keyboard sluggish and types on its own

My wireless iMac keyboard starts acting sluggish, then is unresponsive. The alternate keys screen pops up (showing, for example, glyphs of each letter that have various accent marks) and I can't enter anything from the keyboard. Or the cursor advances without any text showing up. Or it will retype something after a delay.

I've changed the batteries, reconnected the keyboard via bluetooth, shut down the machine. It will be better for a while then start again. What should I try next?