  The theory of Galois orders was introduced by Futorny and Ovsienko. We
introduce the notion of $\mathcal{H}$-Galois $\Lambda$-orders. These are
certain noncommutative orders $F$ in a smash product of the fraction field of a
noetherian integral domain $\Lambda$ by a Hopf algebra $\mathcal{H}$ (or, more
generally, by a coideal subalgebra of a Hopf algebra). They are generalizations
of Webster's principal flag orders. Examples include Cherednik algebras, as
well as examples from Hopf Galois theory. We also define spherical Galois
orders, which are the corresponding generalizations of principal Galois orders
introduced by the author.
  The main results are (1) for every maximal ideal $\mathfrak{m}$ of $\Lambda$
of finite codimension, there exists a simple Harish-Chandra $F$-module in the
fiber of $\mathfrak{m}$; (2) for every character of $\Lambda$ we construct a
canonical simple Harish-Chandra module as a subquotient of the module of local
distributions; (3) if a certain stabilizer coalgebra is finite-dimensional,
then the corresponding fiber of simple Harish-Chandra modules is finite; (4)
centralizers of symmetrizing idempotents are spherical Galois orders and every
spherical Galois order appears that way.
