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I'm looking at the Interstate schematic diagram as I write this to try to help you out.
All tests should be with the ignition on (headlight lighted).
First off, does your horn work? Your horn uses the same fuse as the turn signals. If the horn works, then you know everything up to the fuse is OK. If it doesn't work, then you need to look at the fuse, or the fuse mounting block.
If the horn is working OK, the next thing to do is to check out the flasher. Turn the turn signals on (switch it to left or right) for this test. There are three wires going to the flasher: grey, green, and green with white stripe. There should be +12v at the green/white wire. An easy way to check this is if your brake light works - hit the rear brake pedal and see if your brake light lights up. If it does, then there should be +12v there. If not, then you've got a wiring problem.
Assuming your brake light lights up properly, then let's bypass the flasher to eliminate it as a source of the problem.
Remove the connector from the flasher relay, and insert a paperclip into the connector, so that it shorts out the grey and the green/white wire (NOT the green wire!).
Once the paperclip is inserted, try your turn signals again. If they come on when you switch either left or right, then the problem is a bad flasher. If they don't come on, then we've probably eliminated the flasher as a suspect. Remove the paperclip and replace the connector on the flasher relay.
Assuming the flasher is OK (turn signals did not come on when paperclip was inserted), it's time to check the bulbs. Pull the turn signals out. The right signal has a blue wire and a gree wire. The left signal has an orange wire and a green wire.
Quick note: On GL1100's, green wires are ALWAYS ground.
Put your voltmeter to check between the blue wire and a known ground, then switch your right turn signal on. You should see +12V. Do the same with the orange wire, then switch your left turn signal on. Again you should see +12V. If you do, then do it again, only use the green wire instead of a known ground - if you don't see +12V using the green wire, then the problem is with the ground (green wire). If you do see +12V both ways, then the problem is either the bulbs or the bulb sockets.
If you DON'T see +12V from either the orange or blue wires, then the problem is the turn switch itself, or wiring or connectors somewhere in between the flasher, and the bulb sockets. Being that BOTH died at the same time, I doubt the problem is wiring in between the switch and the bulb sockets. If THAT is the case, then you need to find the turn signal buzzer. It will have three wires on its connector: Green with white stripe, grey, and (what looks to me like) orange with yellow stripe. Check to see if there is +12V on the grey wire. If there is not +12V on the grey wire, then you've got a wiring harness problem somewhere (specifically with that grey wire) between the flasher relay and the turn switch.
If there IS +12V on the grey wire, then you've got a problem either with the turn switch itself, or in the connector that connects the turn switch harness to the main harness. In both cases, it is the grey wire that will be causing the problem.
That's about all I can deduce from the schematic - good luck with it, do let us know how you make out!