The prizes, to be awarded Sunday evening, are the equivalent of Broadway's Tonys and reward achievement in theater, opera and dance.
Law is nominated for best actor for "Hamlet," and Weisz is up for best actress for "A Streetcar Named Desire."
Knightley is a best supporting actress nominee for playing a manipulative movie starlet in "The Misanthrope."
James Earl Jones, Gillian Anderson and Imelda Staunton are also nominated.
Several productions are up for multiple prizes, including rock musical "Spring Awakening," corporate crisis drama "Enron" and the raucous "Jerusalem."
Winners will be announced at London's Grosvenor House Hotel.
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