Although Sinn Fein has in recent years used a strong roster of candidates to build electoral support in the Republic of Ireland, some southern voters who would otherwise be attracted to its broadly center-left policies are still troubled by its previous support for I.R.A. violence under Mr. Adams’s presidency.

Mr. Adams has always denied having been in the I.R.A., but several former republicans have claimed that he had been an active service member in West Belfast in the 1970s and rose to command the organization as chief of staff in 1977, at the height of its violent struggle to end British rule in Northern Ireland.

About 3,500 people died in the Troubles between 1969 and 1998, with the republican side, dominated by the Provisional I.R.A., believed to be responsible for more than 2,000 of these deaths.

Victims of the mainly Roman Catholic I.R.A. included British soldiers, Northern Ireland police officers and Loyalist terrorist opponents, but also many civilian members of the mainly pro-British Protestant majority in Northern Ireland and numerous Catholic civilians who opposed the I.R.A. or were accused of being informers or collaborators.

Mr. Adams has been accused of personally ordering the killings of some of those suspected of being informants. He was questioned in 2014 about the disappearance of Jean McConville, 38, a widowed mother of 10 who was abducted by I.R.A. gunmen in 1972 and found buried on a beach in 2003. He has strongly denied any role in her death and was released without charge.

His successor carries no such historical baggage. Born in a middle-class Dublin neighborhood, Ms. McDonald joined Sinn Fein after the 1998 peace agreement, having previously belonged to Fianna Fail, a conservative establishment party. She is a formidable and aggressive debater, respected by opponents for her communication skills and her grip on policy details.