It is permissible for the imam to lead the people in prayer while he is sitting. This is the view of Abu Ḥanīfa, al-Shāfʿī and Ahmad.

In case the imam starts praying sitting down, then the people behind him should pray sitting down.

This is based on the hadith of Aishah who said, “The Messenger of Allah (peace be upon him) prayed at home when he was sick. He prayed sitting down and some people prayed behind him standing up, but he gestured to them that they should sit. When he finished (the prayer) he said, ‘The imam has been appointed to be followed. When he bows, bow; when he raises his head, raise your heads; and if he prays sitting, then pray sitting.’” (Al-Bukhari)

It was narrated from Anas ibn Malik that the Messenger of Allah (peace and blessings of Allah be upon him) rode a horse and fell, and the right side (of his body) was injured. He offered one of the prayers sitting down, and we prayed behind him sitting down.  When he finished (the prayer) he said, ‘The imam has been appointed to be followed. If he prays standing up, then pray standing up; when he bows, bow; when he raises his head, raise your heads. When he says Sami’ Allaahu liman hamidah (Allah hears the one who praises Him), say Rabbanaa wa laka’l-hamd (Our Lord, to You be praise). If he prays standing, then pray standing, and if he prays sitting, then pray sitting, all of you.’” (Al-Bukhari)

If the imam starts the prayer standing, then something happens to him that makes him unable to stand any more, those who are praying behind him should complete the prayer standing.

The evidence for this is as follows:

It was narrated that Aishah (may Allah be pleased with her) said: “The Prophet (peace and blessings of Allah be upon him) became seriously ill and asked, ‘Have the people prayed?’ We replied, ‘No, they are waiting for you.’ He said, ‘Put some water in a tub for me.’ We did that and he took a bath and tried to get up, but he fainted. Then he (peace and blessings of Allah be upon him) recovered, and again asked, ‘Have the people prayed?’ We said, ‘No, they are waiting for you, O Messenger of Allah.’ He again said, ‘Put some water in a tub for me.’ He sat down and took a bath and tried to get up but fainted again. Then he recovered and said, ‘Have the people prayed?’ We said, ‘No, they are waiting for you, O Messenger of Allaah.’ He again said, ‘Put some water in a tub for me.’ He sat down and took a bath and tried to get up but fainted again. Then he recovered and said, ‘Have the people prayed?’ We said, ‘No, they are waiting for you, O Messenger of Allah.’ The people were in the mosque, waiting for the Prophet to lead them in Isha prayer. The Prophet sent word to Abu Bakr that he should lead the people in prayer. The messenger went to him and said, ‘The Messenger of Allah is telling you to lead the people in prayer.’ Abu Bakr, who was a soft-hearted man, said: ‘O Umar, lead the people in prayer.’ But ‘Umar answered, you have more right to do that.’ So Abu Bakr led the prayers during those days. Then when the Prophet felt a little better, he came out supported by two men, one of whom was al-Abbas, for the Zhuhr prayer. Abu Bakr was leading the people in prayer, but when Abu Bakr saw him, he wanted to step back [to let the Prophet lead the prayer]. But the Prophet gestured to him that he should not step back. He said, ‘Sit me down beside him,’ so they sat him down beside Abu Bakr. Abu Bakr continued praying, but he followed the Prophet in prayer, and the people were following Abu Bakr in the prayer, and the Prophet was sitting down.” (Al-Bukhari)

Almighty Allah knows best.