As said before, make sure you have the
proper drivers installed.
Also, make sure you have all the proper fixes installed of those drivers! On the HP custom support site you will find some articles (eg:
here and
here) about how a certain driver of a certain version for a certain OS has a bug in it which makes that those lights are not properly lit up when they need to. So, it is not uncommon and the same thing might be happening with your keyboard model and driver.
Then it might also be a matter of configuring the monitoring tools for email which might come with the drivers/software of your PC and keyboard. The mail indicator probably needs to be configured in those (eg: configure the correct POP3 settings in HP's own mail monitoring program).