Do I have to pay music partner for live performances?

No. If you're joint authors of the songs, you each have the right to license them, perform them, and create derivative works out of them. But these rights are subject to your obligation to pay your co-author their share of the income. Your co-author does NOT have a right to live performance fees, other than any fees paid by ASCAP and BMI directly to the writers and publishers, if the songs and the payees are registered.

So have you formed a publishing company to publish your share of the songs? Have you registered the songs with a performing rights society as a songwriter and as a publisher? Have you registered the songs for copyrights? I take it you haven't signed any co-publishing agreements with your co-author - now might be a good time to hire a music business lawyer and get your music business affairs in order. Please see my Legal Guides linked below.

Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.