Paramodular groups and theta series

For a paramodular group of any degree and square free level we study the Hecke algebra and the boundary components. We define paramodular theta series and show that for square free level and large enough weight they generate the space of cusp forms (basis problem), using the doubling and pullback of Eisenstein series method. For this we give a new geometric proof of Garrett's double coset decomposition which works in our more general situation.