In the nearly 30 years that it has been on the air, “The Simpsons” has perhaps been more accustomed to dispensing cultural criticism than to receiving it. Yet this long-running Fox animated comedy has found itself at the center of a debate over its depiction of one of its supporting characters, a convenience-store owner named Apu.

Apu, who was introduced on “The Simpsons” in 1990, has been criticized by some viewers as a mocking Indian caricature, one who speaks in a stereotypical accent provided by a voice actor, Hank Azaria, who is Caucasian. This argument has gained visibility from works like “The Problem With Apu,” a documentary created by the comedian Hari Kondabolu, and it was hardly quelled by the show’s apparent response, an April “Simpsons” episode in which the characters briefly attempted to address it.

(Speaking directly to the camera, Lisa Simpson said: “Something that started decades ago and was applauded and inoffensive is now politically incorrect. What can you do?” As Lisa glanced at a picture of Apu, Marge Simpson said, “Some things will be dealt with at a later date.” Lisa added, “If at all.”)

Mr. Azaria said he was “happy to step aside” as the voice of the character. But in an interview with USA Today, the “Simpsons” creator Matt Groening seemed to add fuel to the fire by remarking: “I’m proud of what we do on the show. And I think it’s a time in our culture where people love to pretend they’re offended.”