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While the notion that the E-Verify case will be a precursor to the court’s handling of similar cases may go too far, Mr. Shapiro said, oral arguments may give hints to the proclivities of individual justices. But even that will be limited somewhat by the fact that Justice Kagan will not be participating.

“This case will not be as big a bellwether because of her recusal,” said Jordan Sekulow, a lawyer at the conservative American Center for Law and Justice.

Oral arguments in the case are scheduled for Dec. 8.

The term’s most controversial case, in which Justice Kagan will participate, is scheduled to be heard Wednesday and involves the father of a Marine killed in Iraq who has sued Fred Phelps, the leader of the anti-gay Westboro Baptist Church, for picketing with offensive signs at the Marine’s funeral.

The case, Snyder v. Phelps, will examine whether the protest is protected by the First Amendment or amounts to an invasion of privacy. The court also may address whether the protest violated the First Amendment’s freedom-of-religion protections of those gathered at the funeral.

The case was brought by Albert Snyder, whose son Marine Lance Cpl. Matthew Snyder was killed in 2006. Mr. Phelps and several followers picketed outside Cpl. Snyders’ funeral at a Roman Catholic church in Maryland. They held signs saying “Thank God for Dead Soldiers” and “You’re Going to Hell,” as well as vulgar signs attacking gays and the Catholic Church.

Consisting of no more than about 100 people in Kansas, the Westboro Baptist Church has made a habit of protesting at the funerals of soldiers. The group says it does so because of the nation’s permissive attitude toward homosexuality; it also says it is anti-Catholic.

The protest had a detrimental effect on the Marine’s father, according to his lawyers, who said in court papers that he will “never have another opportunity to bury his son, and he will associate memories of his son with the Phelpses’ hateful epithets and conduct for the rest of his life.”

Mr. Snyder sued Mr. Phelps for invasion of privacy and intentional emotional distress and won a $5 million verdict. But the 4th U.S. Circuit Court of Appeals set aside the verdict, saying the protest included “hyperbolic” speech protected by the First Amendment.

One analyst says it will be difficult to predict how the high court will rule.

“This could actually be the rare case where the traditional wings on the court splinter and we’ll learn a lot more about the differences between the justices on the conservative wing and differences between the justices on the liberal wing, instead of the differences between the conservative and liberal justices,” said Mark Graber, constitutional law scholar and associate dean at the University of Maryland School of Law.

Mr. Graber said Mr. Phelps is neither extreme on the left or right: “If ever there was a plaintiff that most people conservative or liberal would want to shut up it’s Phelps.”

Scheduled for a hearing that same day is another case in which Justice Kagan will participate and involves the stunning story of a Louisiana man who was only a month away from execution when it was discovered that prosecutors withheld evidence that ultimately led to his exoneration.

The case, Connick v. Thompson, is, at its root, about how corrupt prosecutors should be held accountable, said John Hollway, a lawyer who wrote a book about it.

“There needs to be some way for us as a community to say, ‘No, this is wrong and we’re not going to allow it to happen,’” said Mr. Hollway.

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