The federal investigation that prompted the search of Rep. Ilhan Omar’s D.C. residence on Monday centers on claims that she or her late father may have lied on decades-old immigration paperwork to secure legal status and eventually U.S. citizenship, three people familiar with the matter said.

Omar - a Minnesota Democrat who was born in Somalia and fled civil war there with her family before eventually immigrating to the United States in 1995 - has repeatedly rejected such allegations, which have circulated in conservative media for years. She dismissed them again in a statement Monday as “baseless lies and smear campaigns.”

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It is extremely unusual for federal agents to take public investigative actions rel...