Once they were mentor and protégé. Now Daniel Och and James Levin are trapped in a battle for the future of one of New York’s biggest investment firms.

In the late 1990s, Mr. Levin was working at a summer camp in Wisconsin, teaching Mr. Och’s son how to water ski. By last year, the younger man was in line to succeed Mr. Och as chief executive of Och-Ziff Capital Management LLC, the largest publicly traded hedge fund in the U.S. with $33 billion in assets under management. To entice him to stick around, Mr. Och handed Mr. Levin,...