Green Lantern and Power Girl hooked up in the weirdest possible way. Hal Jordan has always been something of a womanizer, and in his very first appearance in Showcase #22 he was pretty aggressive in pursuing his boss, Carol Ferris. By the end of the Silver Age, Green Lantern had dated more girlfriends than any of his Silver Age comrades.
Not all of these romances were particularly tasteful. Take the example of Arisia Rrab, a fellow member of the Green Lantern Corps, who was only a fourteen-year-old by her race's standards, although she insisted she was 28 in Earth years. Hal initially turned Arisia down, and the heartbroken teen used her Power Ring to age her body into that of a young woman. For some reason, this brought an end to Hal's moral objections, and the two had a serious (and highly sexual) relationship for quite some time. It's easily one of the most controversial comic book romances of all time.
One of Green Lantern's strangest hook-ups, though, was with Power Girl. The "Red Winter" plotline saw the sonic manipulator Sonar bring the Justice League to its knees, rendering Hal Jordan amnesiac. Although Power Girl attempted to help him remember, she clearly found Hal's new vulnerability attractive, and the chemistry between the two was evident. Finally, in Justice League Europe #50 they became trapped in one of Green Lantern's power bubbles, with Sonar pounding upon it with sonic waves. The close proximity proved irresistible, and the two ignored the supervillain for a make-out session - and then some.
Writer Gerard Jones moved away from the scene after those panels, but it was pretty clear the two had other things on their minds than saving the world. When they next turned up, Hal had used his Power Ring to restore his memories, and he and Power Girl participated in the final battle against Sonar and his forces. It was unclear quite how far the two had gone until a few issues later Power Girl shocked the team by revealing she was pregnant. An irate Hal approached Power Girl, assuming Aquaman was the father and complained that Kara stopped things because she wanted a "real relationship," not just casual sex. "Well, it never seemed to me that you and Aquaman had a 'real--'," he complained. Power Girl interrupted him with a well-deserved punch that knocked Green Lantern off his feet.
For the record, Aquaman wasn't the father, and neither - of course - was Green Lantern. It was all eventually revealed to be a magical event, with the pregnancy induced by Power Girl's own grandfather. The sparks between Power Girl and Green Lantern were forgotten, Hal's ardor clearly doused by the powerhouse punch he'd received when he'd been a jerk.