First off, what bike do you have? It's worth clarifying this at the start of the thread because the wiring system is different for the various models.
Assuming the bike is a pre-2004 F650GS, the indicators are normally powered up via the load relief relay - that is, when the ignition switch is turned on, this relay is energised and this switches power to the indicator circuit and to the headlight circuit and brake circuits.
The first question is: when you find the LH indicator on with the ignition switch off, is anything else active such as the headlight or the brake light switches? Also are any of the dash indicator bulbs on.
With the ignition on, does the indicator switch do anything or does the LH indicator simply stay on and the RH indicator do nothing?
2006 F650GS in Black