Dear Uhcia,
I refer to the sentence in the third paragraph. It says, "If she wants to leave without any fuss of investigation."
I have a difference of opinion here. Please note that the investigation is not a "fuss" as such. Conducting the investigation is an administrative procedure. Once the employee makes an allegation or accusation in the letter of resignation, it merits investigation. It cannot be at the discretion of the employee.
Conducting an investigation helps in identifying the shortcomings or gaps in the administration. These need to be plugged. However, these cannot be conducted subject to the satisfaction of the person who has made the allegations.
A sentence in the first paragraph of your post says, "Sometimes employees throw words like harassment, discrimination, etc., so as to not serve the notice period, get exempt from notice recovery pay, preemptive action on manager, etc."
The investigation helps in identifying the intention behind making the allegation. If it emerges that the allegations are false, then the administration can reserve the right to take action against the employee for making false allegations.
Conducting an investigation also helps in sending a signal to one and all that the company promotes a culture of justice and fairness, and the issues are not brushed under the carpet.
Thanks,
Dinesh Divekar