Even phones have silent and vibrate modes. I don't know why anyone would assume HUDs can't have similarly minimal modes. Instead of an icon indicating email constantly there at the edge of vision (possibly annoying or distracting like a gnat), in "don't distract me" mode, maybe you only see icons, or possibly better still just simple colored dots, when you tilt your head back.

Also of use might be some way for the device to show to other people (not the user) that it isn't distracting the user at the moment. So it doesn't give off a "they aren't paying attention to me" vibe. Perhaps just normal light leakage, or lack thereof, while the display is active/not will be enough for this signal.