Paula Abdul claims one of the "American Idol" executive producers, Nigel Lythgoe, sexually assaulted her multiple times while she was judging 'Idol' and "So You Think You Can Dance" ... according to a new lawsuit she's filed.

Paula filed the lawsuit Friday in L.A. County ... and in the docs, obtained by TMZ, she alleges the first assault happened during "one of Idol's initial seasons" while they were on the road for regional auditions. Paula claims she and Lythgoe got into an elevator at their hotel, and once the doors closed he shoved her against the wall and grabbed her genitals and breasts ... and tried to tongue-kiss her.

In the suit, Paula says she tried to push him away, and then ran to her room once the elevator doors opened. Paula says she immediately called one of her reps to inform them about the alleged incident -- but fearing she'd be fired in retaliation, she decided to keep quiet.

She says the second alleged encounter happened after she signed a new contract in 2015 to be a judge on 'SYTYCD' -- which was also executive produced by Lythgoe. According to the docs, at some point during her 2 seasons on the show, Lythgoe invited Paula to his house to discuss work opportunities ... and she says she accepted because she thought it was a professional situation.

However, she claims the evening ended with Lythgoe forcing himself on top of her while she was on a couch ... and trying to kiss her. In the suit, she says he was telling her they would make an "excellent power couple." Paula says she pushed him off of her, said she was not interested and immediately left.

Paula alleges there was another instance during her 'SYTYCD' that happened in 2015 -- she claims she witnessed Lythgoe grope one of her assistants.

Aside from the disturbing allegations of assault, Paula also claims Lythgoe "taunted" her by calling her at some point to say they should celebrate because it had been "7 years and the statute of limitations had run."

More broadly, Paula alleges she was the victim of harassment and discrimination -- in terms of compensation -- during her time judging 'Idol.'

She is suing Lythgoe, 19 Entertainment and FremantleMedia North America for sexual assault and battery, sexual harassment, gender violence and negligence.

Paula says she's coming forward now due to California's Sexual Abuse and Cover Up Accountability Act ... which gives plaintiffs a window to file claims that would otherwise be barred by the statute of limitations.

We have reached out to Lythgoe for comment, but no word back yet.