What are the two things on the left and right sides of the ancient official hat? Does it have any meaning?

You are referring to the two wings on the left and right sides of the official hat

The inventor of the long-winged hat was Zhao Kuangyin, the founding emperor of the Song Dynasty. After sitting on the dragon throne, he felt very worried about his colleagues who had conquered the world together. I especially hate the fact that civil and military ministers whisper to each other in the court and comment on government affairs. One day, Zhao Kuangyin went to court in the morning and while listening to a memorial from a certain minister, he found that many officials on both sides were whispering, which was very rude. Zhao Kuangyin was a little annoyed, but he didn't show it. After retiring from the court, he came up with a way and ordered his subordinates to add long wings to the back of the futou gauze hat. The long wings are made of iron sheets and bamboo strips as the skeleton. The iron wings on both sides of a hat are more than a foot each (and will become longer and longer in the future). This kind of hat must be worn except for formal events in court and officialdom, and is not worn on ordinary occasions. Because wearing it makes walking on the street extremely inconvenient. Officials can only talk face to face, and it is difficult to sit side by side. From then on, when ministers went to court, it was difficult for them to line up together and whisper to each other, which would affect the seriousness of the court.