arch·duch·y 
(ärch-dŭchē)
(ärch-dŭchē)n. pl. arch·duch·ies
The territory over which an archduke or an archduchess has authority.
[French archiduché, from obsolete French archeduché : arche-, arch- (from Old French; see ARCH-1) + duché, duchy (from Old French duche; see DUCHY).]