Hollywood's biggest worry about Kagan might be her philosophy on intellectual property matters. As dean of Harvard Law School from 2003 to 2009, she was instrumental in beefing up the school's Berkman Center for Internet & Society by recruiting Lawrence Lessig and others who take a strongly liberal position on "fair use" in copyright disputes.
Most notably, during those years, Professor Charles Nesson at the Berkman Center represented accused file-sharer Joel Tenenbaum in the defense of a lawsuit by the RIAA. Professor Nesson led his cyberlaw class in alleging that "the RIAA is abusing law and the civil process" with excessive damage claims in piracy cases. It was Kagan herself who wrote a personal letter to the U.S. District Court to help certify the students.
http://thresq.hollywoodreporter.com/2010/05/kagan-supreme-court-hollywood.html