Atkin-Lehner Decompositions for Quaternionic modular forms

In this paper, we obtain Atkin–Lehner decompositions for spaces of modular forms on definite quaternion algebras. Similar to Casselman’s approach our methods are representation theoretic. Using Jacquet–Langlands correspondence we also obtain isomorphisms between spaces of quaternionic modular forms and corresponding spaces of Hilbert modular forms.