A spokesperson for the Indianapolis Metropolitan Police Department tells us ... officers did end up taking a case report on May 2 after Beverley hurled a basketball not once, but twice, at Indiana fans toward the end of the Bucks' playoff game with the Pacers at Gainbridge Fieldhouse.

According to the rep, the report has since been forwarded to detectives, who are now trying to gain access to more video from the arena ... as well as interview the involved parties.

The spokesperson added that detectives are expected to send their findings off to prosecutors as soon as they finish the probe.

Beverley has not yet publicly addressed the criminal investigation ... although earlier this week, on his "The Pat Bev Podcast with Rone" podcast, he did express remorse for his actions -- calling his behavior "inexcusable."

As for why he did it, he said fans' trash talk in the arena crossed lines. But, again, he reiterated he shouldn't have heaved the ball into the crowd.

"It doesn't matter what was said," he said. "I have to be better and I will be better."