Yes, Elesin is a tragic character. As the now-dead king's "horseman" he is expected to committ suicide in loyalty to his former leige. Here is an excerpt from the character page at eNotes (link below):
The play opens on the evening of Elesin’s last day of life; at midnight he will die. He says repeatedly that he is ready to give his life, and he knows the importance of fulfilling his responsibility. But Elesin, well known for his many sexual conquests, sees a young woman of great beauty and demands that he be allowed to take her to bed before he dies. Just after leaving the wedding chamber, Elesin begins his passage into the next world, and dances in a hypnotic dream-like trance. But when Simon’s men come to arrest Elesin, he cannot summon the strength to resist them and continue through the transitional state into the next world. Instead, he lives, and brings shame to himself and chaos to the world.