The gemara darshens that one is not allowed to appoint a ger to a position of authority, unless the mother is a jew. The Rambam (Melachim 1:4) writes that this applies not only to a first generation ger, but for all future generations as well (just at the heter for a ger to marry a mamzeres applies to future generations, not only to the first generation, as the gemara says 74b). Kesef Mishneh writes that a Jewish father will also work to allow him to assume a position of leadership. The gemara means that even a Jewish mother would give him enough yichus to assume such a position, but certainly a father would also work. Based on this he explains that Rechav'am the son of Shlomo was able to be king even though his mother was Na'amah Ha'Amonis. However, the Hagahos Maimon says that having his mother from a yisroel (or father) only works for other positions of leadership, but to be king he must be completely from yisroel because we require "muvchar she'biachicha". That is why Agrifos (Sotah 41b) was unfit to be king even though his mother was a jew. Based on this there would be a question how Rechav'am was able to be king.
R' Moshe (Igros Y.D. 4:26) was asked whether a ger can be a rosh yeshiva. He rejects the proof from Shemaya V'avtalyon who were descendents from sancheirev and were leaders, because it could be that they have yichus from a jewish parent, or that since they were so much greater than anyone else, it was a hora'as sha'ah.
R' Moshe then says a wild chiddush. Part of the mitzvah of "v'ahavtem es ha'ger" would demand that we be lenient in these areas whenever possible. Based on this, he develops a sevara to be meikil. Being a Rosh Yeshiva doesn't qualify as a position of authority, because the purpose of the yeshiva is to teach students who choose to enroll on their own. The Rosh Yeshiva has no authority over people against their will. His authority is only similar to that of a business owner over his employees, who chose to work for him in the first place. Although in an earlier teshuva he discusses the notion of a woman being a mashgiach and considers it a position of authority, that is only because the mashgiach has power to go against the ba'al habayis to maintain the kashrus standard, and therefore qualifies as a position of authority.
R' Moshe also suggests that the issur is only on the people to accept him as a leader, but there is no prohibition on the ger to assume a leadership position on his own. Perhaps this is how Shemaya Avtalyon and Devorah became leaders - they just assumed a position due to their greatness without being formally appointed.