To be honest and attempt at being objective there are many possibilities here:
1) One of Madonna's "people" actually did send an email, but without Madonna's consent; hence the miscommunication.
2) Madonna didn't send the email and Gaga just fabricated the story about it.
3) Madonna did send the email, but told her representative to deny it after Gaga broke the news.
There are other possibilities as well, but since no one actually knows what happened it's difficult to speculate when there is no conclusive evidence and celebria and the media are inherently pretty unreliable and self-interested when it comes to this messy ****.
Based on Madonna's history of simply being malicious towards anyone remotely in competition with her, I'd have to say it's completely plausible that she would send such an email and then later deny to make Gaga obviously look bad. Based on the supposed rep's claims, it's also plausible that Gaga simply lied about it.
It's also completely plausible that this whole debacle was an issue of miscommuniation, in which Gaga somehow did receive an email from either (a) one of Madonna's reps, or (b) someone claiming to be one of Madonna's reps, and naturally said that she received her blessing on Jay Leno. Meanwhile, Madonna herself never sent anything and told her rep to deny the existence of any such email.
Trying to be objective, I don't see any particular reason why Gaga would lie on national television about it when she is smart enough to know that she would receive huge backlash if she were exposed as lying. I also would like to believe that Madonna isn't malicious enough to send the email and then deny its existence, so that last possibility of miscommunication is the only one that gives both of them the benefit of the doubt and the one I believe in.
Just my opinion and attempt at exploring all the possibilities, though, so don't stone me
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