of the MC channel, the optimal beamforming for the multiple antennas at the AP is known to be the semi-definite relaxation (SDR) approach [1, 2], which may cause excessive computational load as the transmit antenna number at the AP grows. For the multi-cell environment, coordinated beamforming supports multi-group (MG) MC as well [3]. The optimization of MC for massive-scale antenna array systems in millimeter-wave bands is considered in [4, 5], including the MG MC cases [6, 7, 8]. To improve the wireless transmission channels, multiple input, multiple output (MIMO) techniques with multiple antennas have long been considered and applied in cellular standards. These techniques have recently extended into massive MIMO systems with large numbers of antenna elements, where hybrid analog-and-digital beamforming (BF) techniques [9, 10, 11, 12, 13] are suggested to compromise the implementation cost and complexity. Hardware-intensive radio frequency (RF) chains are limited to few channels for digital BF, which are tied to analogue BF arrays with phase shifters for the large number of antenna elements. Thesehybrid BF systems are required to reside in the transmitters and receivers, and thus may still lead to an excessive amount of hardware and associated cost. On the other hand, intelligent reflecting surfaces (IRSs), intentionally installed near wireless transmission channels, form artificial transmission paths [14; 15; 16; 17; 18; 19; 20] and are introduced to improve wireless links by reflecting incoming signals and providing additional signal paths toward the user terminals. In effect, the analogue BF arrays of hybrid BF systems are pushed to the far corners of the networks in IRSs so that they may be comparable to relay systems. Unlike the signal-processing-intensive and power-hungry relay systems, however, IRS-assisted transmissions achieve very close performance to that of relay networks in removing the shadowed areas [21; 22; 23; 24] of cellular networks when there are enough reflecting elements deployed at the IRS. Note that this remarkable benefit comes at the cost of installing devices around the networks with relatively light computational resources and power consumption. Transmission systems with training methods are presented in [25; 26], and application in the millimeter band is considered in [27]. Multicasting systems with IRS support are considered in [28; 29]. Analogous to the BF in multi-antenna systems, the adjustment of the phase shifters of IRS reflecting elements is the key technical challenge regarding IRSs [14; 15; 16; 25; 26]. An alternative approach for the optimal phase adjustment of a single user link is presented in [16], where the alignment of all phases of the paths through the IRS to the phase of the direct channel from the base station to the user is shown to be optimal. The collaborations of the IRS and various relay systems are considered [21; 22; 23; 24]. The outage performance of an IRS-assisted multicasting system is presented with a random phase adjustment at the IRS in [28]. Here, we consider an MC network, where the user links from the AP are assisted by the reflections from an IRS with a set of joint BF at the AP and the phase alignments at the IRS. The BF adjusts the complex weights of the multiple antennas such that the AP transmission is optimized for a certain metric while the IRS reflection is controlled by determining the phase shifter values of reflecting elements. Just like the analogue BF in a hybrid BF system, the phase shifters of the IRS are constrained to be complex phasers, and this fact makes the associated optimizations hardly convex.