Senatorial is not the person who sits in a senate. That person is a senator. Senatorial is the adjective: it means ‘of a senator’ or ‘of the senate’ — the smaller, upper house that helps make laws in places like the United States. A senatorial election is a race for a senate seat. Senatorial duties are the jobs a senator is supposed to do. The word feels formal and a bit official; you meet it in news and history, not in chat with friends. In American English it almost always points to the national Senate. Other countries have a senate too, and writers can use senatorial for those as well. People sometimes stretch it to mean ‘like a senator’: slow, grave, a little grand. That looser use is rarer, and it can sound old-fashioned.