(and then here tell them what you enjoy about the play, if you have seen it, or if not you would really like to see it. You can also mention any articles, reviews, videos etc.. you have seen on the play). And then request the autographed picture of the cast. I just make the paragraph short, because I imagine a secretary is reading it, and it probably never reaches the cast.
(something like if you heard a report about the musical and people said it was good and/or I wish I could see it but it's a shame I don't live there or maybe I'm gonna see it soon can't wait to see a GREAT show) Request autograph.
normally I would end it with THANKS FOR READING MY LETTER
Good Luck on your letter and I hope you get the Autograph you want