My partner keeps inviting his ex to everything

I (29F) have been with my boyfriend (30M) for two years. We live together and things are generally good. However there's this one issue that keeps popping up. He's still really good friends with his ex girlfriend Jane. They dated for five years before me and apparently it was a really messy breakup on her end. He claims they're just friends now and there's nothing romantic at all but it feels like she's always around.

He invites her to every single social gathering we have. Family dinners friend get-togethers even...