A duplexer is a tunable filter that allows a repeater’s transmit and receive frequencies to share one common antenna. Without a duplexer, the repeater would require two antenna systems, one for the transmitter and one for the receiver.
A duplexer is necessary when working in DIGITAL mode. From experience, a repeater in ANALOG mode will work fine when using one antenna system for the receiver and one for the transmitter. In DIGITAL mode, however, the repeater will work erratically without a duplexer. Sometimes it will repeat and other times it will not.