The southeast court was called "Nazarite." This was where those who had fulfilled the vow of Nazarites shaved their heads (See Num 6, 13; Acts 18). The Mishnah records that the Nazirites Chamber was in the southeast corner of the Court of the Women. In this chamber the Nazirites would prepare their peace offerings, shave their hair, burning their shaven hair under the pot. A portion of the peace offering was also given to the priest.
Some Commentary Notes:
Edersheim - The Chambers. In each of the four corners of the Court of the Women were chambers, or rather unroofed courts, each said to have been 60 feet long. In that at the right hand (on the north-east), the priests who were unfit for other than menial services on account of bodily blemishes, picked the worm-eaten wood from that destined for the altar. In the court at the farther angle (north-west) the purified lepers washed before presenting themselves to the priests at the Gate of Nicanor. At the left (south-east) the Nazarites polled their hair, and cooked their peace-offerings; while in a fourth court (at the south-west) the oil and wine were kept for the drink-offerings. The musical instruments used by the Levites were deposited in two rooms under the Court of the Israelites, to which the access was from the Court of the Women. Of course the western colonnade of this court was open. Thence fifteen easy steps led through the so-called Gate of Nicanor into the Court of Israel. On these steps the Levites were wont on the Feast of Tabernacles to sing the fifteen 'Psalms of Degrees,' or ascent (Psalms 120 to 134), whence some have derived their name. Here, or, rather, in the Gate of Nicanor, all that was ordered to be done 'before the Lord' took place. There the cleansed leper and the women coming for purification presented themselves to the priests, and there also the 'water of jealousy' was given to the suspected wife.