Write down the date

[LewisJian]

I've already signed up the badminton game on your behalf. Please write down the date on your schedule lest you should forget.

I've signed up for a friend and I want her to keep the date of the game in mind. Is it correct to say the above?

 

[emsr2d2]

I've [already] signed you up to the badminton game [on your behalf]. Please write down the date on your schedule in your diary lest you should forget.

See above. You don't need "on your behalf" once you've said "I've signed you up to the ...". In the UK, we tend to use "diary" or "calendar" rather than "schedule". "Lest you forget" (no "should") is quite old-fashioned and formal. We don't use it much these days. I'd say "... so you don't forget".
I think you probably didn't sign the person up to a single game of badminton (one game to 21 points). You probably mean "match", "tournament" or "competition".

 

[Skrej]

AmE would require 'sign you up for the game', rather than 'to the game' which would be ungrammatical in AmE.

 

[jutfrank]

Only 'for' works for me too.