Which simplyconnected period domains are contractible?
Examples. Siegel upperhalf space? Poincare upperhalf plane? Universal cover of a Shimura variety?
Are these contractible?
Which simplyconnected period domains are contractible? Examples. Siegel upperhalf space? Poincare upperhalf plane? Universal cover of a Shimura variety? Are these contractible? 


The universal cover of a Shimura variety is a bounded symmetric domain (that's part of the definition), and these are listed in the Wikipedia (under Hermitian symmetric spaces), so you can check this for yourself. 

