Does a pilgrim go to Jannah if he dies while performing the Hajj?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

If someone dies while performing an act of obedience then we hope that he will enter paradise, as the Prophet said: "When Allaah wants good for a person, He uses him." The companions asked: "How does He use him, O Prophet of Allaah?" The Prophet replied: "He enables him to perform a good deed before his death." [Ahmad]. The Prophet also said: "Actions are judged according to their end [i.e. if one performs actions of the dwellers of paradise and then dies, then it is hoped that he will enter paradise, and if one performs the actions of the inmates of Hell and dies, then is subject to entering Hellfire.]".

Whoever dies while performing Hajj or when praying or while performing any other good deed, then we hope that he will be among the people of paradise.

However, the belief of Ahlus-Sunnah Wal-Jamaa'ah is that they do not confirm that a specific person is among the people of paradise except the ten companions whom the Prophet informed that they will enter paradise, and Thaabit Ibn Qays Ibn Shammaas, and 'Ukkaashah Ibn Mihsan and others whom the Prophet mentioned by name that they are among the people of paradise. As regards other believers, we cannot be definitely sure that they are among the people of paradise. 

Imaam At-Tahaawi said: 'It is hoped that Allaah will forgive the righteous believers and make them enter paradise with His mercy, but we cannot say with absolute certainty that they are among the people of paradise.'  

Allaah Knows best.