An alternative proof of a theorem about local newforms for GSp(4)

The work [RS] presents a theory of local newand oldforms for representations of GSp(4, F ) with trivial central character for F a non-archimedean field of characteristic zero. This theory considers vectors fixed by the paramodular groups K(p) as defined in [RS]. Let (π, V ) be an irreducible, admissible representation of GSp(4, F ) with trivial central character. One of the main theorems of [RS] asserts that if V contains a non-zero vector fixed by some paramodular group K(p), i.e., π is paramodular, and Nπ is the smallest such n, then the space V (Nπ) of K(pπ ) fixed vectors in V is one-dimensional. If π is paramodular, then any non-zero element V (Nπ) is called a newform. Other theorems of [RS] describe the information carried by newforms. In particular, it is proven in [RS] that if π is generic, then π is paramodular, and there exists a newform whose zeta integral is the L-factor L(s, π). In this work we will give an alternative proof of the following theorem. See the introduction of [RS] for an extensive summary of the contents and proofs of [RS].