Musicians: do you get "performance jitters" and how do you get over it?

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I have been playing the piano for 35 years, but I still can only play alone. If anyone else is in the room (except for very close family members) I get nervous and screw up. We recently went to a piano club meeting and I played "Claire de Lune" which I could probably play in my sleep - yet I was nervous and missed a lot of notes, and left feeling completely embarassed and humiliated. What causes this anxiety and how do you get over it?
Thanks.

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1. I thought I was the only one.

I've been playing for 39 years - and that's exactly how I feel.
I can only play okay if I'm completely improvising - even then I make mistakes if there are people around, or even if I'm by myself and IMAGINE there are people around.