What should we do with an Imam who prolongs his bowing (rukuʿ) and prostrations (sujud), when there are elderly and sick people among the worshippers, and he does not respond to the gestures directed at him? And is there a defect in his concealing the sound of his supplication (duʿaʾ) after the prayer?

It is from the Sunnah for the Imam not to prolong the prayer for the people, out of compassion for the weak and those with needs. This is supported by the Prophet’s (peace be upon him) saying: "If any of you leads the people in prayer, he should lighten it, for among them are the weak, the sick, and the old," and his saying: "O people, indeed there are among you those who repel others! So whoever among you leads the people in prayer, let him be brief, for behind him are the old, the weak, and those with needs."

Therefore, the Imam should be advised to lighten the prayer. If he does not respond, his matter can be escalated to the officials. As for supplication after the last tashahhud, whispering it is legislated, but it should not be prolonged in a way that causes hardship to the congregants.